diff --git a/build/a.html b/build/a.html index 621c5b5..ccb378f 100644 --- a/build/a.html +++ b/build/a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1606,6 +1606,8 @@ Table of contents
  • Literaturea.l
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/a_l.html b/build/a_l.html index 4894259..d7f6d38 100644 --- a/build/a_l.html +++ b/build/a_l.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1689,6 +1689,8 @@ Wiley, 1986 [2nd edition] +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c.html b/build/c.html index 78f9a0e..de376c6 100644 --- a/build/c.html +++ b/build/c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1606,6 +1606,8 @@ Table of contents
  • Mathematicsc.m
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m.html b/build/c_m.html index c5fa77d..023a9eb 100644 --- a/build/c_m.html +++ b/build/c_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1612,6 +1612,8 @@ Table of contents
  • Useful Formulasc.m.uf
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_cs.html b/build/c_m_cs.html index 3f874a6..bb877f0 100644 --- a/build/c_m_cs.html +++ b/build/c_m_cs.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1608,6 +1608,8 @@ Table of contents
  • Hyperbolic Coordinatesc.m.cs.hyp
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_cs_cyl.html b/build/c_m_cs_cyl.html index ca99cbc..ada05cb 100644 --- a/build/c_m_cs_cyl.html +++ b/build/c_m_cs_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1643,14 +1643,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
    Gradient
    -
    +

    -
    +
    • Gr4(1.79)
    @@ -1671,14 +1671,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
    Divergence
    -
    +

    -
    +
    • Gr4(2.21)
    @@ -1699,14 +1699,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
    Curl
    -
    +

    -
    +
    • Gr4(2.21)
    @@ -1740,6 +1740,8 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_cs_hyp.html b/build/c_m_cs_hyp.html index eff6a67..3a11a4b 100644 --- a/build/c_m_cs_hyp.html +++ b/build/c_m_cs_hyp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1601,6 +1601,8 @@ Table of contents
    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html index f17b837..9d537d8 100644 --- a/build/c_m_cs_sph.html +++ b/build/c_m_cs_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1714,6 +1714,8 @@ Infinitesimal surface element: depends on situation. +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dc.html b/build/c_m_dc.html index 8110bf0..1dfbb33 100644 --- a/build/c_m_dc.html +++ b/build/c_m_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1611,6 +1611,8 @@ Table of contents
  • Second Derivativesc.m.dc.d2
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dc_curl.html b/build/c_m_dc_curl.html index b769578..b41e9d7 100644 --- a/build/c_m_dc_curl.html +++ b/build/c_m_dc_curl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1613,6 +1613,8 @@ v_x & v_y & v_z \end{array} \right| \nonumber \\ +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dc_d2.html b/build/c_m_dc_d2.html index b405947..b8d19c3 100644 --- a/build/c_m_dc_d2.html +++ b/build/c_m_dc_d2.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1599,9 +1599,9 @@ 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 @@ -1610,36 +1610,36 @@ 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} @@ -1651,6 +1651,8 @@ This always vanishes.

    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dc_del.html b/build/c_m_dc_del.html index 601c287..7e1a7c2 100644 --- a/build/c_m_dc_del.html +++ b/build/c_m_dc_del.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1612,6 +1612,8 @@ The del/grad/nabla operator is defined as +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dc_div.html b/build/c_m_dc_div.html index ad4c6b8..c3d8f89 100644 --- a/build/c_m_dc_div.html +++ b/build/c_m_dc_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1609,6 +1609,8 @@ Table of contents +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dc_g.html b/build/c_m_dc_g.html index 42bfd3b..2525122 100644 --- a/build/c_m_dc_g.html +++ b/build/c_m_dc_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1633,6 +1633,8 @@ is a vector called the gradient of \(T\). +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dc_pr.html b/build/c_m_dc_pr.html index 7b44fca..62fc37f 100644 --- a/build/c_m_dc_pr.html +++ b/build/c_m_dc_pr.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1606,6 +1606,8 @@ Six product rules (on inside front cover of Gr). +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dd.html b/build/c_m_dd.html index 9539924..c943655 100644 --- a/build/c_m_dd.html +++ b/build/c_m_dd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1608,6 +1608,8 @@ Table of contents
  • The Three-Dimensional Delta Functionc.m.dd.3d
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dd_1d.html b/build/c_m_dd_1d.html index 75c7d73..b1f39ed 100644 --- a/build/c_m_dd_1d.html +++ b/build/c_m_dd_1d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1620,6 +1620,8 @@ Consequences: for any smooth differentiable function \(f(x)\), +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dd_3d.html b/build/c_m_dd_3d.html index 72749d4..99cd00a 100644 --- a/build/c_m_dd_3d.html +++ b/build/c_m_dd_3d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1620,14 +1620,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox: More generally,

    -
    +

    -
    +
    • Gr (1.100)
    @@ -1646,14 +1646,14 @@ More generally, Since

    -
    +

    -
    +
    • Gr (1.101)
    @@ -1669,15 +1669,32 @@ Since

    we have that

    +
    +

    + + + + +

    +
    +
      +
    • Gr (1.102)
      • +
    + +
    + +
    \begin{equation} {\boldsymbol \nabla}^2 \left( \frac{1}{|{\bf r} - {\bf r}'|} \right) = -4\pi \delta^{(3)} ({\bf r} - {\bf r}') -\label{Gr(1.102)} +\tag{Lap1or}\label{Lap1or} \end{equation}
    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_dd_div.html b/build/c_m_dd_div.html index 086b891..7835dc3 100644 --- a/build/c_m_dd_div.html +++ b/build/c_m_dd_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1626,6 +1626,8 @@ Problem: in Gr(1.84), we've divided by zero when \(r = 0\). +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_ic.html b/build/c_m_ic.html index 6765a9f..c67e307 100644 --- a/build/c_m_ic.html +++ b/build/c_m_ic.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1611,6 +1611,8 @@ Table of contents
  • Integration by Partsc.m.ic.ip
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_ic_ftc.html b/build/c_m_ic_ftc.html index 348de4b..e55664f 100644 --- a/build/c_m_ic_ftc.html +++ b/build/c_m_ic_ftc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1609,6 +1609,8 @@ Table of contents +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_ic_ftg.html b/build/c_m_ic_ftg.html index f94a53e..b8e64d2 100644 --- a/build/c_m_ic_ftg.html +++ b/build/c_m_ic_ftg.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1618,6 +1618,8 @@ Table of contents +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_ic_gauss.html b/build/c_m_ic_gauss.html index 398e609..07747cc 100644 --- a/build/c_m_ic_gauss.html +++ b/build/c_m_ic_gauss.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1612,6 +1612,8 @@ This is know either as Gauss' theorem, Green's theorem or the d +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_ic_ip.html b/build/c_m_ic_ip.html index d335992..8e42591 100644 --- a/build/c_m_ic_ip.html +++ b/build/c_m_ic_ip.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1619,6 +1619,8 @@ or in other words
    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_ic_lsv.html b/build/c_m_ic_lsv.html index 6a82b7f..3672e9f 100644 --- a/build/c_m_ic_lsv.html +++ b/build/c_m_ic_lsv.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1599,9 +1599,9 @@ Table of contents
    -
    -
    Line Integrals
    -
    +
    +
    Line Integrals
    +

    \[ {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l} @@ -1630,9 +1630,9 @@ Integral over a closed loop:

    -
    -
    Surface Integrals
    -
    +
    +
    Surface Integrals
    +

    \[ \int_{\cal S} {\bf v} \cdot d{\bf a} @@ -1652,9 +1652,9 @@ Over a closed surface:

    -
    -
    Volume Integrals
    -
    +
    +
    Volume Integrals
    +

    \[ \int_{\cal V} T d\tau @@ -1680,6 +1680,8 @@ d\tau = dx dy dz +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_ic_stokes.html b/build/c_m_ic_stokes.html index 636a606..7231030 100644 --- a/build/c_m_ic_stokes.html +++ b/build/c_m_ic_stokes.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1597,14 +1597,14 @@ Table of contents c.m.ic.stokes
    -
    +

    -
    +
    • Gr (1.57)
    @@ -1634,6 +1634,8 @@ the boundary shrinks to a point. +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_uf.html b/build/c_m_uf.html index 98dfc70..b476e42 100644 --- a/build/c_m_uf.html +++ b/build/c_m_uf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1608,6 +1608,8 @@ Table of contents
  • Vector identitiesc.m.uf.vi
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_uf_cyl.html b/build/c_m_uf_cyl.html index 9ae6d3d..354f254 100644 --- a/build/c_m_uf_cyl.html +++ b/build/c_m_uf_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,6 +1621,8 @@ Table of contents +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_uf_sph.html b/build/c_m_uf_sph.html index ae2626d..a6fcb67 100644 --- a/build/c_m_uf_sph.html +++ b/build/c_m_uf_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,6 +1621,8 @@ Table of contents +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_uf_vi.html b/build/c_m_uf_vi.html index 427c2af..54535ce 100644 --- a/build/c_m_uf_vi.html +++ b/build/c_m_uf_vi.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,6 +1624,8 @@ Table of contents
    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_va.html b/build/c_m_va.html index dcadc72..34c6b66 100644 --- a/build/c_m_va.html +++ b/build/c_m_va.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1610,6 +1610,8 @@ Table of contents
  • Position, Displacement and Separation Vectorsc.m.va.pds
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_va_cp.html b/build/c_m_va_cp.html index 19dd05c..0363996 100644 --- a/build/c_m_va_cp.html +++ b/build/c_m_va_cp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1639,6 +1639,8 @@ this relation making plain that +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_va_n.html b/build/c_m_va_n.html index fd411dc..3043320 100644 --- a/build/c_m_va_n.html +++ b/build/c_m_va_n.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1634,6 +1634,8 @@ Component notation (in \({\mathbb R}^3\) with cartesian coordinates): +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_va_pds.html b/build/c_m_va_pds.html index b3018c3..0e225dc 100644 --- a/build/c_m_va_pds.html +++ b/build/c_m_va_pds.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1668,6 +1668,8 @@ Separation vector: for two position vectors \({\bf r}_a, {\bf r}_b\),
    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_va_sp.html b/build/c_m_va_sp.html index 10de89b..ef27cba 100644 --- a/build/c_m_va_sp.html +++ b/build/c_m_va_sp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1634,6 +1634,8 @@ In a general coordinate system with metric \(g\), +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_va_tp.html b/build/c_m_va_tp.html index 71c53b8..f4dd625 100644 --- a/build/c_m_va_tp.html +++ b/build/c_m_va_tp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1681,6 +1681,8 @@ All higher vector products can be reduced to combinations of single vector produ +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_vf.html b/build/c_m_vf.html index 0bcd5d3..185e94c 100644 --- a/build/c_m_vf.html +++ b/build/c_m_vf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1607,6 +1607,8 @@ Table of contents
  • Potentialsc.m.vf.pot
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_vf_helm.html b/build/c_m_vf_helm.html index c5cb70c..7f93a9e 100644 --- a/build/c_m_vf_helm.html +++ b/build/c_m_vf_helm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1626,6 +1626,8 @@ where +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/c_m_vf_pot.html b/build/c_m_vf_pot.html index ff702df..1164164 100644 --- a/build/c_m_vf_pot.html +++ b/build/c_m_vf_pot.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1653,6 +1653,8 @@ The following conditions are equivalent:
    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d.html b/build/d.html index 2052fab..4d2e6c9 100644 --- a/build/d.html +++ b/build/d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,8 +1598,8 @@ Table of contents d
    -
    - +
    + Objectives @@ -1648,6 +1648,8 @@ I have little doubt you'll pass the course.
  • Diagnostics: Relativistic Electrodynamicsd.red
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_emd.html b/build/d_emd.html index 8f9ebad..43556b5 100644 --- a/build/d_emd.html +++ b/build/d_emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1628,6 +1628,8 @@ As a strict minimum, you should be able to: +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_emd_ce.html b/build/d_emd_ce.html index ea8e504..d7e9225 100644 --- a/build/d_emd_ce.html +++ b/build/d_emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1623,6 +1623,8 @@ As a strict minimum, you should be able to: +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_emd_emw.html b/build/d_emd_emw.html index a07c838..5bbfe2a 100644 --- a/build/d_emd_emw.html +++ b/build/d_emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1626,6 +1626,8 @@ As a strict minimum, you should be able to: +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_emf.html b/build/d_emf.html index 90c8676..772b1aa 100644 --- a/build/d_emf.html +++ b/build/d_emf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1623,6 +1623,8 @@ As a strict minimum, you should be able to: +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_ems.html b/build/d_ems.html index 0cebd8f..70566ff 100644 --- a/build/d_ems.html +++ b/build/d_ems.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1645,6 +1645,8 @@ As a strict minimum, you should be able to: +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_ems_ca.html b/build/d_ems_ca.html index 82e0e8c..f6f78ed 100644 --- a/build/d_ems_ca.html +++ b/build/d_ems_ca.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1628,6 +1628,8 @@ a point source charge \(q\) at \({\bf r}_s\) with an infinite grounded conductin +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_ems_ms.html b/build/d_ems_ms.html index 4670f93..160828f 100644 --- a/build/d_ems_ms.html +++ b/build/d_ems_ms.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1625,6 +1625,8 @@ As a strict minimum, you should be able to: +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_emsm.html b/build/d_emsm.html index 0185289..1faaf48 100644 --- a/build/d_emsm.html +++ b/build/d_emsm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1628,6 +1628,8 @@ As a strict minimum, you should be able to: +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_emsm_msm.html b/build/d_emsm_msm.html index efc93f1..822a4af 100644 --- a/build/d_emsm_msm.html +++ b/build/d_emsm_msm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1626,6 +1626,8 @@ As a strict minimum, you should be able to: +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_m.html b/build/d_m.html index 5167f4c..f7f2770 100644 --- a/build/d_m.html +++ b/build/d_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1636,6 +1636,8 @@ Things you should be able to do (ideally: from scratch, on a blank sheet of pape +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/d_red.html b/build/d_red.html index b80a386..5dffa11 100644 --- a/build/d_red.html +++ b/build/d_red.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1630,6 +1630,8 @@ As a strict minimum, you should be able to:
    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd.html b/build/emd.html index eef2d71..0486941 100644 --- a/build/emd.html +++ b/build/emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,8 +1598,8 @@ Table of contents emd
    -
    - +
    + Prerequisites
      @@ -1608,8 +1608,8 @@ Prerequisites
    -
    - +
    + Objectives +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_Fl.html b/build/emd_Fl.html index 1403300..3792472 100644 --- a/build/emd_Fl.html +++ b/build/emd_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1609,6 +1609,8 @@ Table of contents
  • Energy in Magnetic Fieldsemd.Fl.e
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_Fl_Fl.html b/build/emd_Fl_Fl.html index c386afc..4b295c1 100644 --- a/build/emd_Fl_Fl.html +++ b/build/emd_Fl_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,7 +1638,7 @@ 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), \[ @@ -1660,7 +1660,7 @@ to the rate of change of the magnetic flux, \] so we obtain

    -
    +

    Faraday's law (integral form N.B.: for a stationary loop) \[ @@ -1678,7 +1678,7 @@ for any loop (on a wire or not). Using Stokes' theorem, \] we obtain

    -
    +

    Faraday's law (differential form) \[ @@ -1700,6 +1700,8 @@ to an opposing counter-reaction. +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_Fl_e.html b/build/emd_Fl_e.html index dbb1b2f..a4cd06a 100644 --- a/build/emd_Fl_e.html +++ b/build/emd_Fl_e.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1657,7 +1657,7 @@ 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

    -
    +

    \[ W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2 @@ -1678,7 +1678,7 @@ 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} -

    +

    \paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\). Find energy stored in section of length \(l\). @@ -1702,6 +1702,8 @@ Note: gives easy way to find inductance, since \(W = \frac{1}{2} L I^2\).

    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_Fl_i.html b/build/emd_Fl_i.html index 33b2198..17fd043 100644 --- a/build/emd_Fl_i.html +++ b/build/emd_Fl_i.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1637,7 +1637,7 @@ M_{12} = M_{21} \]

    -
    +

    \paragraph{Example 7.10:} short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside @@ -1687,7 +1687,7 @@ Inductance: measured in {\bf henries} (\(H\)). \(H = V s/A\).

    -
    +

    \paragraph{Example 7.11:} find self-inductance of toroidal coil with rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\)) @@ -1714,7 +1714,7 @@ Total flux: \(N\) times this, so self-inductance is Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of {\bf back EMF}.

    -
    +

    \paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\). What is the current ? @@ -1738,6 +1738,8 @@ where \(\tau \equiv L/R\) is the {\bf time constant} of the circuit. +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_Fl_ief.html b/build/emd_Fl_ief.html index 4ca544b..6065345 100644 --- a/build/emd_Fl_ief.html +++ b/build/emd_Fl_ief.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,7 +1621,7 @@ law in integral form: -
    +

    {\bf Example 7.7:} \({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ? @@ -1637,7 +1637,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.

    -
    +

    {\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 \(a < b\), @@ -1671,7 +1671,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with 'slow enough' phenomena.

    -
    +

    {\bf Example 7.9:} infinitely long straight wire carries \(I(t)\). Find induced \({\bf E}\) field as a function of distance \(s\) from wire. @@ -1704,6 +1704,8 @@ Reason: in this case, we've overstepped the quasistatic limit. We need +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_Me.html b/build/emd_Me.html index 8a96e47..3aeae23 100644 --- a/build/emd_Me.html +++ b/build/emd_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1609,6 +1609,8 @@ Table of contents
  • Magnetic Chargeemd.Me.mc
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_Me_Me.html b/build/emd_Me_Me.html index 2b5aea1..f106b08 100644 --- a/build/emd_Me_Me.html +++ b/build/emd_Me_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1600,7 +1600,7 @@ Table of contents

    Full set of equations for the electromagnetic field:

    -
    +

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

    @@ -1616,7 +1616,7 @@ Full set of equations for the electromagnetic field:

    Complement:

    -
    +

    {\bf Force law} \[ @@ -1640,7 +1640,7 @@ take divergence of \((iv)\).

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

    -
    +
    \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, \\ @@ -1655,6 +1655,8 @@ Better way of writing: all fields on left, all sources on right, +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_Me_dc.html b/build/emd_Me_dc.html index 70ddd53..fa28e02 100644 --- a/build/emd_Me_dc.html +++ b/build/emd_Me_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1607,7 +1607,7 @@ the continuity equation as \] The extra term would thus be eliminated if we were to put

    -
    +

    \[ {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} @@ -1631,7 +1631,7 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation

    Maxwell baptized this term the

    -
    +

    {\bf Displacement current} \[ @@ -1664,6 +1664,8 @@ Flat surface: OK, \(E = 0\) and \(I_{\mbox{enc}} = I\). Balloon surface: \(I +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_Me_ebM.html b/build/emd_Me_ebM.html index e7fc2a1..b1ab166 100644 --- a/build/emd_Me_ebM.html +++ b/build/emd_Me_ebM.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1634,6 +1634,8 @@ So for non-steady currents, the 'current enclosed by a loop' is ill-defined. +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_Me_mc.html b/build/emd_Me_mc.html index dabc031..eb68165 100644 --- a/build/emd_Me_mc.html +++ b/build/emd_Me_mc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1632,6 +1632,8 @@ Maxwell's equations {\bf beg} for magnetic charges. But we've never found any!
    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_ce.html b/build/emd_ce.html index ca82e79..4ef5613 100644 --- a/build/emd_ce.html +++ b/build/emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,8 +1598,8 @@ Table of contents emd.ce
    -
    - +
    + Prerequisites
      @@ -1607,8 +1607,8 @@ Prerequisites
    -
    - +
    + Objectives +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_ce_amom.html b/build/emd_ce_amom.html index 22c553e..159c131 100644 --- a/build/emd_ce_amom.html +++ b/build/emd_ce_amom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1600,7 +1600,7 @@ Table of contents

    The angular momentum of EM fields is directly given by

    -
    +

    {\bf Angular momentum of EM fields} \[ @@ -1615,6 +1615,8 @@ The angular momentum of EM fields is directly given by

    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_ce_ce.html b/build/emd_ce_ce.html index 6965d77..c354912 100644 --- a/build/emd_ce_ce.html +++ b/build/emd_ce_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1619,7 +1619,7 @@ This means that \] Since this is true for any volume, we have (re)derived the

    -
    +

    {\bf Continuity equation} \[ @@ -1645,6 +1645,8 @@ imposes a functional constraint on these sources: not {\it any} \(\rho\) and +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html index 023a93b..0c09c19 100644 --- a/build/emd_ce_mom.html +++ b/build/emd_ce_mom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1612,7 +1612,7 @@ 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

    -
    +

    {\bf Momentum density in the EM fields} \[ @@ -1624,7 +1624,7 @@ 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

    -
    +

    {\bf Continuity equation for EM momentum} \[ @@ -1638,6 +1638,8 @@ In a region in which the mechanical momentum is not changing due to external inf +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html index da47ab1..450f4c6 100644 --- a/build/emd_ce_mst.html +++ b/build/emd_ce_mst.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1654,7 +1654,7 @@ and similarly for \({\boldsymbol B}\). We thus get

    This expression can be greatly simplified by introducing the

    -
    +

    {\bf Maxwell stress tensor} \[ @@ -1677,7 +1677,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction

    We then obtain

    -
    +

    {\bf EM force per unit volume} \[ @@ -1689,7 +1689,7 @@ We then obtain

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

    -
    +

    {\bf Total force on charges in volume} \[ @@ -1703,6 +1703,8 @@ where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html index 41c8b1d..690101d 100644 --- a/build/emd_ce_poy.html +++ b/build/emd_ce_poy.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1666,7 +1666,7 @@ so we get Substituting this in \ref{Gr(8.6)} and using the divergence theorem, we obtain

    -
    +

    {\bf Poynting's theorem} \[ @@ -1691,7 +1691,7 @@ 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} \[ @@ -1704,7 +1704,7 @@ Energy per unit time, per unit area carried by EM fields:

    We can thus express Poynting's theorem more compactly:

    -
    +

    {\bf Poynting's theorem} \[ @@ -1717,7 +1717,7 @@ We can thus express Poynting's theorem more compactly:

    where we have defined the total

    -
    +

    {\bf Energy in electromagnetic fields} \[ @@ -1740,7 +1740,7 @@ Then, \] so we get the

    -
    +

    {\bf Poynting theorem (differential form)} \[ @@ -1757,7 +1757,7 @@ 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 @@ -1787,6 +1787,8 @@ and the value is as expected. +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_emw.html b/build/emd_emw.html index 977478c..d4653e1 100644 --- a/build/emd_emw.html +++ b/build/emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,8 +1598,8 @@ Table of contents emd.emw
    -
    - +
    + Prerequisites
      @@ -1608,8 +1608,8 @@ Prerequisites
    -
    - +
    + Objectives +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html index fca0c76..c1cf235 100644 --- a/build/emd_emw_ep.html +++ b/build/emd_emw_ep.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1628,7 +1628,7 @@ so for a monochromatic EM plan wave, \] or more succinctly:

    -
    +

    {\bf Poynting vector of a monochromatic EM wave} \[ @@ -1644,7 +1644,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w

    Similary, we get the

    -
    +

    {\bf Momentum density of a monochromatic EM wave} \[ @@ -1680,6 +1680,8 @@ The {\it radiation pressure} is the momentum transfer per unit area per unit of

    +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html index fa31848..ca73c1b 100644 --- a/build/emd_emw_mpw.html +++ b/build/emd_emw_mpw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1629,7 +1629,7 @@ 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

    -
    +

    {\bf E and B fields for a monochromatic EM plane wave} \[ @@ -1658,6 +1658,8 @@ or if you prefer explicit real parts (adding a possible phase shift \(\delta\)): +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html index c97e01e..828c8e0 100644 --- a/build/emd_emw_we.html +++ b/build/emd_emw_we.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1626,7 +1626,7 @@ 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

    -
    +

    {\bf Wave equations for electric and magnetic fields in vacuum} \[ @@ -1667,6 +1667,8 @@ the actual electric and magnetic fields are given by the real part. +

    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emdm.html b/build/emdm.html index 3149315..d896018 100644 --- a/build/emdm.html +++ b/build/emdm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1607,6 +1607,8 @@ Table of contents
  • Electromagnetic Waves in Matteremdm.emwm
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emdm_Me.html b/build/emdm_Me.html index cc39ac5..5e7b45c 100644 --- a/build/emdm_Me.html +++ b/build/emdm_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1607,6 +1607,8 @@ Table of contents
  • Boundary Conditionsemdm.Me.bc
    • +
    +
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-10 Thu 08:32

    +

    Created: 2022-02-13 Sun 21:20

    diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html index a01206c..c15aaeb 100644 --- a/build/emdm_Me_Mem.html +++ b/build/emdm_Me_Mem.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia \] We therefore have the

    -
    +

    {\bf Polarization current density} \[ @@ -1651,7 +1651,7 @@ 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:

    -