diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html
index 698ac1c..8ce53cd 100644
--- a/build/c_m_cs_sph.html
+++ b/build/c_m_cs_sph.html
@@ -1,7 +1,7 @@
-
+
Pre-Quantum Electrodynamics
@@ -1607,14 +1607,14 @@ which \(r\) is the distance from the chosen origin,
The usual Cartesian coordinates relate to spherical coordinates
according to
-
+
-
+
@@ -1639,14 +1639,14 @@ A generic vector can be expressed as
where the explicit relation between spherical and
Cartesian unit vectors is
-
+
-
+
@@ -1669,14 +1669,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
An infinitesimal displacement \(d{\bf l}\) can be written as
-
+
-
+
@@ -1692,14 +1692,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
Infinitesimal volume element:
-
+
-
+
@@ -1720,14 +1720,14 @@ Infinitesimal surface element: depends on situation.
\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) 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.
diff --git a/build/emd_Fl_Fl.html b/build/emd_Fl_Fl.html
index 8b94a11..4156263 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)
\[
@@ -1715,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Author: Jean-Sébastien Caux
-
Created: 2022-02-14 Mon 20:35
+
Created: 2022-02-15 Tue 10:14
diff --git a/build/emd_Fl_e.html b/build/emd_Fl_e.html
index de21585..206e0c8 100644
--- a/build/emd_Fl_e.html
+++ b/build/emd_Fl_e.html
@@ -1,7 +1,7 @@
-
+
Pre-Quantum Electrodynamics
@@ -1657,7 +1657,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
\]
We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
-
\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 ?
@@ -1753,7 +1753,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Author: Jean-Sébastien Caux
-
Created: 2022-02-14 Mon 20:35
+
Created: 2022-02-15 Tue 10:14
diff --git a/build/emd_Fl_ief.html b/build/emd_Fl_ief.html
index 9bd61af..94cfff0 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:
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+
{\bf Example 7.7:}
\({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ?
@@ -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.
@@ -1719,7 +1719,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
diff --git a/build/emd_Me_dc.html b/build/emd_Me_dc.html
index 6e8e452..e2f1ef1 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
-
The angular momentum of EM fields is directly given by
-
+
{\bf Angular momentum of EM fields}
\[
@@ -1630,7 +1630,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Author: Jean-Sébastien Caux
-
Created: 2022-02-14 Mon 20:35
+
Created: 2022-02-15 Tue 10:14
diff --git a/build/emd_ce_ce.html b/build/emd_ce_ce.html
index 394fa7c..463ee80 100644
--- a/build/emd_ce_ce.html
+++ b/build/emd_ce_ce.html
@@ -1,7 +1,7 @@
-
+
Pre-Quantum Electrodynamics
@@ -1619,7 +1619,7 @@ This means that
\]
Since this is true for any volume, we have (re)derived the
-
diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html
index 827009c..0964faf 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}
\[
@@ -1653,7 +1653,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Author: Jean-Sébastien Caux
-
Created: 2022-02-14 Mon 20:35
+
Created: 2022-02-15 Tue 10:14
diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html
index 15e712a..1348c7b 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}
\[
@@ -1718,7 +1718,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Author: Jean-Sébastien Caux
-
Created: 2022-02-14 Mon 20:35
+
Created: 2022-02-15 Tue 10:14
diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html
index 27c8d9a..0bd7651 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)}
\[
@@ -1757,7 +1757,7 @@ and has a similar for to the continuity equation
-
+
\paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting.
Assuming that the field is uniform, the electric field parallel to the wire is
@@ -1802,7 +1802,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
@@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Author: Jean-Sébastien Caux
-
Created: 2022-02-14 Mon 20:35
+
Created: 2022-02-15 Tue 10:14
diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html
index e1c0f6a..b1d8001 100644
--- a/build/emd_emw_ep.html
+++ b/build/emd_emw_ep.html
@@ -1,7 +1,7 @@
-
+
Pre-Quantum Electrodynamics
@@ -1628,7 +1628,7 @@ so for a monochromatic EM plan wave,
\]
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}
\[
@@ -1695,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Author: Jean-Sébastien Caux
-
Created: 2022-02-14 Mon 20:35
+
Created: 2022-02-15 Tue 10:14
diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html
index 8cc5467..4fa7ad7 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}
\[
@@ -1673,7 +1673,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Author: Jean-Sébastien Caux
-
Created: 2022-02-14 Mon 20:35
+
Created: 2022-02-15 Tue 10:14
diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html
index 2e24f76..4c0b161 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}
\[
@@ -1682,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html
index 2ccb94c..c393f2b 100644
--- a/build/emdm_Me_Mem.html
+++ b/build/emdm_Me_Mem.html
@@ -1,7 +1,7 @@
-
+
Pre-Quantum Electrodynamics
@@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\]
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: