From 3c40f5bfe86108d122a30039e08edd8d92c349cf Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jean-S=C3=A9bastien?= Date: Wed, 9 Feb 2022 07:44:58 +0100 Subject: [PATCH] Update 2022-02-09 07:44 --- build/a.html | 21 +++++++++++---- build/a_l.html | 21 +++++++++++---- build/c.html | 21 +++++++++++---- build/c_m.html | 21 +++++++++++---- build/c_m_cs.html | 21 +++++++++++---- build/c_m_cs_cyl.html | 33 +++++++++++++++-------- build/c_m_cs_hyp.html | 21 +++++++++++---- build/c_m_cs_sph.html | 21 +++++++++++---- build/c_m_dc.html | 21 +++++++++++---- build/c_m_dc_curl.html | 21 +++++++++++---- build/c_m_dc_d2.html | 51 ++++++++++++++++++++++-------------- build/c_m_dc_del.html | 21 +++++++++++---- build/c_m_dc_div.html | 21 +++++++++++---- build/c_m_dc_g.html | 21 +++++++++++---- build/c_m_dc_pr.html | 21 +++++++++++---- build/c_m_dd.html | 21 +++++++++++---- build/c_m_dd_1d.html | 21 +++++++++++---- build/c_m_dd_3d.html | 25 +++++++++++++----- build/c_m_dd_div.html | 21 +++++++++++---- build/c_m_ic.html | 21 +++++++++++---- build/c_m_ic_ftc.html | 21 +++++++++++---- build/c_m_ic_ftg.html | 21 +++++++++++---- build/c_m_ic_gauss.html | 21 +++++++++++---- build/c_m_ic_ip.html | 21 +++++++++++---- build/c_m_ic_lsv.html | 39 +++++++++++++++++---------- build/c_m_ic_stokes.html | 25 +++++++++++++----- build/c_m_uf.html | 21 +++++++++++---- build/c_m_uf_cyl.html | 21 +++++++++++---- build/c_m_uf_sph.html | 21 +++++++++++---- build/c_m_uf_vi.html | 21 +++++++++++---- build/c_m_va.html | 21 +++++++++++---- build/c_m_va_cp.html | 21 +++++++++++---- build/c_m_va_n.html | 21 +++++++++++---- build/c_m_va_pds.html | 21 +++++++++++---- build/c_m_va_sp.html | 21 +++++++++++---- build/c_m_va_tp.html | 21 +++++++++++---- build/c_m_vf.html | 21 +++++++++++---- build/c_m_vf_helm.html | 21 +++++++++++---- build/c_m_vf_pot.html | 21 +++++++++++---- build/d.html | 25 +++++++++++++----- build/d_emd.html | 21 +++++++++++---- build/d_emd_ce.html | 21 +++++++++++---- build/d_emd_emw.html | 21 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+++++++++++---- build/emsm_msm.html | 29 +++++++++++++------- build/emsm_msm_H.html | 21 +++++++++++---- build/emsm_msm_H_A.html | 29 +++++++++++++------- build/emsm_msm_a.html | 21 +++++++++++---- build/emsm_msm_fmo.html | 21 +++++++++++---- build/emsm_msm_fmo_bc.html | 31 +++++++++++++++------- build/emsm_msm_fmo_fim.html | 21 +++++++++++---- build/emsm_msm_fmo_pibc.html | 21 +++++++++++---- build/emsm_msm_lnlm.html | 21 +++++++++++---- build/emsm_msm_lnlm_fm.html | 21 +++++++++++---- build/emsm_msm_lnlm_sp.html | 23 +++++++++++----- build/emsm_msm_m.html | 21 +++++++++++---- build/emsm_msm_m_dpf.html | 25 +++++++++++++----- build/emsm_msm_m_fdi.html | 25 +++++++++++++----- build/in.html | 19 +++++++++++--- build/in_p.html | 21 +++++++++++---- build/in_t.html | 21 +++++++++++---- build/in_t_c.html | 49 ++++++++++++++++++++-------------- build/in_t_l.html | 21 +++++++++++---- build/index.html | 19 +++++++++++--- build/qed.html | 21 +++++++++++---- build/qed_t.html | 21 +++++++++++---- build/red.html | 29 +++++++++++++------- build/red_rem.html | 21 +++++++++++---- build/red_rem_Fmunu.html | 27 +++++++++++++------ build/red_rem_Ltf.html | 23 +++++++++++----- build/red_rem_Me.html | 33 +++++++++++++++-------- build/red_rem_mre.html | 23 +++++++++++----- build/red_rm.html | 21 +++++++++++---- build/red_rm_Mf.html | 27 +++++++++++++------ build/red_rm_pt.html | 25 +++++++++++++----- build/red_rm_rme.html | 29 +++++++++++++------- build/red_sr.html | 21 +++++++++++---- build/red_sr_4v.html | 27 +++++++++++++------ build/red_sr_Lt.html | 23 +++++++++++----- build/red_sr_p.html | 25 +++++++++++++----- build/style.css | 14 +++++++--- 209 files changed, 3663 insertions(+), 1367 deletions(-) diff --git a/build/a.html b/build/a.html index a1c9672..3ae8e8f 100644 --- a/build/a.html +++ b/build/a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents - +

Appendices @@ -1631,10 +1631,21 @@ Table of contents
  • Literaturea.l
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    diff --git a/build/a_l.html b/build/a_l.html index 46771ad..1465dae 100644 --- a/build/a_l.html +++ b/build/a_l.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents -
    + -
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    diff --git a/build/c.html b/build/c.html index 3f1a057..2151581 100644 --- a/build/c.html +++ b/build/c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents - +
    • Compendium

    Compendium @@ -1631,10 +1631,21 @@ Table of contents
  • Mathematicsc.m
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    +
    + +Creative Commons License + +Except where otherwise noted, all content is licensed under a +Creative Commons Attribution 4.0 International License. +
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    diff --git a/build/c_m.html b/build/c_m.html index 1e87a4c..f129529 100644 --- a/build/c_m.html +++ b/build/c_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents - +

    Mathematics @@ -1637,10 +1637,21 @@ Table of contents
  • Useful Formulasc.m.uf
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    +
    + +Creative Commons License + +Except where otherwise noted, all content is licensed under a +Creative Commons Attribution 4.0 International License. +
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    diff --git a/build/c_m_cs.html b/build/c_m_cs.html index c1751dd..d231674 100644 --- a/build/c_m_cs.html +++ b/build/c_m_cs.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents - +

    Coordinate Systems @@ -1633,10 +1633,21 @@ Table of contents
  • Hyperbolic Coordinatesc.m.cs.hyp
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    +
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    diff --git a/build/c_m_cs_cyl.html b/build/c_m_cs_cyl.html index ac082c7..5742373 100644 --- a/build/c_m_cs_cyl.html +++ b/build/c_m_cs_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents -
    +
    Cylindrical Coordinates @@ -1668,14 +1668,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
    Gradient
    -
    +

    -
    +
    • Gr4(1.79)
    @@ -1696,14 +1696,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
    Divergence
    -
    +

    -
    +
    • Gr4(2.21)
    @@ -1724,14 +1724,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
    Curl
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    +

    -
    +
    • Gr4(2.21)
    @@ -1765,10 +1765,21 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in -
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    diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html index c854513..c06e092 100644 --- a/build/c_m_cs_sph.html +++ b/build/c_m_cs_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents -
    +
    Spherical Coordinates @@ -1739,10 +1739,21 @@ Infinitesimal surface element: depends on situation. -
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    +
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    Differential Calculus @@ -1636,10 +1636,21 @@ Table of contents
  • Second Derivativesc.m.dc.d2
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    diff --git a/build/c_m_dc_curl.html b/build/c_m_dc_curl.html index fb804be..0fb3e77 100644 --- a/build/c_m_dc_curl.html +++ b/build/c_m_dc_curl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents -
    +
    The Curl @@ -1638,10 +1638,21 @@ v_x & v_y & v_z \end{array} \right| \nonumber \\ -
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    diff --git a/build/c_m_dc_d2.html b/build/c_m_dc_d2.html index f286af4..b9024f9 100644 --- a/build/c_m_dc_d2.html +++ b/build/c_m_dc_d2.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents -
    +
    Second Derivatives @@ -1624,9 +1624,9 @@ 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 @@ -1635,36 +1635,36 @@ given by the Laplacian of the corresponding vector elements.

    -
    -
    Curl of a gradient
    -
    +
    +
    Curl of a gradient
    +

    This always vanishes.

    -
    -
    Gradient of the divergence
    -
    +
    +
    Gradient of the divergence
    +

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

    -
    -
    Divergence of a curl
    -
    +
    +
    Divergence of a curl
    +

    This always vanishes.

    -
    -
    Curl of curl
    -
    +
    +
    Curl of curl
    +

    \[ {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v} @@ -1676,10 +1676,21 @@ This always vanishes.

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    +
    The \({\boldsymbol \nabla}\) Operator @@ -1637,10 +1637,21 @@ The del/grad/nabla operator is defined as -
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    +
    The Divergence @@ -1634,10 +1634,21 @@ Table of contents -
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    diff --git a/build/c_m_dc_g.html b/build/c_m_dc_g.html index 32a09f6..8028591 100644 --- a/build/c_m_dc_g.html +++ b/build/c_m_dc_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents -
    +
    Gradient @@ -1658,10 +1658,21 @@ is a vector called the gradient of \(T\). -
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    +
    Product Rules @@ -1631,10 +1631,21 @@ Six product rules (on inside front cover of Gr). -
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    Dirac delta Distribution @@ -1633,10 +1633,21 @@ Table of contents
  • The Three-Dimensional Delta Functionc.m.dd.3d
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    \[ {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l} @@ -1655,9 +1655,9 @@ Integral over a closed loop:

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    \[ \int_{\cal S} {\bf v} \cdot d{\bf a} @@ -1677,9 +1677,9 @@ Over a closed surface:

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    Volume Integrals
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    Vector Fields @@ -1632,10 +1632,21 @@ Table of contents
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    Diagnostics: Electromagnetodynamics @@ -1653,10 +1653,21 @@ As a strict minimum, you should be able to: -
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    Diagnostics: Electromagnetostatics @@ -1670,10 +1670,21 @@ As a strict minimum, you should be able to: -
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    Diagnostics: Magnetostatics in Matter @@ -1651,10 +1651,21 @@ As a strict minimum, you should be able to: -
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    Diagnostics: Mathematical Preliminaries @@ -1661,10 +1661,21 @@ Things you should be able to do (ideally: from scratch, on a blank sheet of pape -
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    • Electromagnetodynamics

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    Induction: Faraday's Law @@ -1634,10 +1634,21 @@ Table of contents
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    Energy in Magnetic Fields @@ -1682,7 +1682,7 @@ 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

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    \[ W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2 @@ -1703,7 +1703,7 @@ W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int \hspace{2cm} \mbox{(7.31 and 7.34)} \end{align} -

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    \paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\). Find energy stored in section of length \(l\). @@ -1727,10 +1727,21 @@ Note: gives easy way to find inductance, since \(W = \frac{1}{2} L I^2\).

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    Inductance @@ -1662,7 +1662,7 @@ M_{12} = M_{21} \]

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    \paragraph{Example 7.10:} short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside @@ -1712,7 +1712,7 @@ Inductance: measured in {\bf henries} (\(H\)). \(H = V s/A\).

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    \paragraph{Example 7.11:} find self-inductance of toroidal coil with rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\)) @@ -1739,7 +1739,7 @@ Total flux: \(N\) times this, so self-inductance is Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of {\bf back EMF}.

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    \paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\). What is the current ? @@ -1763,10 +1763,21 @@ where \(\tau \equiv L/R\) is the {\bf time constant} of the circuit. -

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    The Induced Electric Field @@ -1646,7 +1646,7 @@ law in integral form: -
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    {\bf Example 7.7:} \({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ? @@ -1662,7 +1662,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.

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    {\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim. Uniform magnetic field \({\bf B}_0\) in central region up to \(a < b\), @@ -1696,7 +1696,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with 'slow enough' phenomena.

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    {\bf Example 7.9:} infinitely long straight wire carries \(I(t)\). Find induced \({\bf E}\) field as a function of distance \(s\) from wire. @@ -1729,10 +1729,21 @@ Reason: in this case, we've overstepped the quasistatic limit. We need -

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    Maxwell's Equations @@ -1634,10 +1634,21 @@ Table of contents
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    Maxwell's Equations @@ -1625,7 +1625,7 @@ Table of contents

    Full set of equations for the electromagnetic field:

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    {\bf Maxwell's equations} {\it (in vacuum)}

    @@ -1641,7 +1641,7 @@ Full set of equations for the electromagnetic field:

    Complement:

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    {\bf Force law} \[ @@ -1665,7 +1665,7 @@ take divergence of \((iv)\).

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

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    \begin{align} (i) &{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, &(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\ @@ -1680,10 +1680,21 @@ Better way of writing: all fields on left, all sources on right, -
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    Maxwell's Correction to Ampère's Law; the Displacement Current @@ -1632,7 +1632,7 @@ the continuity equation as \] The extra term would thus be eliminated if we were to put

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    \[ {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} @@ -1656,7 +1656,7 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation

    Maxwell baptized this term the

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    {\bf Displacement current} \[ @@ -1689,10 +1689,21 @@ Flat surface: OK, \(E = 0\) and \(I_{\mbox{enc}} = I\). Balloon surface: \(I -

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    Electrodynamics Before Maxwell @@ -1659,10 +1659,21 @@ So for non-steady currents, the 'current enclosed by a loop' is ill-defined. -
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    Charge and Energy Flows emd.ce

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    Angular Momentum @@ -1625,7 +1625,7 @@ Table of contents

    The angular momentum of EM fields is directly given by

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    {\bf Angular momentum of EM fields} \[ @@ -1640,10 +1640,21 @@ The angular momentum of EM fields is directly given by

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    The Continuity Equation @@ -1644,7 +1644,7 @@ This means that \] Since this is true for any volume, we have (re)derived the

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    {\bf Continuity equation} \[ @@ -1670,10 +1670,21 @@ imposes a functional constraint on these sources: not {\it any} \(\rho\) and -

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    Momentum @@ -1637,7 +1637,7 @@ 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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    {\bf Momentum density in the EM fields} \[ @@ -1649,7 +1649,7 @@ 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

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    {\bf Continuity equation for EM momentum} \[ @@ -1663,10 +1663,21 @@ In a region in which the mechanical momentum is not changing due to external inf -

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    Maxwell's Stress Tensor @@ -1679,7 +1679,7 @@ and similarly for \({\boldsymbol B}\). We thus get

    This expression can be greatly simplified by introducing the

    -
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    {\bf Maxwell stress tensor} \[ @@ -1702,7 +1702,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction

    We then obtain

    -
    +

    {\bf EM force per unit volume} \[ @@ -1714,7 +1714,7 @@ We then obtain

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

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    {\bf Total force on charges in volume} \[ @@ -1728,10 +1728,21 @@ where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the -

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    Poynting's Theorem; the Poynting Vector @@ -1691,7 +1691,7 @@ so we get Substituting this in \ref{Gr(8.6)} and using the divergence theorem, we obtain

    -
    +

    {\bf Poynting's theorem} \[ @@ -1716,7 +1716,7 @@ 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:

    -
    +

    {\bf Poynting vector} \[ @@ -1729,7 +1729,7 @@ Energy per unit time, per unit area carried by EM fields:

    We can thus express Poynting's theorem more compactly:

    -
    +

    {\bf Poynting's theorem} \[ @@ -1742,7 +1742,7 @@ We can thus express Poynting's theorem more compactly:

    where we have defined the total

    -
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    {\bf Energy in electromagnetic fields} \[ @@ -1765,7 +1765,7 @@ Then, \] so we get the

    -
    +

    {\bf Poynting theorem (differential form)} \[ @@ -1782,7 +1782,7 @@ and has a similar for to the continuity equation -

    +

    \paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting. Assuming that the field is uniform, the electric field parallel to the wire is @@ -1812,10 +1812,21 @@ and the value is as expected. -

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    Electromagnetic waves in vacuum emd.emw

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    Energy and Momentum @@ -1653,7 +1653,7 @@ so for a monochromatic EM plan wave, \] or more succinctly:

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    {\bf Poynting vector of a monochromatic EM wave} \[ @@ -1669,7 +1669,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w

    Similary, we get the

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    {\bf Momentum density of a monochromatic EM wave} \[ @@ -1705,10 +1705,21 @@ The {\it radiation pressure} is the momentum transfer per unit area per unit of

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    Monochromatic Plane Waves @@ -1654,7 +1654,7 @@ 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

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    {\bf E and B fields for a monochromatic EM plane wave} \[ @@ -1683,10 +1683,21 @@ or if you prefer explicit real parts (adding a possible phase shift \(\delta\)): -

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    The Wave Equation @@ -1651,7 +1651,7 @@ 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

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    {\bf Wave equations for electric and magnetic fields in vacuum} \[ @@ -1692,10 +1692,21 @@ the actual electric and magnetic fields are given by the real part. -

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    diff --git a/build/emdm.html b/build/emdm.html index 0b6b77c..532e030 100644 --- a/build/emdm.html +++ b/build/emdm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents - +
    • Electromagnetodynamics in Matter

    Electromagnetodynamics in Matter @@ -1632,10 +1632,21 @@ Table of contents
  • Electromagnetic Waves in Matteremdm.emwm
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    Author: Jean-Sébastien Caux

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    diff --git a/build/emdm_Me.html b/build/emdm_Me.html index 37b1302..3cb351b 100644 --- a/build/emdm_Me.html +++ b/build/emdm_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents - +

    Maxwell's Equations in Matter @@ -1632,10 +1632,21 @@ Table of contents
  • Boundary Conditionsemdm.Me.bc
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    diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html index c187471..99df269 100644 --- a/build/emdm_Me_Mem.html +++ b/build/emdm_Me_Mem.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ Table of contents -
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    Maxwell's Equations in Matter @@ -1658,7 +1658,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia \] We therefore have the

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    {\bf Polarization current density} \[ @@ -1676,7 +1676,7 @@ 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:

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