diff --git a/build/a.html b/build/a.html index 83e5b5a..f6c2424 100644 --- a/build/a.html +++ b/build/a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-03-07 Mon 20:38

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Created: 2022-03-15 Tue 08:10

diff --git a/build/a_l.html b/build/a_l.html index 9901ba4..b1f2e59 100644 --- a/build/a_l.html +++ b/build/a_l.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1720,7 +1720,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c.html b/build/c.html index ccd8f54..7054128 100644 --- a/build/c.html +++ b/build/c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m.html b/build/c_m.html index 8c9d9f4..7e563a4 100644 --- a/build/c_m.html +++ b/build/c_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_cs.html b/build/c_m_cs.html index 4ae3d23..dfe3b1d 100644 --- a/build/c_m_cs.html +++ b/build/c_m_cs.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_cs_cyl.html b/build/c_m_cs_cyl.html index ca0e39c..78fd440 100644 --- a/build/c_m_cs_cyl.html +++ b/build/c_m_cs_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1659,14 +1659,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
Gradient
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+

-
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  • Gr4(1.79)
@@ -1687,14 +1687,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
Divergence
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+

-
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  • Gr4(2.21)
@@ -1715,14 +1715,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
Curl
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+

-
+
  • Gr4(2.21)
@@ -1771,7 +1771,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_cs_hyp.html b/build/c_m_cs_hyp.html index bafae8c..a5ad65e 100644 --- a/build/c_m_cs_hyp.html +++ b/build/c_m_cs_hyp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1632,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html index 22f3871..29dce12 100644 --- a/build/c_m_cs_sph.html +++ b/build/c_m_cs_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1623,14 +1623,14 @@ which \(r\) is the distance from the chosen origin, The usual Cartesian coordinates relate to spherical coordinates according to

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@@ -1655,14 +1655,14 @@ A generic vector can be expressed as where the explicit relation between spherical and Cartesian unit vectors is

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@@ -1685,14 +1685,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).

An infinitesimal displacement \(d{\bf l}\) can be written as

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@@ -1708,14 +1708,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin

Infinitesimal volume element:

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@@ -1736,14 +1736,14 @@ Infinitesimal surface element: depends on situation.
Gradient
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Divergence
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Curl
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Laplacian
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+

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@@ -1851,7 +1851,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_dc.html b/build/c_m_dc.html index 76bcb30..83ba3fe 100644 --- a/build/c_m_dc.html +++ b/build/c_m_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1642,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_dc_curl.html b/build/c_m_dc_curl.html index d8fa22d..394bf5b 100644 --- a/build/c_m_dc_curl.html +++ b/build/c_m_dc_curl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1644,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_dc_d2.html b/build/c_m_dc_d2.html index 95efce6..248c1a9 100644 --- a/build/c_m_dc_d2.html +++ b/build/c_m_dc_d2.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1615,9 +1615,9 @@ Table of contents
-
-
Divergence of gradient
-
+
+
Divergence of gradient
+

\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the Laplacian of the scalar field \(T\). The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components @@ -1626,44 +1626,44 @@ given by the Laplacian of the corresponding vector elements.

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Curl of a gradient
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+
Curl of a gradient
+

This always vanishes.

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Gradient of the divergence
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+
Gradient of the divergence
+

\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.

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Divergence of a curl
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Divergence of a curl
+

This always vanishes.

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Curl of curl
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Curl of curl
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+

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@@ -1694,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_dc_del.html b/build/c_m_dc_del.html index 37f9ecb..3bfe77d 100644 --- a/build/c_m_dc_del.html +++ b/build/c_m_dc_del.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_dc_div.html b/build/c_m_dc_div.html index 93a76c6..effcd20 100644 --- a/build/c_m_dc_div.html +++ b/build/c_m_dc_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_dc_g.html b/build/c_m_dc_g.html index dd7fe9b..069c74a 100644 --- a/build/c_m_dc_g.html +++ b/build/c_m_dc_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1664,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/c_m_dc_pr.html b/build/c_m_dc_pr.html index 8f782e9..50a52c1 100644 --- a/build/c_m_dc_pr.html +++ b/build/c_m_dc_pr.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1622,14 +1622,14 @@ explicited as follows:

Gradient of a product:

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+

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  • Gr (3)
  • W (1-111)
    • @@ -1649,14 +1649,14 @@ explicited as follows:

    Gradient of a scalar product:

    -
    +

    -
    +
    -
    + -
     + + +

    @@ -1661,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd.html b/build/emd.html index 53658a4..59ac15c 100644 --- a/build/emd.html +++ b/build/emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1614,8 +1614,8 @@ Table of contents emd
    -
    - +
    + Prerequisites
      @@ -1624,8 +1624,8 @@ Prerequisites
    -
    - +
    + Objectives
      @@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_Fl.html b/build/emd_Fl.html index 77d0342..21a9ed1 100644 --- a/build/emd_Fl.html +++ b/build/emd_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_Fl_Fl.html b/build/emd_Fl_Fl.html index 80f81e3..2eb510b 100644 --- a/build/emd_Fl_Fl.html +++ b/build/emd_Fl_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1644,18 +1644,18 @@ Empirically: the changing magnetic field induces an electric current around the circuit. This current is really driven by an electric field having a component along the wire. The line integral of this field is called the

    -
    +

    Electromotive force (or electromotance),

    -
    +

    -
    +
    • Gr (7.9)
    @@ -1680,14 +1680,14 @@ The precise statement associated to Faraday's observations is that the electromotive force is proportional to the rate of change of the magnetic flux,

    -
    +

    -
    +
    • Gr (7.14)
    @@ -1702,18 +1702,18 @@ to the rate of change of the magnetic flux, \] so we obtain

    -
    +

    Faraday's law (integral form N.B.: for a stationary loop)

    -
    +

    -
    +
    • Gr (7.15)
    @@ -1737,15 +1737,15 @@ for any loop (on a wire or not). Using Stokes' theorem, \] we obtain

    -
    -
    +
    +

    -
    +
    • Gr (7.16)
    @@ -1790,7 +1790,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_Fl_e.html b/build/emd_Fl_e.html index f4e18dd..47d03b3 100644 --- a/build/emd_Fl_e.html +++ b/build/emd_Fl_e.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1639,14 +1639,14 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo \] Generalization to volume currents:

    -
    +

    -
    +
    • Gr (7.31)
    @@ -1681,15 +1681,15 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold \] We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with

    -
    -
    +
    +

    -
    +
    • Gr (7.34)
    @@ -1718,7 +1718,7 @@ W_{mag} &= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &= \frac{1}{2 which are equations W_vcd, W_intEsq, W_intAJ and W_intBsq.

    -
    +

    Example: energy in coaxial cable

    @@ -1768,7 +1768,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_Fl_i.html b/build/emd_Fl_i.html index ae40779..388ff54 100644 --- a/build/emd_Fl_i.html +++ b/build/emd_Fl_i.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1641,14 +1641,14 @@ so \] and we can write the mutual inductance as the Neumann formula,

    -
    +

    -
    +
    • Gr (7.22)
    @@ -1665,14 +1665,14 @@ M_{21} = \frac{\mu_0}{4\pi} \oint_{{\cal P}_1} \oint_{{\cal P}_2} \frac{d{\bf l} Two things: first, \(M_{21}\) is purely geometrical. Second,

    -
    +

    -
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    • Gr (7.23)
    @@ -1687,7 +1687,7 @@ M_{12} = M_{21} \]

    -
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    Example: solenoid in solenoid

    @@ -1735,14 +1735,14 @@ What if we vary current in loop 1? Flux in 2 will vary. Induces EMF in loop 2: \] Changing current also induces EMF in the source loop itself:

    -
    +

    -
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    • Gr (7.25)
    @@ -1765,7 +1765,7 @@ Inductance: measured in henries (\(H\)). \(H = V s/A\).

    -
    +

    Example: self-inductance of toroidal coil

    @@ -1804,7 +1804,7 @@ Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of back EMF.

    -
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    Example: circuit

    @@ -1853,7 +1853,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_Fl_ief.html b/build/emd_Fl_ief.html index df729de..3de193e 100644 --- a/build/emd_Fl_ief.html +++ b/build/emd_Fl_ief.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1637,7 +1637,7 @@ law in integral form: -
    +

    Example: loop with time-dependent flux

    @@ -1664,7 +1664,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
    -
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    Example: wheel with charged rim traversed by flux

    @@ -1705,7 +1705,7 @@ called the quasistatic approximation, and works provided we deal with slow enough phenomena.

    -
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    Example: field from wire with time-dependent current

    @@ -1764,7 +1764,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_Me.html b/build/emd_Me.html index f73f980..14162d3 100644 --- a/build/emd_Me.html +++ b/build/emd_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_Me_Me.html b/build/emd_Me_Me.html index 9ca2b1f..447b9ef 100644 --- a/build/emd_Me_Me.html +++ b/build/emd_Me_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1616,18 +1616,18 @@ Table of contents

    Full set of equations for the electromagnetic field:

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    Maxwell's equations (in vacuum)

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    -
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    @@ -1644,7 +1644,7 @@ Full set of equations for the electromagnetic field:

    Complement:

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    Force law LorFo \[ @@ -1667,15 +1667,15 @@ take divergence of \((iv)\) and use \((i)\).

    Better way of writing: all fields on left, all sources on right,

    -
    -
    +
    +

    -
    +
    • Gr (7.42)
    @@ -1713,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_Me_dc.html b/build/emd_Me_dc.html index 8f5848a..4d54e09 100644 --- a/build/emd_Me_dc.html +++ b/build/emd_Me_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1622,15 +1622,15 @@ The term which should be zero (but isn't) in -
    +
    +

    -
    +
    • Gr (7.36)
    @@ -1659,18 +1659,18 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation

    Maxwell baptized this term the

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    Displacement current

    -
    +

    -
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    • Gr (7.37)
    @@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_Me_ebM.html b/build/emd_Me_ebM.html index 1eabbfe..1c8d345 100644 --- a/build/emd_Me_ebM.html +++ b/build/emd_Me_ebM.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1630,14 +1630,14 @@ Fatal inconsistency: div of curl must always vanish. Check on \((iii)\): \] But: try same with \((iv)\):

    -
    +

    -
    +
    • Gr (7.35)
    @@ -1682,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_Me_mc.html b/build/emd_Me_mc.html index d8cd56f..be16b63 100644 --- a/build/emd_Me_mc.html +++ b/build/emd_Me_mc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_ce.html b/build/emd_ce.html index dc3af2c..b9a793e 100644 --- a/build/emd_ce.html +++ b/build/emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1614,8 +1614,8 @@ Table of contents emd.ce
    -
    - +
    + Prerequisites
      @@ -1623,8 +1623,8 @@ Prerequisites
    -
    - +
    + Objectives
      @@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_ce_amom.html b/build/emd_ce_amom.html index b8e2700..950723c 100644 --- a/build/emd_ce_amom.html +++ b/build/emd_ce_amom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1616,18 +1616,18 @@ Table of contents

    The angular momentum of EM fields is directly given by

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    Angular momentum of EM fields

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    +

    -
    +
    @@ -1661,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_ce_ce.html b/build/emd_ce_ce.html index ac74f3b..f559746 100644 --- a/build/emd_ce_ce.html +++ b/build/emd_ce_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1635,7 +1635,7 @@ This means that \] Since this is true for any volume, we have (re)derived the

    -
    +

    Continuity equation conteq \[ @@ -1675,7 +1675,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html index afae5c8..aa90916 100644 --- a/build/emd_ce_mom.html +++ b/build/emd_ce_mom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1628,18 +1628,18 @@ in which the first integral can be interpreted as the momentum stored in the EM

    This is thus simply a conservation law for momentum, with

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    Momentum density in the EM fields

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    +

    -
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    @@ -1655,18 +1655,18 @@ This is thus simply a conservation law for momentum, with

    In a region in which the mechanical momentum is not changing due to external influences, we then have the

    -
    +

    Continuity equation for EM momentum

    -
    +

    -
    +
    @@ -1699,7 +1699,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html index 7ea61ba..90f337a 100644 --- a/build/emd_ce_mst.html +++ b/build/emd_ce_mst.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1667,18 +1667,18 @@ and similarly for \({\boldsymbol B}\). We thus get

    This expression can be greatly simplified by introducing the

    -
    +

    Maxwell stress tensor

    -
    +

    -
    +
    @@ -1699,18 +1699,18 @@ The element \(T_{ij}\) represents the force per unit area in the \(i\) direction

    We then obtain the

    -
    +

    EM force per unit volume

    -
    +

    -
    +
    @@ -1726,18 +1726,18 @@ We then obtain the

    where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the

    -
    +

    Total force on charges in volume

    -
    +

    -
    +
    @@ -1770,7 +1770,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html index 78992fe..cada692 100644 --- a/build/emd_ce_poy.html +++ b/build/emd_ce_poy.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1617,7 +1617,6 @@ Table of contents Earlier: work done to assemble a static charge distribution: \[ W_e = \frac{\varepsilon_0}{2} \int d\tau ~E^2 -\tag{\ref{eq:Energy_as_int_E2}} \] Work necessary to get currents going: \[ @@ -1637,14 +1636,14 @@ done by EM forces? From Lorentz force law: Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus,

    -
    +

    -
    +
    • Gr (8.6)
    @@ -1684,18 +1683,18 @@ so we get Substituting this in dWdt_intEJ and using the divergence theorem, we obtain

    -
    +

    Poynting's theorem

    -
    +

    -
    +
    • Gr (8.9)
    @@ -1721,18 +1720,18 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.

    Energy per unit time, per unit area carried by EM fields: given by the

    -
    +

    Poynting vector

    -
    +

    -
    +
    • Gr (8.10)
    @@ -1751,18 +1750,18 @@ Energy per unit time, per unit area carried by EM fields: given by the

    We can thus express Poynting's theorem more compactly:

    -
    +

    Poynting's theorem (integral form)

    -
    +

    -
    +
    • Gr (8.11)
    @@ -1781,18 +1780,18 @@ We can thus express Poynting's theorem more compactly:

    where we have defined the total

    -
    +

    Energy in electromagnetic fields

    -
    +

    -
    +
    • Gr (8.5)
    @@ -1821,18 +1820,18 @@ Then, \] so we get the

    -
    +

    Poynting theorem (differential form)

    -
    +

    -
    +
    • Gr (8.14)
    @@ -1855,7 +1854,7 @@ and has a similar for to the continuity equation -
    +

    Example: Joule heating

    @@ -1881,7 +1880,7 @@ wire of radius \(a\), \] Poynting: \[ -{\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s} +{\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol r} \] and points radially inwards. Energy per unit time passing surface of wire: \[ @@ -1912,7 +1911,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_emw.html b/build/emd_emw.html index aae625f..e88cc8b 100644 --- a/build/emd_emw.html +++ b/build/emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1614,8 +1614,8 @@ Table of contents emd.emw
    -
    - +
    + Prerequisites
      @@ -1624,21 +1624,14 @@ Prerequisites
    -
    - +
    + Objectives
    • Understand how to obtain the wave equation from Maxwell's equations
    • Understand monochromatic plane waves in vacuum
    • Understand their energy and momentum
      • -
    • Understand EM waves in linear media
      • -
    • Understand reflection and transmission of EM waves at an interface
      • -
    • Know the laws of reflection and refraction
      • -
    • Know Fresnel's equations (parallel polarization)
      • -
    • Know Brewster's angle
      • -
    • Understand EM waves in conductors
      • -
    • Understand simple waveguides and coaxial cables
    @@ -1666,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html index b765d78..7f90b97 100644 --- a/build/emd_emw_ep.html +++ b/build/emd_emw_ep.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1644,18 +1644,18 @@ so for a monochromatic EM plan wave, \] or more succinctly:

    -
    +

    Poynting vector of a monochromatic EM wave

    -
    +

    -
    +
    • Gr (9.57)
    @@ -1678,7 +1678,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w

    Similary, we get the

    -
    +

    Momentum density of a monochromatic EM wave \[ @@ -1729,7 +1729,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html index e12754f..6b5f01b 100644 --- a/build/emd_emw_mpw.html +++ b/build/emd_emw_mpw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1636,14 +1636,14 @@ From Faraday: \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia \] so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and

    -
    +

    -
    +
    • Gr (9.47)
    @@ -1662,18 +1662,18 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0. Generalizing to propagation in the direction of an arbitrary wavevector \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the

    -
    +

    E and B fields for a monochromatic EM plane wave

    -
    +

    -
    +
    • Gr (9.49)
    @@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html index cbaae90..34cd8e8 100644 --- a/build/emd_emw_we.html +++ b/build/emd_emw_we.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1641,18 +1641,18 @@ These take the form of coupled first-order partial differential equations for \( Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\), we get the

    -
    +

    Wave equations for electric and magnetic fields in vacuum

    -
    +

    -
    +
    • Gr (9.41)
    @@ -1714,7 +1714,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emdm.html b/build/emdm.html index 50e92df..d8eab27 100644 --- a/build/emdm.html +++ b/build/emdm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emdm_Me.html b/build/emdm_Me.html index c3fdfbe..60938aa 100644 --- a/build/emdm_Me.html +++ b/build/emdm_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html index 9443516..42906a3 100644 --- a/build/emdm_Me_Mem.html +++ b/build/emdm_Me_Mem.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1616,13 +1616,12 @@ Table of contents

    Maxwell's equations in vacuum:

    -\begin{align} +\begin{align*} (i)~~ &{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, \hspace{1cm} &\mbox{Gauss}, \nonumber \\ (ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \hspace{1cm} &\mbox{anonymous} \nonumber \\ (iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &\mbox{Faraday}, \nonumber \\ (iv)~~ &{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &\mbox{Ampère + Maxwell}. -\label{Gr(7.39)} -\end{align} +\end{align*}

    are complete as they stand. In presence of matter: more convenient to write sources in terms of free charges and currents. @@ -1642,25 +1641,25 @@ and magnetization \({\bf M}\) produces bound current density sigmab. If \(P\) increases a bit, charge increases, giving net current \[ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partial t} da_{\perp}. \] We therefore have the

    -
    +

    Polarization current density

    -
    +

    -
    +
    • Gr (7.48)
    @@ -1677,14 +1676,14 @@ We therefore have the

    -otherwise simply called the {\bf polarization current}. +otherwise simply called the polarization current. This has nothing to do with the bound current \({\bf J}_b\) (the latter is associated to magnetization; the polarization current is the result of linear motion of charge when polarization changes). We can check consistency with the continuity equation associated to the conservation of bound charges:

    -
-
- +
+ Prerequisites
    @@ -1627,8 +1627,8 @@ Prerequisites
-
- +
+ Objectives
    @@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_fe.html b/build/ems_ca_fe.html index 6317781..d38b3c6 100644 --- a/build/ems_ca_fe.html +++ b/build/ems_ca_fe.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1624,7 +1624,7 @@ A generic configuration of static charges coupled via the Coulomb interaction defines an electrostatic problem, whose solution is in principle obtained from calculating either the field according to E_vcd

-
+

@@ -1638,7 +1638,7 @@ from calculating either the field according to p_vcd

-
+

@@ -1658,7 +1658,7 @@ condition curlE0 can be expressed as th 🐟

-
+

@@ -1674,7 +1674,7 @@ condition curlE0 can be expressed as th

In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation Lap

-
+

@@ -1712,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_fe_L.html b/build/ems_ca_fe_L.html index de5b709..7c40eaf 100644 --- a/build/ems_ca_fe_L.html +++ b/build/ems_ca_fe_L.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1630,14 +1630,14 @@ In one dimension, the potential is a single-variable function \(\phi (x)\) and the Laplace equation reads

-
+

-
+
@@ -1652,14 +1652,14 @@ function \(\phi (x)\) and the Laplace equation reads

The solution to this is

-
+

-
+
  • Gr (3.6)
@@ -1718,14 +1718,14 @@ In two dimensions, the potential becomes a function of two variables (here: \(x\) and \(y\)), so Laplace's equation now reads

-
+

-
+
@@ -1778,14 +1778,14 @@ a point equals its value averaged over a sphere \(S_R({\bf r})\) of any radius \(R\) centered on this point (and of course not containing any charges),

-
+

-
+
@@ -1797,8 +1797,8 @@ a point equals its value averaged over a sphere \]

-
- +
+ Physicist's proof

@@ -1860,8 +1860,8 @@ proving the theorem.

-
- +
+ Formal proof @@ -1911,14 +1911,14 @@ we get the following general

Theorem:

-
+

-
+
@@ -1971,19 +1971,19 @@ are necessarily positive, we thus require \(f_x > 0\), \(f_y > 0\) and \(f of the \(f_x + f_y + f_z = 0\) condition above.

-
+

-
+
-
+

Earnshaw's theorem (physical version)

@@ -2102,7 +2102,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_fe_g.html b/build/ems_ca_fe_g.html index 5eaafab..6b9c3a4 100644 --- a/build/ems_ca_fe_g.html +++ b/build/ems_ca_fe_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents
@@ -1613,11 +1613,11 @@ Table of contents ems.ca.fe.g
-
+

George Green

-
 @@ -1635,14 +1635,14 @@ This yields The first term on the left-hand side vanishes since \(\Phi\) satisfies Laplace. The right-hand side can be made to vanish if \(\Phi\) obeys either

-
+

-
+
@@ -1654,14 +1654,14 @@ The right-hand side can be made to vanish if \(\Phi\) obeys either

or

-
+

-
+
@@ -1689,19 +1689,19 @@ additive constant.

We can thus finally state the

-
+

-
+
-
+

Uniqueness Theorem

@@ -1740,7 +1740,7 @@ Reading other books, you might be misled into thinking that there are numerous c and corollaries and that things are

-
+

Comment/warning: uniqueness theorem on uniqueness theorems
Do not be misled: there is a unique uniqueness theorem for the @@ -1767,7 +1767,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_me.html b/build/ems_ca_me.html index 9e620ee..d64bdc8 100644 --- a/build/ems_ca_me.html +++ b/build/ems_ca_me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1645,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_me_Ed.html b/build/ems_ca_me_Ed.html index 3bf1cbc..a43dbdf 100644 --- a/build/ems_ca_me_Ed.html +++ b/build/ems_ca_me_Ed.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1634,14 +1634,14 @@ E_\phi &= -\frac{1}{r \sin \theta} \frac{\partial \phi}{\partial \varphi} =

we get

-
+

-
+
  • Gr (3.103)
@@ -1663,14 +1663,14 @@ E_d ({\bf r}) = - \frac{1}{4\pi \varepsilon_0} \nabla \frac{{\bf p} \cdot {\bf r \] immediately yielding the coordinate-free expression

-
+

-
+
  • Gr (3.104)
@@ -1710,7 +1710,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_me_Eq.html b/build/ems_ca_me_Eq.html index be340e8..a095f02 100644 --- a/build/ems_ca_me_Eq.html +++ b/build/ems_ca_me_Eq.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_me_a.html b/build/ems_ca_me_a.html index f163564..812de72 100644 --- a/build/ems_ca_me_a.html +++ b/build/ems_ca_me_a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1628,14 +1628,14 @@ By Taylor expanding, we get Formally, we could do this for any vector \({\bf r}_s\) such that \(|{\bf r}_s| < |{\bf r}|\) by Taylor expanding with the \({\boldsymbol \nabla}\) operator,

-
+

-
+
@@ -1653,14 +1653,14 @@ the potential takes the form of the general solution of Laplace's equation -
+

-
+
  • Gr(3.94)
@@ -1682,14 +1682,14 @@ we can expand the potential at a point \({\bf r}\) outside \({\cal V}\) accordin (here, we put the origin of our coordinate system closer to all points in \({\cal V}\) than to \({\bf r}\) to ensure convergence)

-
+

-
+
  • Gr (3.95)
@@ -1752,14 +1752,14 @@ For \(r \gg d\), we can expand (immediately dropping terms of order \(d^2/r^2\)) Putting things together, the leading term in the expansion p_Leg for the potential of the physical dipole is given by

-
+

-
+
  • Gr (3.90)
@@ -1806,7 +1806,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_me_h.html b/build/ems_ca_me_h.html index b457342..603c6cd 100644 --- a/build/ems_ca_me_h.html +++ b/build/ems_ca_me_h.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1621,19 +1621,19 @@ The next terms in the expansion are obtained similarly: the quadrupole te \] and can be rewritten as

-
+

-
+
-
+

\[ \phi_q ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} @@ -1645,19 +1645,19 @@ and can be rewritten as

in terms of the quadrupole moment

-
+

-
+
-
+

\[ Q_{ab} = \int_{\cal V} d\tau_s (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s). @@ -1676,19 +1676,19 @@ This is a symmetric, traceless rank \(2\) tensor: \(Q_{ab} = Q_{ba}\) and

Our expansion for the potential, with terms up to quadrupole ones, thus looks like

-
+

-
+
-
+

\[ \phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \left( \frac{Q}{r} + \sum_a \frac{r_a}{r^3} p_a + \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} + ... \right) @@ -1709,7 +1709,7 @@ the leading nonvanishing multipole moment is independent of the chosen location the origin of the coordinate system (see Jackson Prob. 4.4).

-
+

A consistent nomenclature for the multipole expansion?

@@ -1858,7 +1858,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_me_md.html b/build/ems_ca_me_md.html index 04ae605..838f20b 100644 --- a/build/ems_ca_me_md.html +++ b/build/ems_ca_me_md.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1617,14 +1617,14 @@ Table of contents The series p_Leg is organized in increasing powers of inverse distance. The leading term is called the *monopole term, and is

-
+

-
+
  • Gr (3.97)
@@ -1632,7 +1632,7 @@ The leading term is called the *monopole term, and is
-
+

\[ \phi_m ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r}, \hspace{1cm} @@ -1653,19 +1653,19 @@ The next term is the dipole term: by using \(P_1 (x) = x\), we have \] This can be written

-
+

-
+
-
+

\[ \phi_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{\hat{\bf r} \cdot {\bf p}}{r^2}, @@ -1715,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_mi.html b/build/ems_ca_mi.html index e5e9271..8d96c7f 100644 --- a/build/ems_ca_mi.html +++ b/build/ems_ca_mi.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1646,14 +1646,14 @@ by distance \(d\). For definiteness, we put a charge \(q\) at coordinate

By superposition, we have that

-
+

-
+
  • FLS II (6.8)
  • Gr (3.9)
    • @@ -1711,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_mi_fe.html b/build/ems_ca_mi_fe.html index b25424e..3ac3967 100644 --- a/build/ems_ca_mi_fe.html +++ b/build/ems_ca_mi_fe.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1617,14 +1617,14 @@ Table of contents The force between the point charge and the plane is the same as in the case of two point charges:

-
+

-
+
  • Gr (3.12)
@@ -1677,7 +1677,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_mi_isc.html b/build/ems_ca_mi_isc.html index 2451f85..227a040 100644 --- a/build/ems_ca_mi_isc.html +++ b/build/ems_ca_mi_isc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1617,14 +1617,14 @@ Table of contents Use scd_cond, with the normal direction now being \(\hat{\bf z}\):

-
+

-
+
  • Gr (3.10)
@@ -1645,14 +1645,14 @@ The total induced charge can be obtained by simple integration as Using planar coordinates, \(\sigma(r) = \frac{-qd}{4\pi (r^2 + (d/2)^2)^{3/2}}\), so

-
+

-
+
  • Gr (3.11)
@@ -1693,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_mi_o.html b/build/ems_ca_mi_o.html index 018d383..33ada8d 100644 --- a/build/ems_ca_mi_o.html +++ b/build/ems_ca_mi_o.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1676,7 +1676,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_sv.html b/build/ems_ca_sv.html index a08f45c..99586c0 100644 --- a/build/ems_ca_sv.html +++ b/build/ems_ca_sv.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1656,7 +1656,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_sv_car.html b/build/ems_ca_sv_car.html index a293243..a2869a2 100644 --- a/build/ems_ca_sv_car.html +++ b/build/ems_ca_sv_car.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1613,7 +1613,7 @@ Table of contents ems.ca.sv.car
-
+

@@ -1659,14 +1659,14 @@ This thus falls back onto a 2d problem. We need to solve the 2d Laplace equatio

Let us look for solutions in the form

-
+

-
+
  • Gr (3.23)
@@ -1685,14 +1685,14 @@ individual term in the Laplace equation equals a constant, and that these constants add up to zero. We can thus put (the sign choice anticipates the solution somewhat)

-
+

-
+
  • Gr (3.26)
@@ -1716,14 +1716,14 @@ Let's look first of all at the solutions of +

-
+
  • Gr (3.27)
@@ -1780,14 +1780,14 @@ C_n = \frac{2}{a} \int_0^a dy ~\phi_0(y) \sin(n\pi y/a)

Specific example: say that \(\phi_0(y) = \phi_0\), i.e. just a constant. Then,

-
+

-
+
  • Gr (3.35)
@@ -1841,7 +1841,7 @@ The solution for the specific case \(\phi_0 (y) = \phi_0\) is thus

-
+

Example: rectangular pipe

@@ -1898,14 +1898,14 @@ The full solution is then a linear combination of complete set of functions, The coefficients must be chosen such that \((iii)\) is fulfilled, \(\phi(b,y) = \phi_0\). This simple case of a constant value \(\phi_0\) gives us the same relation as Cn, so

-
+

-
+
  • Gr (3.42)
@@ -1941,7 +1941,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_sv_cyl.html b/build/ems_ca_sv_cyl.html index a52c13f..9fc7110 100644 --- a/build/ems_ca_sv_cyl.html +++ b/build/ems_ca_sv_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1633,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_ca_sv_sph.html b/build/ems_ca_sv_sph.html index edacf52..da95e36 100644 --- a/build/ems_ca_sv_sph.html +++ b/build/ems_ca_sv_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1617,14 +1617,14 @@ Table of contents In spherical coordinates, the Laplace equation takes the following form (using sph_Lap):

-
+

-
+
  • Gr (3.53)
  • W (11-86)
    • @@ -1633,7 +1633,7 @@ In spherical coordinates, the Laplace equation takes the following form
-
+

@@ -1650,14 +1650,14 @@ In spherical coordinates, the Laplace equation takes the following form If you are dealing with a problem having azimuthal symmetry, \(\phi\) is independent of \(\varphi\) and the equation simplifies to:

-
+

-
+
  • Gr (3.54)
  • W (11-87)
    • @@ -1755,7 +1755,7 @@ This equation is solved by Legendre polynomials of the variable \(\cos \t

    -
    +

    Legendre polynomials

    @@ -1776,14 +1776,14 @@ and conveniently defined (for trigonometric arguments) to obey the orthogonality relationship (the reason for the normalization on the right-hand side will become clear later)

    -
    +

    -
    +
    @@ -1798,14 +1798,14 @@ relationship (the reason for the normalization on the right-hand side will becom

    This same relation can be more simply written by using the variable \(x = \cos \theta\),

    -
    +

    -
    +
    @@ -1823,14 +1823,14 @@ To get started, we need to define the "seed" polynomial (carrying label \(l=0\)) To make life easy, we set \(P_0 (x) = 1\). Higher polynomials are then sought in the form of power series in \(x\). This leads to the first few Legendre polynomials being:

    -
    +

    -
    +
    @@ -1850,14 +1850,14 @@ P_5 (x) &= \frac{1}{8} (63x^5 - 70x^3 + 15x). The prefactor (and thus the factor in the orthogonality relations Leg_orth_trig and the equivalent Leg_orth is chosen for convenience such that each polynomial takes the value \(1\) when evaluated at argument \(x = 1\),

    -
    +

    -
    +
    @@ -1873,14 +1873,14 @@ P_l(1) = 1

    The Legendre polynomial \(P_l\) obeys the differential equation

    -
    +

    -
    +
    @@ -1892,14 +1892,14 @@ The Legendre polynomial \(P_l\) obeys the differential equation \] or equivalently

    -
    +

    -
    +
    @@ -1915,14 +1915,14 @@ or equivalently A particularly convenient formula for deriving \(P_l(x)\) is the Rodrigues formula:

    -
    +

    -
    +
    @@ -1938,14 +1938,14 @@ P_l(x) = \frac{1}{2^l l!} \left( \frac{d}{dx} \right)^l (x^2 - 1)^l

    Actually, a more practical formula is Bonnet's recursion relation

    -
    +

    -
    +
    @@ -1981,14 +1981,14 @@ We therefore come to the culmination of our efforts here, and write the general solution to any problem with azimuthal symmetry (for which the potential takes a finite value for \(\theta = 0, \pi\)) as

    -
    +

    -
    +
    • Gr (3.65)
    @@ -1996,7 +1996,7 @@ the general solution to any problem with azimuthal symmetry
    -
    +

    \[ \phi(r,\theta) = \sum_{l=0}^{\infty} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \theta) @@ -2008,7 +2008,7 @@ the general solution to any problem with azimuthal symmetry -

    +

    Example: potential inside a hollow sphere

    @@ -2091,7 +2091,7 @@ Thus, \(A_0 = k/2\), \(A_1 = -k/2\), and all others are zero, so
    -
    +

    Example: surface charge density on sphere

    @@ -2221,14 +2221,14 @@ A_1^i = \frac{k}{2\varepsilon_0} \int_0^\pi d\theta \sin \theta [P_l(\cos \theta

    The potential inside/outside the sphere is then

    -
    +

    -
    +
    @@ -2261,7 +2261,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/ems_es.html b/build/ems_es.html index aa6ade2..d1b5824 100644 --- a/build/ems_es.html +++ b/build/ems_es.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1620,8 +1620,8 @@ Table of contents
  • PM 1
-
- +
+ Prerequisites
    @@ -1629,8 +1629,8 @@ Prerequisites
-
- +
+ Objectives
    @@ -1672,7 +1672,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_c.html b/build/ems_es_c.html index 7e1755e..32c552a 100644 --- a/build/ems_es_c.html +++ b/build/ems_es_c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1645,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_c_cap.html b/build/ems_es_c_cap.html index 721427c..4240173 100644 --- a/build/ems_es_c_cap.html +++ b/build/ems_es_c_cap.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1635,7 +1635,7 @@ C \equiv \frac{Q}{V} \]

-
+

Concentric shells

@@ -1686,7 +1686,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_c_ic.html b/build/ems_es_c_ic.html index 7ee1d75..93bc562 100644 --- a/build/ems_es_c_ic.html +++ b/build/ems_es_c_ic.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1618,7 +1618,7 @@ Charge in hollow conductor. Induced surface charges on inside and outside surfaces.

-
+

Field outside spherical conductor with irregular cavity

@@ -1669,7 +1669,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_c_p.html b/build/ems_es_c_p.html index 1912026..a502192 100644 --- a/build/ems_es_c_p.html +++ b/build/ems_es_c_p.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1613,7 +1613,7 @@ Table of contents ems.es.c.p
-
+

Basic Properties of a Conductor

@@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_c_sc.html b/build/ems_es_c_sc.html index 0596fc7..be3419d 100644 --- a/build/ems_es_c_sc.html +++ b/build/ems_es_c_sc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1618,14 +1618,14 @@ For a conductor, we can exploit the fact that electrical fields vanish on the in get a proper boundary condition for the potential. Namely, here, boundary condition Edisc yields

-
+

-
+
  • Gr (2.48)
@@ -1643,14 +1643,14 @@ get a proper boundary condition for the potential. Namely, here, boundary condi

which in terms of potential reads

-
+

-
+
  • Gr (2.49)
@@ -1712,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_e.html b/build/ems_es_e.html index b6e4c3b..44ef56e 100644 --- a/build/ems_es_e.html +++ b/build/ems_es_e.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1634,14 +1634,14 @@ W_u = -\int_{{\bf a}}^{{\bf b}} {\bf E} \cdot d{\bf l} For a point charge distribution, the total work required to assemble it can also be rewritten in terms of the potential by using p,

-
+

-
+
  • Gr (2.42)
@@ -1660,14 +1660,14 @@ W = \frac{1}{2} \sum_{i=1}^m q_i \phi({\bf r}_i) For a volume charge distribution, this becomes

-
+

-
+
  • Gr (2.43)
@@ -1709,14 +1709,14 @@ W = \frac{\varepsilon_0}{2} \left( \int_{\cal V} E^2 d\tau + \oint_{\cal S} \phi Integrating over all space (and thus dropping the surface integrale whose integrand is assumed to vanish at infinity), we get

-
+

-
+
  • Gr (2.45)
@@ -1724,7 +1724,7 @@ to vanish at infinity), we get
-
+

@@ -1738,7 +1738,7 @@ W = \frac{\varepsilon_0}{2} \int E^2 d\tau
-
+

Energy of spherical shell

@@ -1811,7 +1811,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_ec.html b/build/ems_es_ec.html index d8670db..afa7ad6 100644 --- a/build/ems_es_ec.html +++ b/build/ems_es_ec.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1645,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_ec_b.html b/build/ems_es_ec_b.html index 8276803..c170c42 100644 --- a/build/ems_es_ec_b.html +++ b/build/ems_es_ec_b.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_ec_c.html b/build/ems_es_ec_c.html index 271517e..4c97c76 100644 --- a/build/ems_es_ec_c.html +++ b/build/ems_es_ec_c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_ec_q.html b/build/ems_es_ec_q.html index 721cfc8..a8ea181 100644 --- a/build/ems_es_ec_q.html +++ b/build/ems_es_ec_q.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_ec_s.html b/build/ems_es_ec_s.html index 724a6dd..767423b 100644 --- a/build/ems_es_ec_s.html +++ b/build/ems_es_ec_s.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1635,7 +1635,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_ef.html b/build/ems_es_ef.html index 29271b7..4c98ff2 100644 --- a/build/ems_es_ef.html +++ b/build/ems_es_ef.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1645,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_ef_Gl.html b/build/ems_es_ef_Gl.html index 1779a07..1798291 100644 --- a/build/ems_es_ef_Gl.html +++ b/build/ems_es_ef_Gl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1658,14 +1658,14 @@ sphere of radius \(r\) around the charge,

so by superposition, we obtain

-
+

-
+
  • Gr (2.13)
@@ -1673,7 +1673,7 @@ so by superposition, we obtain
-
+

Gauss' law (in integral form)

@@ -1705,14 +1705,14 @@ By applying the divergence theorem, and using \(Q_{\mbox{enc}} = \int_{\cal V} \rho d\tau\), and using the fact the the choice of volume is arbitrary, we get

-
+

-
+
  • Gr (2.14)
@@ -1720,7 +1720,7 @@ is arbitrary, we get
-
+

Gauss' law in differential form

@@ -1779,7 +1779,7 @@ cylindrical or plane symmetry. Gaussian surfaces: respectively, concentric sphere, coaxial cylinder, pillbox.

-
+ -
+

Example 2.3: infinitely long cylinder carrying charge density \(\rho = k s\) for some constant \(k\). Find \({\bf E}\) within the cylinder.

@@ -1853,7 +1853,7 @@ Therefore,
-
+

Example 2.4: infinite plane (defined by \(z = 0\)) with uniform surface charge density \(\sigma\). Find \({\bf E}\).

@@ -1879,7 +1879,7 @@ where \(\hat{\bf n}\) is a unit vector extending away from the plane. Independe
-
+

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_ef_cE.html b/build/ems_es_ef_cE.html index ce6101f..cca0ffe 100644 --- a/build/ems_es_ef_cE.html +++ b/build/ems_es_ef_cE.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1617,14 +1617,14 @@ Table of contents Since the work done when moving on a closed path vanishes, we have

-
+

-
+
  • Gr (2.19)
@@ -1632,7 +1632,7 @@ Since the work done when moving on a closed path vanishes, we have
-
+

@@ -1650,14 +1650,14 @@ Since the work done when moving on a closed path vanishes, we have which by Stokes' theorem implies that the electrostatic field is curlless,

-
+

-
+
  • Gr (2.20)
@@ -1665,7 +1665,7 @@ which by Stokes' theorem implies that
-
+

@@ -1711,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_ef_ccd.html b/build/ems_es_ef_ccd.html index 5f095ce..779961e 100644 --- a/build/ems_es_ef_ccd.html +++ b/build/ems_es_ef_ccd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1619,14 +1619,14 @@ calculated from Coulomb's law using the superposition principle. Since each inf volume element \(d\tau' = dx' dy' dz'\) contains a charge \(dq' = \rho({\bf r}') d\tau'\), we have

-
+

-
+
  • Gr4 (2.8)
@@ -1634,7 +1634,7 @@ volume element \(d\tau' = dx' dy' dz'\) contains a charge \(dq' = \rho({\bf r}')
-
+

@@ -1654,14 +1654,14 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S} \(\sigma({\bf r})\), we have over an infinitesimal area \(da'\) a charge \(dq' = \sigma({\bf r}') da'\), so

-
+

-
+
  • Gr4(2.7)
@@ -1669,7 +1669,7 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
-
+

@@ -1686,14 +1686,14 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S} Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf r}')\),

-
+

-
+
  • Gr (2.6)
@@ -1701,7 +1701,7 @@ Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf
-
+

@@ -1715,7 +1715,7 @@ Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf
-
+

Example

@@ -1749,7 +1749,7 @@ most easily by observing that \(\frac{d}{dx} \left( \frac{x}{\sqrt{z^2 + x^2}} \ = \frac{1}{\sqrt{z^2 + x^2}} - \frac{x^2}{(z^2 + x^2)^{3/2}} = \frac{z^2}{(z^2 + x^2)^{3/2}}\), leading to

-

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_ef_pc.html b/build/ems_es_ef_pc.html index 89430aa..0d6fb41 100644 --- a/build/ems_es_ef_pc.html +++ b/build/ems_es_ef_pc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1627,14 +1627,14 @@ an electric field originating from the source charge. Invoking the superposition principle, we can thus write

-
+

-
+
  • Gr (2.3)
@@ -1642,7 +1642,7 @@ Invoking the superposition principle, we can thus write
-
+

@@ -1658,14 +1658,14 @@ Invoking the superposition principle, we can thus write

with the electric field of a point charge distribution being

-
+

-
+
  • Gr (2.4)
@@ -1673,7 +1673,7 @@ with the electric field of a point charge distribution being
-
+

@@ -1691,7 +1691,7 @@ The electric field \({\bf E} ({\bf r})\) is thus the force per unit charge that be exerted if you put a test charge at position \({\bf r}\).

-
+

Example

@@ -1735,7 +1735,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_efo.html b/build/ems_es_efo.html index b044dab..e8ad618 100644 --- a/build/ems_es_efo.html +++ b/build/ems_es_efo.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1646,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-07 Mon 20:38

+

Created: 2022-03-15 Tue 08:10

diff --git a/build/ems_es_efo_cl.html b/build/ems_es_efo_cl.html index e4f3bac..2377102 100644 --- a/build/ems_es_efo_cl.html +++ b/build/ems_es_efo_cl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1619,14 +1619,14 @@ Table of contents
  • PM 1.4
    • -
    +

    -
    +
    • FLS II (4.9)
    • Gr (2.1)
      • @@ -1638,7 +1638,7 @@ Table of contents
    -
    +

    The force \(F_{t\leftarrow s}\) exerted by a point source charge \(q_s\) sitting at \({\bf r}_s\) on a point test charge \(q_t\) sitting at \({\bf r}_t\) is given by Coulomb's law, @@ -1655,7 +1655,7 @@ on a point test charge \(q_t\) sitting at \({\bf r}_t\) is given by Coulomb's la

    -
    -
    - +
    + Objectives

    @@ -1646,8 +1646,8 @@ Objectives for this part: what you should learn by reading this

    -
    - +
    + Core

    @@ -1655,8 +1655,8 @@ Core material: you have to know this by heart, and how to use it.

    -
    - +
    + Main

    @@ -1664,8 +1664,8 @@ Main matter which you should know how to use.

    -
    - +
    + Example

    @@ -1673,8 +1673,8 @@ Example of the concepts just covered.

    -
    - +
    + Info

    @@ -1682,8 +1682,8 @@ Additional (contextual) information.

    -
    - +
    + Historical Context

    @@ -1709,7 +1709,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/in_t_l.html b/build/in_t_l.html index 88a8c68..ad19a5a 100644 --- a/build/in_t_l.html +++ b/build/in_t_l.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1673,7 +1673,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/index.html b/build/index.html index 11b09e5..8950ce3 100644 --- a/build/index.html +++ b/build/index.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1964,7 +1964,6 @@ Table of contents
  • Diagnosticsd
  • @@ -2091,7 +2091,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    • Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/qed.html b/build/qed.html index b4c6b0d..3771f87 100644 --- a/build/qed.html +++ b/build/qed.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/qed_t.html b/build/qed_t.html index 689f631..b0d9469 100644 --- a/build/qed_t.html +++ b/build/qed_t.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1632,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red.html b/build/red.html index 1dbadc6..90680f4 100644 --- a/build/red.html +++ b/build/red.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1614,8 +1614,8 @@ Table of contents red
    -
    - +
    + Prerequisites
      @@ -1623,8 +1623,8 @@ Prerequisites
    -
    - +
    + Objectives
      @@ -1667,7 +1667,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_rem.html b/build/red_rem.html index cc6ef06..326f459 100644 --- a/build/red_rem.html +++ b/build/red_rem.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_rem_Fmunu.html b/build/red_rem_Fmunu.html index 02150e0..319cf0d 100644 --- a/build/red_rem_Fmunu.html +++ b/build/red_rem_Fmunu.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1616,7 +1616,7 @@ Table of contents

    We have seen that a four-vector transforms according to \[ - \bar{a}^\mu = \Lambda_\nu^\mu a^\nu + \bar{a}^\mu = \Lambda^\mu{}_\nu a^\nu \] in which \(\Lambda\) is a matrix representing the Lorentz transformation. The form this matrix takes depends on the actual transformation: for the specific @@ -1628,11 +1628,11 @@ case of motion in the \(x\) direction with velocity \(v\), 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right). \] -A four-vector is synonymous to a {\bf rank-one tensor}. +A four-vector is synonymous to a rank-one tensor. Higher-rank tensors are simply objects carrying more indices. -For example, a {\bf rank-two tensor} transforms as +For example, a rank-two tensor transforms as \[ - \bar{t}^{\mu \nu} = \Lambda^\mu_\lambda \Lambda^\nu_\sigma t^{\lambda \sigma}. + \bar{t}^{\mu \nu} = \Lambda^\mu{}_\lambda \Lambda^\nu{}_\sigma t^{\lambda \sigma}. \] Such a rank-two tensor can be represented similarly to a matrix: \[ @@ -1656,59 +1656,89 @@ Under the Lorentz transformation along \(x\) defined above, we can work out how the nonvanishing elements of an antisymmetric tensor transform:

    \begin{align} - \bar{t}_a^{01} &= \Lambda^0_\mu \Lambda^1_\nu t_a^{\mu \nu} - = \Lambda^0_0 \Lambda^1_0 t_a^{00} + \Lambda^0_1 \Lambda^1_0 t_a^{10} - + \Lambda^0_0 \Lambda^1_1 t_a^{01} + \Lambda^0_1 \Lambda^1_1 t_a^{11} \\ - & = (\Lambda^0_0 \Lambda^1_1 - \Lambda^0_1 \Lambda^1_0) t_a^{01} + \bar{t}_a^{01} &= \Lambda^0{}_\mu \Lambda^1{}_\nu t_a^{\mu \nu} + = \Lambda^0{}_0 \Lambda^1{}_0 t_a^{00} + \Lambda^0{}_1 \Lambda^1{}_0 t_a^{10} + + \Lambda^0{}_0 \Lambda^1{}_1 t_a^{01} + \Lambda^0{}_1 \Lambda^1{}_1 t_a^{11} \\ + & = (\Lambda^0{}_0 \Lambda^1{}_1 - \Lambda^0{}_1 \Lambda^1{}_0) t_a^{01} = \gamma^2 (1 - \beta^2) t_a^{01} = t_a^{01}, \\ - \bar{t}_a^{02} &= \Lambda^0_\mu \Lambda^2_\nu t_a^{\mu \nu} - = \Lambda^0_0 \Lambda^2_2 t_a^{02} + \Lambda^0_1 \Lambda^2_2 t_a^{12} + \bar{t}_a^{02} &= \Lambda^0{}_\mu \Lambda^2{}_\nu t_a^{\mu \nu} + = \Lambda^0{}_0 \Lambda^2{}_2 t_a^{02} + \Lambda^0{}_1 \Lambda^2{}_2 t_a^{12} = \gamma (t_a^{02} - \beta t_a^{12}), \\ - \bar{t}_a^{03} &= \Lambda^0_\mu \Lambda^3_\nu t_a^{\mu \nu} - = \Lambda^0_0 \Lambda^3_3 t_a^{03} + \Lambda^0_1 \Lambda^3_3 t_a^{13} + \bar{t}_a^{03} &= \Lambda^0{}_\mu \Lambda^3{}_\nu t_a^{\mu \nu} + = \Lambda^0{}_0 \Lambda^3{}_3 t_a^{03} + \Lambda^0{}_1 \Lambda^3{}_3 t_a^{13} = \gamma (t_a^{03} - \beta t_a^{13}), \\ - \bar{t}_a^{12} &= \Lambda^1_\mu \Lambda^2_\nu t_a^{\mu \nu} - = \Lambda^1_0 \Lambda^2_2 t_a^{02} + \Lambda^1_1 \Lambda^2_2 t_a^{12} + \bar{t}_a^{12} &= \Lambda^1{}_\mu \Lambda^2{}_\nu t_a^{\mu \nu} + = \Lambda^1{}_0 \Lambda^2{}_2 t_a^{02} + \Lambda^1{}_1 \Lambda^2{}_2 t_a^{12} = \gamma (t_a^{12} - \beta t_a^{02}), \\ - \bar{t}_a^{13} &= \Lambda^1_\mu \Lambda^3_\nu t_a^{\mu \nu} - = \Lambda^1_0 \Lambda^3_3 t_a^{03} + \Lambda^1_1 \Lambda^3_3 t_a^{13} + \bar{t}_a^{13} &= \Lambda^1{}_\mu \Lambda^3{}_\nu t_a^{\mu \nu} + = \Lambda^1{}_0 \Lambda^3{}_3 t_a^{03} + \Lambda^1{}_1 \Lambda^3{}_3 t_a^{13} = \gamma (t_a^{13} - \beta t_a^{03}), \\ - \bar{t}_a^{23} &= \Lambda^2_\mu \Lambda^3_\nu t_a^{\mu \nu} - = \Lambda^2_2 \Lambda^3_3 t_a^{23} + \bar{t}_a^{23} &= \Lambda^2{}_\mu \Lambda^3{}_\nu t_a^{\mu \nu} + = \Lambda^2{}_2 \Lambda^3{}_3 t_a^{23} = t_a^{23}. \end{align}

    -Comparing with the transformation rules for the electromagnetic field which we obtained in (\ref{eq:EMFieldsLorentzTransfo}), we can define the +Comparing with the transformation rules for the electromagnetic field which we obtained in EMtr, we can define the

    -
    +
    +

    +Electromagnetic Field Tensor +

    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf Electromagnetic Field Tensor} \[ - F^{\mu \nu} = \left( \begin{array}{cccc} - 0 & E_x/c & E_y/c & E_z/c \\ - -E_x/c & 0 & B_z & -B_y \\ - -E_y/c & -B_z & 0 & B_x \\ - -E_z/c & B_y & -B_x & 0 \end{array} \right) - \] +F^{\mu \nu} = \left( \begin{array}{cccc} +0 & E_x/c & E_y/c & E_z/c \\ +-E_x/c & 0 & B_z & -B_y \\ +-E_y/c & -B_z & 0 & B_x \\ +-E_z/c & B_y & -B_x & 0 \end{array} \right) +\tag{Fmunu}\label{Fmunu} +\]

    together with the handy

    -
    +
    +

    +Dual Field Tensor +

    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf Dual Field Tensor} \[ - G^{\mu \nu} = \left( \begin{array}{cccc} - 0 & B_x & B_y & B_z \\ - -B_x & 0 & -E_z/c & E_y/c \\ - -B_y & E_z/c & 0 & -E_x/c \\ - -B_z & -E_y/c & E_x/c & 0 \end{array} \right) - \] +G^{\mu \nu} = \left( \begin{array}{cccc} +0 & B_x & B_y & B_z \\ +-B_x & 0 & -E_z/c & E_y/c \\ +-B_y & E_z/c & 0 & -E_x/c \\ +-B_z & -E_y/c & E_x/c & 0 \end{array} \right) +\tag{Gmunu}\label{Gmunu} +\]

    @@ -1723,12 +1753,27 @@ obtained from the field tensor by the substitution

    Our electromagnetic field transformation laws then become the simple

    -
    +
    +

    +Lorentz Transformation Rules for EM Fields +

    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf Lorentz Transformation Rules for EM Fields} \[ - \bar{F}^{\mu \nu} = \Lambda^\mu_\lambda \Lambda^\nu_\sigma F^{\lambda \sigma} - \] +\bar{F}^{\mu \nu} = \Lambda^\mu{}_\lambda \Lambda^\nu{}_\sigma F^{\lambda \sigma} +\tag{LorF}\label{LorF} +\]

    @@ -1753,7 +1798,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_rem_Ltf.html b/build/red_rem_Ltf.html index 935bd3b..4107cba 100644 --- a/build/red_rem_Ltf.html +++ b/build/red_rem_Ltf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1663,8 +1663,8 @@ density, we get Going back to our setup with plates in the \(xz\) plane which we started from, in the moving frame, there is now a magnetic field due to surface currents: \[ - {\boldsymbol K}_{\mbox{\tiny top}} = \sigma v_0 \hat{\boldsymbol x} - = -{\boldsymbol K}_{\mbox{\tiny bot}}. + {\boldsymbol K}_{\mbox{top}} = \sigma v_0 \hat{\boldsymbol x} + = -{\boldsymbol K}_{\mbox{bot}}. \] This magnetic field between the plates is thus \[ @@ -1736,10 +1736,22 @@ These factors cancel so \(\bar{B}_x = B_x\).

    We thus obtain the

    -
    +

    -{\bf EM field transformation laws (motion along \(x\) with velocity \(v\))} +EM field transformation laws (motion along \(x\) with velocity \(v\))

    +
    +

    + + + + +

    +
    + +
    + +
    \begin{align} \bar{E}_x &= E_x, \hspace{10mm} & \bar{E}_y &= \gamma (E_y - v B_z), \hspace{10mm} & @@ -1747,7 +1759,7 @@ We thus obtain the \bar{B}_x &= B_x, & \bar{B}_y &= \gamma \left( B_y + \frac{v}{c^2} E_z \right), & \bar{B}_z &= \gamma \left( B_z - \frac{v}{c^2} E_y \right) - \label{eq:EMFieldsLorentzTransfo} + \tag{EMtr}\label{EMtr} \end{align}
    @@ -1757,7 +1769,7 @@ Two special cases can be mentioned:

    -\paragraph{If \({\boldsymbol B} = 0\) in \({\cal S}\):} +If \({\boldsymbol B} = 0\) in \({\cal S}\): Then, \(\bar{\boldsymbol B} = \gamma \frac{v}{c^2} (E_z \hat{\boldsymbol y} - E_y \hat{\boldsymbol z}) = \frac{v}{c^2} (\bar{E}_z \hat{\boldsymbol y} - \bar{E}_y \hat{\boldsymbol z})\) so \[ \bar{\boldsymbol B} = -\frac{1}{c^2} {\boldsymbol v} \times \bar{\boldsymbol E}. @@ -1765,7 +1777,7 @@ Then, \(\bar{\boldsymbol B} = \gamma \frac{v}{c^2} (E_z \hat{\boldsymbol y} - E_

    -\paragraph{If \({\boldsymbol E} = 0\) in \({\cal S}\):} +If \({\boldsymbol E} = 0\) in \({\cal S}\): Then, \(\hat{\boldsymbol E} = -\gamma v (B_z \hat{\boldsymbol y} - B_y \hat{\boldsymbol z}) = -v (\bar{B}_z \hat{\boldsymbol y} - \bar{B}_y \hat{\boldsymbol z})\) so \[ @@ -1793,7 +1805,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_rem_Me.html b/build/red_rem_Me.html index c886817..1f0d8a7 100644 --- a/build/red_rem_Me.html +++ b/build/red_rem_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1637,39 +1637,84 @@ We thus obtain \] Recognizing the proper velocity, we thus get that charge density and current density can together form the

    -
    +
    +

    +Current density 4-vector +

    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf Current density 4-vector} \[ - J^\mu = \rho_0 \eta^\mu, \hspace{10mm} - J^\mu = \left( c\rho, J_x, J_y, J_z \right) - \] +J^\mu = \rho_0 \eta^\mu, \hspace{10mm} +J^\mu = \left( c\rho, J_x, J_y, J_z \right) +\tag{Jmu}\label{Jmu} +\]

    -The continuity equation (\ref{eq:continuity}) takes the simple form +The continuity equation conteq takes the simple form

    -
    +
    +

    +Continuity equation +

    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf Continuity equation} \[ - \frac{\partial J^\mu}{\partial x^\mu} = 0 - \] +\frac{\partial J^\mu}{\partial x^\mu} = 0 +\tag{conteq_rel}\label{conteq_rel} +\]

    while similarly the notation simplifies for

    -
    +
    +

    +Maxwell's equations +

    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf Maxwell's equations} \[ - \frac{\partial F^{\mu \nu}}{\partial x^\nu} = \mu_0 J^\mu, - \hspace{10mm} - \frac{\partial G^{\mu \nu}}{\partial x^\nu} = 0 - \] +\frac{\partial F^{\mu \nu}}{\partial x^\nu} = \mu_0 J^\mu, +\hspace{10mm} +\frac{\partial G^{\mu \nu}}{\partial x^\nu} = 0 +\tag{Max_rel}\label{Max_rel} +\]

    @@ -1677,35 +1722,49 @@ while similarly the notation simplifies for

    In terms of \(F^{\mu \nu}\) and the proper velocity \(\eta^\mu\), we also have the

    -
    +
    +

    +Minkowski force on a charge \(q\) +

    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf Minkowski force on a charge \(q\)} \[ - K^\mu = q F^{\mu \nu} \eta_\nu - \] +K^\mu = q F^{\mu \nu} \eta_\nu +\tag{MinkF_rel}\label{MinkF_rel} +\]

    whose vector components are \[ - {\boldsymbol K} = \frac{q}{\sqrt{1 - u^2/c^2}} \left( - {\boldsymbol E} + {\boldsymbol u} \times {\boldsymbol B} \right) +{\boldsymbol K} = \frac{q}{\sqrt{1 - u^2/c^2}} \left( +{\boldsymbol E} + {\boldsymbol u} \times {\boldsymbol B} \right) \] -which becomes the Lorentz force law when remembering -(\ref{eq:MinkowskiForce}). +which becomes the Lorentz force law when remembering MinkF.

    We had \[ - {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}, \hspace{10mm} + {\boldsymbol E} = -{\boldsymbol \nabla} \phi - \frac{\partial {\boldsymbol A}}{\partial t}, \hspace{10mm} {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}. \] We can group the potentials together into a 4-vector: \[ - A^\mu = \left( V/c, A_x, A_y, A_z \right). + A^\mu = \left( \phi/c, A_x, A_y, A_z \right). \] The field tensor is then expressed as \[ @@ -1721,18 +1780,46 @@ We can now exploit gauge invariance A^\mu ~~\longrightarrow~~ {A^\mu}^\prime = A^\mu + \frac{\partial \lambda}{\partial x_\mu} \] which leaves \(F^{\mu \nu}\) invariant. In particular, we can choose -the Lorenz gauge (\ref{eq:InhomogeneousMaxwellLorenzGauge}) here expressed as +the Lorenz gauge LorenzG here expressed as +

    +
    +

    + + + + +

    +
    + +
    + +
    +

    \[ \frac{\partial A^\mu}{\partial x^\mu} = 0. +\tag{LorenzG_4v}\label{LorenzG_4v} \] -Defining the +Using the d'Alembertian operator introduced in dAl +here rewritten as

    -
    +
    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf d'Alembertian operator} \[ - \square^2 \equiv \frac{\partial}{\partial x_\nu} \frac{\partial}{\partial x^\nu} = {\boldsymbol \nabla}^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \] +\square^2 \equiv \frac{\partial}{\partial x_\nu} \frac{\partial}{\partial x^\nu} = {\boldsymbol \nabla}^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} +\tag{dAl_4v}\label{dAl_4v} +\]

    @@ -1740,12 +1827,27 @@ Defining the we obtain the final form of

    -
    +
    +

    +Maxwell's equations (Lorenz gauge, 4-vector notation) +

    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf Maxwell's equations (Lorenz gauge, 4-vector notation)} \[ - \square^2 A^\mu = -\mu_0 J^\mu - \] +\square^2 A^\mu = -\mu_0 J^\mu +\tag{Max_Lor_4v}\label{Max_Lor_4v} +\]

    @@ -1772,7 +1874,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_rem_mre.html b/build/red_rem_mre.html index bf1c2c2..7aae8b6 100644 --- a/build/red_rem_mre.html +++ b/build/red_rem_mre.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1622,7 +1622,7 @@ implies the existence of magnetism, and vice-versa.

    To illustrate this, we take a simple example

    -

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_rm.html b/build/red_rm.html index d0b3ca2..7a25f14 100644 --- a/build/red_rm.html +++ b/build/red_rm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_rm_Mf.html b/build/red_rm_Mf.html index 8ed2c11..3dccdf3 100644 --- a/build/red_rm_Mf.html +++ b/build/red_rm_Mf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1616,37 +1616,44 @@ Table of contents

    Newton's second law remains valid provided we use the relativistic momentum:

    -
    +

    -{\bf Newton's law (relativistic case)} +Newton's law (relativistic case) \[ - {\boldsymbol F} = \frac{d{\boldsymbol p}}{dt} - \] +{\boldsymbol F} = \frac{d{\boldsymbol p}}{dt} +\]

    -
    +

    -\paragraph{Example 12.10: Motion under a constant force.} A particle +Example: motion under a constant force +

    + +

    +A particle of mass \(m\) is subjected to a constant force \(F\). If it starts at the origin at \(t=0\), what is it's position as a function of time? -\paragraph{Solution:} +

    + +

    +Solution: \[ - \frac{dp}{dt} = F, ~~p(0) = 0 ~~\longrightarrow~~ - p = Ft - \] +\frac{dp}{dt} = F, ~~p(0) = 0 ~~\longrightarrow~~ +p = Ft +\] and thus \[ - p(t) = \frac{m u(t)}{1 - u(t)^2/c^2} = Ft - ~~\longrightarrow~~ u(t) = \frac{(F/m)t}{\sqrt{1 + (Ft/mc)^2}}. - \] +p(t) = \frac{m u(t)}{1 - u(t)^2/c^2} = Ft +~~\longrightarrow~~ u(t) = \frac{(F/m)t}{\sqrt{1 + (Ft/mc)^2}}. +\] Integrating again to get the displacement, \[ - x(t) = \frac{F}{m}\int_0^t dt' \frac{t'}{\sqrt{1 + (Ft'/mc)^2}} - = \frac{mc^2}{F} \left[ \sqrt{1 + (Ft/mc)^2} - 1 \right]. - \] -The particle's world line thus shows {\bf hyperbolic motion}. +x(t) = \frac{F}{m}\int_0^t dt' \frac{t'}{\sqrt{1 + (Ft'/mc)^2}} += \frac{mc^2}{F} \left[ \sqrt{1 + (Ft/mc)^2} - 1 \right]. +\] +The particle's world line thus shows hyperbolic motion.

    @@ -1654,7 +1661,10 @@ The particle's world line thus shows {\bf hyperbolic motion}.

    -\paragraph{Work and energy} +Work and energy +

    + +

    In the context of relativity, work is still the line integral of the force: \[ W \equiv \int {\boldsymbol F} \cdot d{\boldsymbol l} @@ -1682,7 +1692,10 @@ and we thus get

    -\paragraph{Force} +Force +

    + +

    Since \({\boldsymbol F}\) involves a derivative with respect to ordinary time, it does not transform well under Lorentz transformations. For the example of motion along \(\hat{\boldsymbol x}\), @@ -1704,11 +1717,25 @@ whereas the longitudinal component transforms in a complicated way: \] For the specific case where the particle is instantaneously at rest in the original frame, then +

    +
    +

    + + + + +

    +
    + +
    + +
    +

    \[ \bar{\boldsymbol F}_\perp = \frac{1}{\gamma} {\boldsymbol F}_\perp, \hspace{10mm} \bar{F}_\parallel = F_\parallel. - \label{eq:ForceTransfoSimple} +\tag{Ftr0}\label{Ftr0} \]

    @@ -1717,15 +1744,29 @@ The way to avoid complicated transformation rules is to define a four-vector as the derivative of momentum with respect to proper time, which leads to the definition of the

    -
    +
    +

    +Minkowski force +

    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf Minkowski force} \[ - {\boldsymbol K} = \frac{d{\boldsymbol p}}{d\tau} - = \left( \frac{dt}{d\tau} \right) \frac{d{\boldsymbol p}}{dt} - = \frac{1}{\sqrt{1 - u^2/c^2}} {\boldsymbol F} - \label{eq:MinkowskiForce} - \] +{\boldsymbol K} = \frac{d{\boldsymbol p}}{d\tau} += \left( \frac{dt}{d\tau} \right) \frac{d{\boldsymbol p}}{dt} += \frac{1}{\sqrt{1 - u^2/c^2}} {\boldsymbol F} +\tag{MinkF}\label{MinkF} +\]

    @@ -1754,7 +1795,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_rm_pt.html b/build/red_rm_pt.html index 415b9fe..bc1bbf7 100644 --- a/build/red_rm_pt.html +++ b/build/red_rm_pt.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1616,7 +1616,7 @@ Table of contents

    If you move at velocity \(u\), then, as compared to a ground clock, your time will go slower. For a ground clock time interval \(dt\), -your {\bf proper time} interval is +your proper time interval is \[ d\tau = \sqrt{1 - u^2/c^2}~dt. \] @@ -1625,16 +1625,16 @@ of your position \({\boldsymbol l}\) with respect to ground time: \[ {\boldsymbol u} = \frac{d {\boldsymbol l}}{dt} \] -and this is called the {\bf ordinary velocity}. -The {\bf proper velocity} is defined as the hybrid-frame quantity +and this is called the ordinary velocity. +The proper velocity is defined as the hybrid-frame quantity (distance as measured on the ground) divided by (proper time interval):

    -
    +

    -{\bf Proper velocity} +Proper velocity \[ - {\boldsymbol \eta} \equiv \frac{d {\boldsymbol l}}{d\tau} - \] +{\boldsymbol \eta} \equiv \frac{d {\boldsymbol l}}{d\tau} +\]

    @@ -1651,16 +1651,16 @@ By adding the zeroth component \] we can define the

    -
    +

    -{\bf four-velocity} or {\bf proper velocity four-vector} +four-velocity or proper velocity four-vector \[ - \eta^\mu \equiv \frac{dx^\mu}{d\tau} - \] +\eta^\mu \equiv \frac{dx^\mu}{d\tau} +\] which transforms as \[ - \bar{\eta}^\mu = \Lambda^\mu_\nu \eta^\nu - \] +\bar{\eta}^\mu = \Lambda^\mu_\nu \eta^\nu +\]

    @@ -1699,7 +1699,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_rm_rme.html b/build/red_rm_rme.html index 1247d0b..86659f9 100644 --- a/build/red_rm_rme.html +++ b/build/red_rm_rme.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1613,21 +1613,21 @@ Table of contents red.rm.rme
    -
    +

    -The {\bf relativistic momentum} \({\boldsymbol p}\) is defined as +The relativistic momentum \({\boldsymbol p}\) is defined as \[ - {\boldsymbol p} \equiv m {\boldsymbol \eta} = - \frac{m {\boldsymbol u}}{1 - u^2/c^2}. - \] -The {\bf relativistic energy} is defined as +{\boldsymbol p} \equiv m {\boldsymbol \eta} = +\frac{m {\boldsymbol u}}{1 - u^2/c^2}. +\] +The relativistic energy is defined as \[ - E \equiv \frac{m c^2}{\sqrt{1 - u^2/c^2}}. - \] -These can be combined into the {\bf energy-momentum four-vector} +E \equiv \frac{m c^2}{\sqrt{1 - u^2/c^2}}. +\] +These can be combined into the energy-momentum four-vector \[ - p^\mu \equiv m \eta^\mu. - \] +p^\mu \equiv m \eta^\mu. +\]

    @@ -1635,12 +1635,12 @@ These can be combined into the {\bf energy-momentum four-vector}

    When the object is stationary, its energy is the

    -
    +

    -{\bf Rest energy} +Rest energy \[ - E_{\mbox{\tiny rest}} \equiv m c^2. - \] +E_{\mbox{rest}} \equiv m c^2. +\]

    @@ -1648,12 +1648,12 @@ When the object is stationary, its energy is the When moving, the difference between relativistic and rest energies is the

    -
    +

    -{\bf Kinetic energy} +Kinetic energy \[ - E_{\mbox{\tiny kin}} \equiv E - mc^2 = mc^2 \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right). - \] +E_{\mbox{kin}} \equiv E - mc^2 = mc^2 \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right). +\]

    @@ -1661,25 +1661,25 @@ is the For velocities much smaller than the speed of light, we can expand this to \[ - E_{\mbox{\tiny kin}} = \frac{1}{2} mu^2 + \frac{3}{8} \frac{mu^2}{c^2} + ... + E_{\mbox{kin}} = \frac{1}{2} mu^2 + \frac{3}{8} \frac{mu^2}{c^2} + ... \]

    In a closed system,

    -
    +

    -{\bf Total relativistic energy and momentum is conserved} +Total relativistic energy and momentum is conserved \[ - E^2 - c^2 p^2 = m^2 c^4 - \] +E^2 - c^2 p^2 = m^2 c^4 +\]

    -N.B.: don't confuse an {\bf invariant} quantity with a {\bf conserved} quantity. +N.B.: don't confuse an invariant quantity with a conserved quantity.

    @@ -1702,7 +1702,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_sr.html b/build/red_sr.html index f5fb6f9..37f95b5 100644 --- a/build/red_sr.html +++ b/build/red_sr.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_sr_4v.html b/build/red_sr_4v.html index 4fb6045..f0e47c6 100644 --- a/build/red_sr_4v.html +++ b/build/red_sr_4v.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1614,24 +1614,24 @@ Table of contents red.sr.4v

    -\paragraph{Four-vectors.} Let's introduce the standard notations +Four-vectors: let's introduce the standard notations \[ x^0 \equiv ct, \hspace{10mm} \beta \equiv \frac{v}{c}, \hspace{10mm} x^1 = x, ~~x^2 = y, ~~x^3 = z. \] The Lorentz transformation then reads

    -
    +

    -{\bf Lorentz transformation (motion along \(x\) at velocity \(v\))} -\[ +Lorentz transformation (motion along \(x\) at velocity \(v\)) + \[ \bar{x}^0 = \gamma \left( x^0 - \beta x^1 \right), ~~~~\bar{x}^1 = \gamma \left( x^1 - \beta x^0 \right), ~~~~\bar{x}^2 = x^2, ~~~~\bar{x}^3 = x^3 \] -or in matrix form -\[ + or in matrix form + \[ \left( \begin{array}{c} \bar{x}^0 \\ \bar{x}^1 \\ \bar{x}^2 \\ \bar{x}^3 \end{array} \right) = \left( \begin{array}{cccc} @@ -1647,17 +1647,17 @@ or in matrix form

    This can be compactly written as \[ - \bar{x}^\mu = \sum_{\nu = 0}^3 \Lambda^\mu_\nu x^\nu. + \bar{x}^\mu = \sum_{\nu = 0}^3 \Lambda^\mu{}_\nu x^\nu. \]

    -\paragraph{Covariant and contravariant vectors.} Four-vectors with -upper index are called {\it contravariant}. Their lower-index -counterparts are called {\it covariant} vectors and are obtained by +Covariant and contravariant vectors: four-vectors with +upper index are called contravariant. Their lower-index +counterparts are called covariant vectors and are obtained by using the Minkowski metric \(g_{\mu \nu}\) according to

    -
    +

    \[ a_\mu = \sum_{\nu = 0}^3 g_{\mu \nu} a^\nu, \hspace{10mm} @@ -1671,9 +1671,9 @@ using the Minkowski metric \(g_{\mu \nu}\) according to

    -\paragraph{Scalar products} are defined as the in-product of covariant/contravariant four-vectors, +Scalar products are defined as the in-product of covariant/contravariant four-vectors,

    -
    +

    \[ \sum_{\mu = 0}^3 a^\mu b_\mu \equiv a^\mu b_\mu @@ -1683,7 +1683,7 @@ using the Minkowski metric \(g_{\mu \nu}\) according to

    where in the right-hand side we have introduced the -{\bf Einstein summation convention}, namely that any repeated index +Einstein summation convention, namely that any repeated index is implicitly summed over. As you can trivially check, it doesn't matter which vector is co/contravariant: \(a^\mu b_\mu = a_\mu b^\mu\). Scalar products are Lorentz-invariant and thus take the same value in @@ -1691,25 +1691,45 @@ all inertial systems.

    -\paragraph{Invariant intervals.} Generalizing the notion of the norm of +Invariant intervals: generalizing the notion of the norm of a vector, the scalar product of a four-vector with itself is known as the invariant interval. Because of the geometry of spacetime, the invariant can take positive or negative values. The nomenclature goes as follows:

    -\begin{center} - \begin{tabular}{cc} - $a^\mu a_\mu > 0$ & $a^\mu$ is {\it spacelike} \\ - $a^\mu a_\mu < 0$ & $a^\mu$ is {\it timelike} \\ - $a^\mu a_\mu = 0$ & $a^\mu$ is {\it lightlike} - \end{tabular} -\end{center} + + + + +++ ++ + + + + + + + + + + + + + + + + +
    \(a^\mu a_\mu > 0\)\(a^\mu\) is {\it spacelike}
    \(a^\mu a_\mu < 0\)\(a^\mu\) is {\it timelike}
    \(a^\mu a_\mu = 0\)\(a^\mu\) is {\it lightlike}
    +

    For two events \(A\) and \(B\), the difference \[ \Delta x^\mu \equiv x_A^\mu - x_B^\mu \] -is called the {\bf displacement four-vector} and its self-scalar product -is the {\bf invariant interval} between the two events: +is called the displacement four-vector and its self-scalar product +is the invariant interval between the two events: \[ I \equiv \Delta x^\mu \Delta x_\mu = -c^2 \Delta t^2 + |{\boldsymbol x}|^2 \] @@ -1718,12 +1738,12 @@ is their spatial separation vector.

    -\paragraph{Spacetime diagrams.} These are also know as -{\it Minkowski diagrams}. Time is on the vertical axis, space on the +Spacetime diagrams: these are also know as +Minkowski diagrams. Time is on the vertical axis, space on the horizontal one. The trajectory of a particle is known as its -{\bf world line}. Light is represented as propagating at lines at -45 degrees, defining the {\bf forward} and -{\bf backward light cones}. Lorentz transformations, which preserve +world line. Light is represented as propagating at lines at +45 degrees, defining the forward and +backward light cones. Lorentz transformations, which preserve all invariant intervals, move spacetime points around but leave them on the same hyperboloid.

    @@ -1746,7 +1766,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_sr_Lt.html b/build/red_sr_Lt.html index 64cb8ff..26ad2f6 100644 --- a/build/red_sr_Lt.html +++ b/build/red_sr_Lt.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1615,7 +1615,7 @@ Table of contents

    To talk about coordinate transformations, it is necessary to talk about -{\it events}, namely occurrences at a specific point in space and time. +events, namely occurrences at a specific point in space and time.

    @@ -1670,9 +1670,9 @@ where \(\bar{d}\) is the distance from \({\cal O}\) to \({\cal A}\) at time \] Solving these relations yields the dictionary for

    -
    +

    -{\bf Lorentz transformations (motion along \(x\) at velocity \(v\))} +Lorentz transformations (motion along \(x\) at velocity \(v\))

    \begin{align} \bar{t} &= \gamma \left( t - \frac{v}{c^2} x \right), & \bar{y} &= y, \nonumber\\ @@ -1682,7 +1682,7 @@ Solving these relations yields the dictionary for

    -\paragraph{Einstein's velocity addition rule.} Using these rules, +Einstein's velocity addition rule: using these rules, one can show that velocities add as \[ v_{13} = \frac{v_{12} + v_{23}}{1 + v_{12}v_{23}/c^2} @@ -1709,7 +1709,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/red_sr_p.html b/build/red_sr_p.html index e79fc70..cfc9abc 100644 --- a/build/red_sr_p.html +++ b/build/red_sr_p.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1310,10 +1310,6 @@ Table of contents @@ -1631,66 +1631,80 @@ This has a number of important consequences.

    -\paragraph{Relativity of simultaneity:} two events which are simultaneous in one reference frame, are not necessarily simultaneous in another one. +Relativity of simultaneity: two events which are simultaneous in one reference frame, are not necessarily simultaneous in another one.

    -\paragraph{Time dilation:} example of a light ray in a travelling train car. -For the observer inside the car: \(\Delta t_{\mbox{\tiny car}} = h/c\). +Time dilation: example of a light ray in a travelling train car. +For the observer inside the car: \(\Delta t_{\mbox{car}} = h/c\). For an observer on the ground, if the train is moving at velocity \(v\), -then \(\Delta t_{\mbox{\tiny gr}} = \sqrt{h^2 + v^2 \Delta t_{\mbox{\tiny gr}}^2}/c\) so +then \(\Delta t_{\mbox{gr}} = \sqrt{h^2 + v^2 \Delta t_{\mbox{gr}}^2}/c\) so \[ - \Delta t_{\mbox{\tiny gr}} = \frac{h}{c} \frac{1}{\sqrt{1 - v^2/c^2}} + \Delta t_{\mbox{gr}} = \frac{h}{c} \frac{1}{\sqrt{1 - v^2/c^2}} \] and we get \[ - \Delta t_{\mbox{\tiny tr}} = \sqrt{1 - v^2/c^2}~ \Delta t_{\mbox{\tiny gr}} + \Delta t_{\mbox{tr}} = \sqrt{1 - v^2/c^2}~ \Delta t_{\mbox{gr}} \] so the time interval in the train is shorter, namely there is a

    -
    +
    +

    +Time dilation factor +

    +
    +

    + + + + +

    +
    + +
    + +

    -{\bf Time dilation factor} \[ - \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} - \label{eq:Gamma} - \] +\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} +\tag{gamma}\label{gamma} +\]

    -\paragraph{Lorentz contraction:} lengths are also modified. +Lorentz contraction: lengths are also modified. Back to our train, with a mirror on one end. A light signal is sent from the opposite end, and the time for the round-trip of the light is measured. For the observer on the -train, the time is \(\Delta t_{\mbox{\tiny tr}} = 2 \Delta x_{\mbox{\tiny tr}}/c\) -with \(\Delta x_{\mbox{\tiny tr}}\) being the length of the train car. +train, the time is \(\Delta t_{\mbox{tr}} = 2 \Delta x_{\mbox{tr}}/c\) +with \(\Delta x_{\mbox{tr}}\) being the length of the train car. For the observer on the ground, the total time is made up of the back and forth journey of the light, with times \[ - \Delta t_{\mbox{\tiny gr,1}} = \frac{\Delta x_{\mbox{\tiny gr}} + v \Delta t_{\mbox{\tiny gr,1}}}{c}, \hspace{10mm} - \Delta t_{\mbox{\tiny gr,2}} = \frac{\Delta x_{\mbox{\tiny gr}} - v \Delta t_{\mbox{\tiny gr,2}}}{c} + \Delta t_{\mbox{gr,1}} = \frac{\Delta x_{\mbox{gr}} + v \Delta t_{\mbox{gr,1}}}{c}, \hspace{10mm} + \Delta t_{\mbox{gr,2}} = \frac{\Delta x_{\mbox{gr}} - v \Delta t_{\mbox{gr,2}}}{c} \] so \[ - \Delta t_{\mbox{\tiny gr,1}} = \frac{\Delta x_{\mbox{\tiny gr}}}{c-v}, \hspace{10mm} - \Delta t_{\mbox{\tiny gr,2}} = \frac{\Delta x_{\mbox{\tiny gr}}}{c+v} + \Delta t_{\mbox{gr,1}} = \frac{\Delta x_{\mbox{gr}}}{c-v}, \hspace{10mm} + \Delta t_{\mbox{gr,2}} = \frac{\Delta x_{\mbox{gr}}}{c+v} \] and thus \[ - \Delta t_{\mbox{\tiny gr}} = - \Delta t_{\mbox{\tiny gr,1}} + \Delta t_{\mbox{\tiny gr,2}} - = \frac{2 \Delta x_{\mbox{\tiny gr}}}{c} \frac{1}{1 - v^2/c^2}. + \Delta t_{\mbox{gr}} = + \Delta t_{\mbox{gr,1}} + \Delta t_{\mbox{gr,2}} + = \frac{2 \Delta x_{\mbox{gr}}}{c} \frac{1}{1 - v^2/c^2}. \] Using the time dilation relation then gives

    -
    +

    -{\bf Lorentz contraction} +Lorentz contraction \[ - \Delta x_{\mbox{\tiny tr}} = \frac{1}{\sqrt{1 - v^2/c^2}} \Delta x_{\mbox{\tiny gr}} - \] +\Delta x_{\mbox{tr}} = \frac{1}{\sqrt{1 - v^2/c^2}} \Delta x_{\mbox{gr}} +\]

    @@ -1718,7 +1732,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-03-07 Mon 20:38

    +

    Created: 2022-03-15 Tue 08:10

    diff --git a/build/style.css b/build/style.css index a05e53f..7d14731 100644 --- a/build/style.css +++ b/build/style.css @@ -256,6 +256,8 @@ ul.altsecnrs > li:not(:first-child)::before { div.eqlabel { float: right; clear: both; + position: relative; + z-index: 10; } div.eqlabel p { margin: 0;