diff --git a/build/a.html b/build/a.html index ccb378f..df39c7f 100644 --- a/build/a.html +++ b/build/a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/a_l.html b/build/a_l.html index d7f6d38..4676259 100644 --- a/build/a_l.html +++ b/build/a_l.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1704,7 +1704,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c.html b/build/c.html index de376c6..1f266a2 100644 --- a/build/c.html +++ b/build/c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m.html b/build/c_m.html index 023a9eb..472de92 100644 --- a/build/c_m.html +++ b/build/c_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1627,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_cs.html b/build/c_m_cs.html index bb877f0..0288be5 100644 --- a/build/c_m_cs.html +++ b/build/c_m_cs.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1623,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_cs_cyl.html b/build/c_m_cs_cyl.html index ada05cb..06985b4 100644 --- a/build/c_m_cs_cyl.html +++ b/build/c_m_cs_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,11 +1598,11 @@ Table of contents c.m.cs.cyl

-\((r, \phi, z)\). Relation to Cartesian coordinates: +\((r, \varphi, z)\). Relation to Cartesian coordinates:

\begin{equation} -x = r \cos \phi, y = r \sin \phi, z = z +x = r \cos \varphi, y = r \sin \varphi, z = z \label{Gr(1.74)} \end{equation} @@ -1611,8 +1611,8 @@ The unit vectors are

\begin{equation} -\hat{\bf r} = \cos \phi ~\hat{\bf x} + \sin \phi~\hat{\bf y}, -\hat{\boldsymbol \phi} = -\sin \theta ~\hat{\bf x} + \cos \phi~\hat{\bf y}, +\hat{\bf r} = \cos \varphi ~\hat{\bf x} + \sin \varphi~\hat{\bf y}, +\hat{\boldsymbol \varphi} = -\sin \theta ~\hat{\bf x} + \cos \varphi~\hat{\bf y}, \hat{\bf z} = \hat{\bf z}. \label{Gr(1.75)} \end{equation} @@ -1622,7 +1622,7 @@ Infinitesimal displacement:

\begin{equation} -d{\bf l} = dr ~\hat{\bf r} + r d\phi~\hat{\boldsymbol \phi} + dz ~\hat{\bf z}. +d{\bf l} = dr ~\hat{\bf r} + r d\varphi~\hat{\boldsymbol \varphi} + dz ~\hat{\bf z}. \label{Gr(1.77)} \end{equation} @@ -1631,26 +1631,26 @@ Volume element:

\begin{equation} -d\tau = r dr d\phi dz +d\tau = r dr d\varphi dz \label{Gr(1.78)} \end{equation}

-Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in ]-\infty, \infty[\). +Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \in ]-\infty, \infty[\).

Gradient
-
+

-
+
  • Gr4(1.79)
@@ -1661,7 +1661,7 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in \begin{equation} {\boldsymbol \nabla} T = \frac{\partial T}{\partial r}~\hat{\bf r} -+ \frac{1}{r} \frac{\partial T}{\partial \phi}~\hat{\boldsymbol \phi} ++ \frac{1}{r} \frac{\partial T}{\partial \varphi}~\hat{\boldsymbol \varphi} + \frac{\partial T}{\partial z} ~\hat{\bf z} \tag{cylgrad}\label{cylgrad} \end{equation} @@ -1671,14 +1671,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
Divergence
-
+

-
+
  • Gr4(2.21)
@@ -1689,7 +1689,7 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in \begin{equation} {\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r} \frac{\partial}{\partial r} (r v_r) -+ \frac{1}{r} \frac{\partial v_{\phi}}{\partial \phi} + \frac{\partial v_z}{\partial z}. ++ \frac{1}{r} \frac{\partial v_{\varphi}}{\partial \varphi} + \frac{\partial v_z}{\partial z}. \tag{cyl_div} \label{cyl_div} \end{equation} @@ -1699,14 +1699,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
Curl
-
+

-
+
  • Gr4(2.21)
@@ -1716,9 +1716,9 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
\begin{align} -{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \phi} - \frac{\partial v_{\phi}}{\partial z}\right) ~\hat{\bf r} -+ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \phi} \nonumber \\ -+ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\phi}) - \frac{\partial v_r}{\partial \phi} \right) ~\hat{\bf z} +{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \varphi} - \frac{\partial v_{\varphi}}{\partial z}\right) ~\hat{\bf r} ++ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \varphi} \nonumber \\ ++ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\varphi}) - \frac{\partial v_r}{\partial \varphi} \right) ~\hat{\bf z} \tag{cyl_curl} \label{cyl_curl} \end{align} @@ -1730,7 +1730,7 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
\begin{equation} {\boldsymbol \nabla}^2 T = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right) -+ \frac{1}{r^2} \frac{\partial^2 T}{\partial \phi^2} + \frac{\partial^2 T}{\partial z^2} ++ \frac{1}{r^2} \frac{\partial^2 T}{\partial \varphi^2} + \frac{\partial^2 T}{\partial z^2} \label{Gr(1.82)} \end{equation}
@@ -1755,7 +1755,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_cs_hyp.html b/build/c_m_cs_hyp.html index 3a11a4b..d375ecc 100644 --- a/build/c_m_cs_hyp.html +++ b/build/c_m_cs_hyp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,7 +1616,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html index 9d537d8..698ac1c 100644 --- a/build/c_m_cs_sph.html +++ b/build/c_m_cs_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,64 +1598,117 @@ Table of contents c.m.cs.sph

-\((r, \theta, \phi)\). \(\theta\) is the polar angle, \(\phi\) the azimuthal angle. +In this system, we use coordinates \((r, \theta, \varphi)\) in +which \(r\) is the distance from the chosen origin, +\(\theta\) is the polar angle and \(\varphi\) is the azimuthal angle.

+

+The usual Cartesian coordinates relate to spherical coordinates +according to +

+
+

+ + + + +

+
+ +
+ +
+

\[ -x = r \sin \theta \cos \phi, -y = r \sin \theta \sin \phi, +x = r \sin \theta \cos \varphi, \hspace{5mm} +y = r \sin \theta \sin \varphi, \hspace{5mm} z = r \cos \theta. -\label{Gr(1.62)} +\tag{sph_xyz}\label{sph_xyz} \]

-Unit vectors: \(\hat{\boldsymbol r}, \hat{\boldsymbol \theta}, \hat{\boldsymbol \phi}\). -

- -

+The unit vectors are written \(\hat{\boldsymbol r}\), +\(\hat{\boldsymbol \theta}\) and \(\hat{\boldsymbol \varphi}\). +A generic vector can be expressed as \[ -{\bf A} = A_r \hat{\bf r} + A_{\theta} \hat{\bf \theta} + A_{\phi} \hat{\boldsymbol \phi} -\label{Gr(1.63)} +{\bf v} = v_r \hat{\bf r} + v_{\theta} \hat{\bf \theta} + v_{\varphi} \hat{\boldsymbol \varphi} \] +where the explicit relation between spherical and +Cartesian unit vectors is

- +

-In terms of Cartesian unit vectors: + + + +

+
+ +
+ +
\begin{align} -\hat{\boldsymbol r} &= \sin \theta \cos \phi \hat{\bf x} + \sin \theta \sin \phi \hat{\bf y} + \cos \theta \hat{\bf z}, \nonumber \\ -\hat{\boldsymbol \theta} &= \cos \theta \cos \phi \hat{\bf x} + \cos \theta \sin \phi \hat{\bf y} - \sin \theta \hat{\bf z}, \nonumber \\ -\hat{\boldsymbol \phi} &= -\sin \phi \hat{\bf x} + \cos \phi \hat{\bf y}. -\label{Gr(1.64)} +\hat{\boldsymbol r} &= \sin \theta \cos \varphi ~\hat{\bf x} + \sin \theta \sin \varphi ~\hat{\bf y} + \cos \theta ~\hat{\bf z}, \nonumber \\ +\hat{\boldsymbol \theta} &= \cos \theta \cos \varphi ~\hat{\bf x} + \cos \theta \sin \varphi ~\hat{\bf y} - \sin \theta ~\hat{\bf z}, \nonumber \\ +\hat{\boldsymbol \varphi} &= -\sin \varphi ~\hat{\bf x} + \cos \varphi ~\hat{\bf y}. +\tag{sph_uv}\label{sph_uv} \end{align}

-Careful: these unit vectors are direction dependent, i.e. we should really -write \(\hat{\boldsymbol r} (\theta, \phi), \hat{\boldsymbol \theta} (\theta, \phi), \hat{\boldsymbol \phi} (\theta, \phi)\). +Do be careful: these unit vectors are direction dependent, i.e. we should really +write \(\hat{\boldsymbol r} (\theta, \varphi)\), +\(\hat{\boldsymbol \theta} (\theta, \varphi)\) +and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).

-Infinitesimal displacement \(d{\bf l}\): +An infinitesimal displacement \(d{\bf l}\) can be written as

+
+

+ + + + +

+
+ +
+ +

\[ -d{\bf l} = dr \hat{\boldsymbol r} + r d\theta \hat{\boldsymbol \theta} + r\sin \theta d\phi \hat{\boldsymbol \phi}. -\label{Gr(1.68)} +d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin \theta d\varphi ~\hat{\boldsymbol \varphi}. +\tag{sph_dl}\label{sph_dl} \]

Infinitesimal volume element:

+
+

+ + + + +

+
+ +
+ +

\[ -d\tau = dl_r dl_{\theta} dl_{\phi} = r^2 \sin \theta dr d\theta d\phi -\label{Gr(1.69)} +d\tau = dl_r dl_{\theta} dl_{\varphi} = r^2 \sin \theta dr d\theta d\varphi +\tag{sph_dtau}\label{sph_dtau} \]

@@ -1667,10 +1720,23 @@ Infinitesimal surface element: depends on situation.
Gradient
+
+

+ + + + +

+
+ +
+ +
+ \begin{equation} {\boldsymbol \nabla} T = \frac{\partial T}{\partial r} \hat{\boldsymbol r} + \frac{1}{r} \frac{\partial T}{\partial \theta} \hat{\boldsymbol \theta} -+ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \phi} \hat{\boldsymbol \phi}. -\label{Gr(1.70)} ++ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \varphi} \hat{\boldsymbol \varphi}. +\tag{sph_grad}\label{sph_grad} \end{equation}
@@ -1678,10 +1744,23 @@ Infinitesimal surface element: depends on situation.
Divergence
+
+

+ + + + +

+
+ +
+ +
+ \begin{equation} {\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 v_r) + \frac{1}{r\sin \theta} \frac{\partial}{\partial \theta} (\sin\theta v_{\theta}) -+ \frac{1}{r \sin \theta} \frac{\partial v_{\phi}}{\partial \phi} -\label{Gr(1.71)} ++ \frac{1}{r \sin \theta} \frac{\partial v_{\varphi}}{\partial \varphi} +\tag{sph_div}\label{sph_div} \end{equation}
@@ -1689,23 +1768,50 @@ Infinitesimal surface element: depends on situation.
Curl
-\begin{equation} -{\boldsymbol \nabla} \times {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\phi}) - \frac{\partial v_{\theta}}{\partial \phi} \right] \hat{\bf r} -+ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \phi} - \frac{\partial}{\partial r} (r v_{\phi}) \right] \hat{\boldsymbol \theta} -+ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \phi} -\label{Gr(1.72)} -\end{equation} +
+

+ + + + +

+
+ +
+ +
+ +\begin{align} +{\boldsymbol \nabla} \times {\bf v} = &\frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\varphi}) - \frac{\partial v_{\theta}}{\partial \varphi} \right] \hat{\bf r} +\nonumber \\ +&+ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \varphi} - \frac{\partial}{\partial r} (r v_{\varphi}) \right] \hat{\boldsymbol \theta} ++ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \varphi} +\tag{sph_curl}\label{sph_curl} +\end{align}
Laplacian
+
+

+ + + + +

+
+ +
+ +
+ \begin{equation} {\boldsymbol \nabla}^2 T = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial T}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial T}{\partial \theta}\right) -+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \phi^2} -\label{Gr(1.73)} ++ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \varphi^2} +\tag{sph_Lap}\label{sph_Lap} \end{equation}
@@ -1729,7 +1835,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dc.html b/build/c_m_dc.html index 1dfbb33..716b6eb 100644 --- a/build/c_m_dc.html +++ b/build/c_m_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1626,7 +1626,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dc_curl.html b/build/c_m_dc_curl.html index b41e9d7..e955df8 100644 --- a/build/c_m_dc_curl.html +++ b/build/c_m_dc_curl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1628,7 +1628,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dc_d2.html b/build/c_m_dc_d2.html index b8d19c3..32d84c9 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} @@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dc_del.html b/build/c_m_dc_del.html index 7e1a7c2..b6a7aa6 100644 --- a/build/c_m_dc_del.html +++ b/build/c_m_dc_del.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1627,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dc_div.html b/build/c_m_dc_div.html index c3d8f89..718663b 100644 --- a/build/c_m_dc_div.html +++ b/build/c_m_dc_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dc_g.html b/build/c_m_dc_g.html index 2525122..239f3ee 100644 --- a/build/c_m_dc_g.html +++ b/build/c_m_dc_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dc_pr.html b/build/c_m_dc_pr.html index 62fc37f..1945434 100644 --- a/build/c_m_dc_pr.html +++ b/build/c_m_dc_pr.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dd.html b/build/c_m_dd.html index c943655..16dc433 100644 --- a/build/c_m_dd.html +++ b/build/c_m_dd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1623,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dd_1d.html b/build/c_m_dd_1d.html index b1f39ed..2a6b983 100644 --- a/build/c_m_dd_1d.html +++ b/build/c_m_dd_1d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1635,7 +1635,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dd_3d.html b/build/c_m_dd_3d.html index 99cd00a..83b6184 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)
@@ -1669,14 +1669,14 @@ Since

we have that

-
+

-
+
  • Gr (1.102)
@@ -1708,7 +1708,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_dd_div.html b/build/c_m_dd_div.html index 7835dc3..bab088f 100644 --- a/build/c_m_dd_div.html +++ b/build/c_m_dd_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1613,8 +1613,8 @@ Integrate over a sphere of radius \(R\) centered at the origin (Prob. 1.38b): \begin{equation} -\oint \frac{\hat{\bf r}}{r^2} \cdot d{\bf a} = \int \left(\frac{1}{R^2} \hat{\bf r} \right) \cdot \left( R^2 \sin \theta d\theta d\phi ~\hat{\bf r} \right) -= \int_0^{\pi} d\theta \sin \theta \int_0^{2\pi} d\phi = 4\pi +\oint \frac{\hat{\bf r}}{r^2} \cdot d{\bf a} = \int \left(\frac{1}{R^2} \hat{\bf r} \right) \cdot \left( R^2 \sin \theta d\theta d\varphi ~\hat{\bf r} \right) += \int_0^{\pi} d\theta \sin \theta \int_0^{2\pi} d\varphi = 4\pi \label{Gr(1.85)} \end{equation} @@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_ic.html b/build/c_m_ic.html index c67e307..a8fa5ab 100644 --- a/build/c_m_ic.html +++ b/build/c_m_ic.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1626,7 +1626,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_ic_ftc.html b/build/c_m_ic_ftc.html index e55664f..ebb6f85 100644 --- a/build/c_m_ic_ftc.html +++ b/build/c_m_ic_ftc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_ic_ftg.html b/build/c_m_ic_ftg.html index b8e64d2..f12f7d0 100644 --- a/build/c_m_ic_ftg.html +++ b/build/c_m_ic_ftg.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1633,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_ic_gauss.html b/build/c_m_ic_gauss.html index 07747cc..8ef736e 100644 --- a/build/c_m_ic_gauss.html +++ b/build/c_m_ic_gauss.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1627,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_ic_ip.html b/build/c_m_ic_ip.html index 8e42591..e5ae5ae 100644 --- a/build/c_m_ic_ip.html +++ b/build/c_m_ic_ip.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1634,7 +1634,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/c_m_ic_lsv.html b/build/c_m_ic_lsv.html index 3672e9f..3845f5a 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
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Line Integrals
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+
+
Line Integrals
+

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

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Surface Integrals
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+
Surface Integrals
+

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

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-
Volume Integrals
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+
+
Volume Integrals
+

\[ \int_{\cal V} T d\tau @@ -1695,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_ic_stokes.html b/build/c_m_ic_stokes.html index 7231030..adf06db 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
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+

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  • Gr (1.57)
@@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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diff --git a/build/c_m_uf.html b/build/c_m_uf.html index b476e42..2903542 100644 --- a/build/c_m_uf.html +++ b/build/c_m_uf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1623,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_uf_cyl.html b/build/c_m_uf_cyl.html index 354f254..bec70c5 100644 --- a/build/c_m_uf_cyl.html +++ b/build/c_m_uf_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1600,21 +1600,21 @@ Table of contents \begin{align*} \mbox{Gradient:} &{\boldsymbol \nabla} T = \frac{\partial T}{\partial r}~\hat{\bf r} -+ \frac{1}{r} \frac{\partial T}{\partial \phi}~\hat{\boldsymbol \phi} ++ \frac{1}{r} \frac{\partial T}{\partial \varphi}~\hat{\boldsymbol \varphi} + \frac{\partial T}{\partial z} ~\hat{\bf z} \nonumber\\ \mbox{Divergence:} &{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r} \frac{\partial}{\partial r} (r v_r) -+ \frac{1}{r} \frac{\partial v_{\phi}}{\partial \phi} + \frac{\partial v_z}{\partial z}. ++ \frac{1}{r} \frac{\partial v_{\varphi}}{\partial \varphi} + \frac{\partial v_z}{\partial z}. \nonumber\\ \mbox{Curl:} -&{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \phi} - \frac{\partial v_{\phi}}{\partial z}\right) ~\hat{\bf r} -+ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \phi} -+ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\phi}) - \frac{\partial v_r}{\partial \phi} \right) ~\hat{\bf z} +&{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \varphi} - \frac{\partial v_{\varphi}}{\partial z}\right) ~\hat{\bf r} ++ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \varphi} ++ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\varphi}) - \frac{\partial v_r}{\partial \varphi} \right) ~\hat{\bf z} \nonumber\\ \mbox{Laplacian:} &{\boldsymbol \nabla}^2 T = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right) -+ \frac{1}{r^2} \frac{\partial^2 T}{\partial \phi^2} + \frac{\partial^2 T}{\partial z^2} ++ \frac{1}{r^2} \frac{\partial^2 T}{\partial \varphi^2} + \frac{\partial^2 T}{\partial z^2} \end{align*}
@@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_uf_sph.html b/build/c_m_uf_sph.html index a6fcb67..80bf96e 100644 --- a/build/c_m_uf_sph.html +++ b/build/c_m_uf_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1600,21 +1600,21 @@ Table of contents \begin{align*} \mbox{Gradient:} &{\boldsymbol \nabla} T = \frac{\partial T}{\partial r} \hat{\boldsymbol r} + \frac{1}{r} \frac{\partial T}{\partial \theta} \hat{\boldsymbol \theta} -+ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \phi} \hat{\boldsymbol \phi}. ++ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \varphi} \hat{\boldsymbol \varphi}. \nonumber \\ \mbox{Divergence:} &{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 v_r) + \frac{1}{r\sin \theta} \frac{\partial}{\partial \theta} (\sin\theta v_{\theta}) -+ \frac{1}{r \sin \theta} \frac{\partial v_{\phi}}{\partial \phi} ++ \frac{1}{r \sin \theta} \frac{\partial v_{\varphi}}{\partial \varphi} \nonumber\\ \mbox{Curl:} -&{\boldsymbol \nabla} \times {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\phi}) - \frac{\partial v_{\theta}}{\partial \phi} \right] \hat{\bf r} -+ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \phi} - \frac{\partial}{\partial r} (r v_{\phi}) \right] \hat{\boldsymbol \theta} -+ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \phi} +&{\boldsymbol \nabla} \times {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\varphi}) - \frac{\partial v_{\theta}}{\partial \varphi} \right] \hat{\bf r} ++ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \varphi} - \frac{\partial}{\partial r} (r v_{\varphi}) \right] \hat{\boldsymbol \theta} ++ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \varphi} \nonumber\\ \mbox{Laplacian:} &{\boldsymbol \nabla}^2 T = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial T}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial T}{\partial \theta}\right) -+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \phi^2} ++ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \varphi^2} \end{align*}
@@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_uf_vi.html b/build/c_m_uf_vi.html index 54535ce..a9a46d2 100644 --- a/build/c_m_uf_vi.html +++ b/build/c_m_uf_vi.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/c_m_va.html b/build/c_m_va.html index 34c6b66..0b6642c 100644 --- a/build/c_m_va.html +++ b/build/c_m_va.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1625,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/c_m_va_cp.html b/build/c_m_va_cp.html index 0363996..9ccd462 100644 --- a/build/c_m_va_cp.html +++ b/build/c_m_va_cp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1654,7 +1654,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/c_m_va_n.html b/build/c_m_va_n.html index 3043320..6a07d19 100644 --- a/build/c_m_va_n.html +++ b/build/c_m_va_n.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_va_pds.html b/build/c_m_va_pds.html index 0e225dc..71a6e41 100644 --- a/build/c_m_va_pds.html +++ b/build/c_m_va_pds.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1683,7 +1683,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/c_m_va_sp.html b/build/c_m_va_sp.html index ef27cba..6488216 100644 --- a/build/c_m_va_sp.html +++ b/build/c_m_va_sp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/c_m_va_tp.html b/build/c_m_va_tp.html index f4dd625..4688041 100644 --- a/build/c_m_va_tp.html +++ b/build/c_m_va_tp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1696,7 +1696,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/c_m_vf.html b/build/c_m_vf.html index 185e94c..45e3781 100644 --- a/build/c_m_vf.html +++ b/build/c_m_vf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_vf_helm.html b/build/c_m_vf_helm.html index 7f93a9e..3e2e631 100644 --- a/build/c_m_vf_helm.html +++ b/build/c_m_vf_helm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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Created: 2022-02-14 Mon 20:35

diff --git a/build/c_m_vf_pot.html b/build/c_m_vf_pot.html index 1164164..ab54332 100644 --- a/build/c_m_vf_pot.html +++ b/build/c_m_vf_pot.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1668,7 +1668,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/d.html b/build/d.html index 4d2e6c9..db0d9a9 100644 --- a/build/d.html +++ b/build/d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,8 +1598,8 @@ Table of contents d
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- +
+ Objectives @@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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diff --git a/build/d_emd.html b/build/d_emd.html index 43556b5..34f1d74 100644 --- a/build/d_emd.html +++ b/build/d_emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/d_emd_ce.html b/build/d_emd_ce.html index d7e9225..8657e8b 100644 --- a/build/d_emd_ce.html +++ b/build/d_emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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Created: 2022-02-14 Mon 20:35

diff --git a/build/d_emd_emw.html b/build/d_emd_emw.html index 5bbfe2a..56c7264 100644 --- a/build/d_emd_emw.html +++ b/build/d_emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/d_emf.html b/build/d_emf.html index 772b1aa..94503ad 100644 --- a/build/d_emf.html +++ b/build/d_emf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/d_ems.html b/build/d_ems.html index 70566ff..9ff5b9d 100644 --- a/build/d_ems.html +++ b/build/d_ems.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1660,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/d_ems_ca.html b/build/d_ems_ca.html index f6f78ed..f9335ca 100644 --- a/build/d_ems_ca.html +++ b/build/d_ems_ca.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/d_ems_ms.html b/build/d_ems_ms.html index 160828f..eb3c9d5 100644 --- a/build/d_ems_ms.html +++ b/build/d_ems_ms.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/d_emsm.html b/build/d_emsm.html index 1faaf48..25f6b2b 100644 --- a/build/d_emsm.html +++ b/build/d_emsm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-14 Mon 20:35

diff --git a/build/d_emsm_msm.html b/build/d_emsm_msm.html index 822a4af..79fdc6e 100644 --- a/build/d_emsm_msm.html +++ b/build/d_emsm_msm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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Created: 2022-02-14 Mon 20:35

diff --git a/build/d_m.html b/build/d_m.html index f7f2770..30c102c 100644 --- a/build/d_m.html +++ b/build/d_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1651,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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Created: 2022-02-14 Mon 20:35

diff --git a/build/d_red.html b/build/d_red.html index 5dffa11..b13a956 100644 --- a/build/d_red.html +++ b/build/d_red.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1645,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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diff --git a/build/emd.html b/build/emd.html index 0486941..304bd18 100644 --- a/build/emd.html +++ b/build/emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,8 +1598,8 @@ Table of contents emd
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+ Prerequisites
    @@ -1608,8 +1608,8 @@ Prerequisites
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+ Objectives
    @@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_Fl.html b/build/emd_Fl.html index 3792472..d771bba 100644 --- a/build/emd_Fl.html +++ b/build/emd_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_Fl_Fl.html b/build/emd_Fl_Fl.html index 4b295c1..8b94a11 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

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Electromotive force (or electromotance), \[ @@ -1660,7 +1660,7 @@ to the rate of change of the magnetic flux, \] so we obtain

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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

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Faraday's law (differential form) \[ @@ -1715,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_Fl_e.html b/build/emd_Fl_e.html index a4cd06a..de21585 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

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\[ W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2 @@ -1678,18 +1678,18 @@ 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 \(l\). \paragraph{Solution:} from Ampère, \[ - {\bf B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol \phi}, \hspace{1cm} a < s < b, \hspace{1cm} + {\bf B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol \varphi}, \hspace{1cm} a < s < b, \hspace{1cm} {\bf B} = 0, \hspace{1cm} s < a ~\mbox{or}~ s > b. \] Energy is thus \[ - W_{mag} = \frac{1}{2\mu_0} \int_0^{2\pi} d\phi \int_0^l dz \int_a^b s ds \left(\frac{\mu_0 I}{2\pi s}\right)^2 + W_{mag} = \frac{1}{2\mu_0} \int_0^{2\pi} d\varphi \int_0^l dz \int_a^b s ds \left(\frac{\mu_0 I}{2\pi s}\right)^2 = \frac{\mu_0 I^2 l}{4\pi} \ln \frac{b}{a}. \]

@@ -1717,7 +1717,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_Fl_i.html b/build/emd_Fl_i.html index 17fd043..2a4b884 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} \]

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\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} (\(H\)). \(H = V s/A\).

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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 \(h\)) @@ -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}.

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\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

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Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_Fl_ief.html b/build/emd_Fl_ief.html index 6065345..9bd61af 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 \({\bf E}(t)\) ? @@ -1629,7 +1629,7 @@ law in integral form: amperian loop of radius \(s\), apply Faraday: \[ \oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt} - \Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \phi}. + \Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}. \] Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.

@@ -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 \(a < b\), @@ -1646,7 +1646,7 @@ pointing up. Field turned off. What happens ? Faraday: \[ \oint {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt} = - \pi a^2 \frac{dB}{dt} - \Rightarrow {\bf E} = -\frac{a^2}{2b} \frac{dB}{dt} \hat{\boldsymbol \phi}. + \Rightarrow {\bf E} = -\frac{a^2}{2b} \frac{dB}{dt} \hat{\boldsymbol \varphi}. \] Torque on segment \(d{\bf l}\): \(|{\bf r} \times {\bf F}| = b \lambda E dl\). Total torque: @@ -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.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_Me.html b/build/emd_Me.html index 3aeae23..0e9b0a0 100644 --- a/build/emd_Me.html +++ b/build/emd_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_Me_Me.html b/build/emd_Me_Me.html index f106b08..22b4c15 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:

-
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{\bf Force law} \[ @@ -1640,7 +1640,7 @@ 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, \\ @@ -1670,7 +1670,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_Me_dc.html b/build/emd_Me_dc.html index fa28e02..6e8e452 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

-
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\[ {\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

-
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{\bf Displacement current} \[ @@ -1679,7 +1679,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_Me_ebM.html b/build/emd_Me_ebM.html index b1ab166..7787ba0 100644 --- a/build/emd_Me_ebM.html +++ b/build/emd_Me_ebM.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_Me_mc.html b/build/emd_Me_mc.html index eb68165..f78a7d8 100644 --- a/build/emd_Me_mc.html +++ b/build/emd_Me_mc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1647,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_ce.html b/build/emd_ce.html index 4ef5613..fa4964a 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
    @@ -1646,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_ce_amom.html b/build/emd_ce_amom.html index 159c131..2afe358 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} \[ @@ -1630,7 +1630,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_ce_ce.html b/build/emd_ce_ce.html index c354912..394fa7c 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

-
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{\bf Continuity equation} \[ @@ -1660,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html index 0c09c19..827009c 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

-
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{\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

-
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{\bf Continuity equation for EM momentum} \[ @@ -1653,7 +1653,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html index 450f4c6..15e712a 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

-
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{\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

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{\bf EM force per unit volume} \[ @@ -1689,7 +1689,7 @@ 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} \[ @@ -1718,7 +1718,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

+

Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html index 690101d..27c8d9a 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

-
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{\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:

-
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{\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:

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

where we have defined the total

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

-
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{\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 @@ -1767,11 +1767,11 @@ Assuming that the field is uniform, the electric field parallel to the wire is where \(V\) is the potential difference between the ends ald \(L\) is the length. Magnetic field is circumferential: wire of radius \(a\), \[ - {\boldsymbol B} = \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol \phi} + {\boldsymbol B} = \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol \varphi} \] Poynting: \[ - {\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \phi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s} + {\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s} \] and points radially inwards. Energy per unit time passing surface of wire: \[ @@ -1802,7 +1802,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_emw.html b/build/emd_emw.html index d4653e1..fa98403 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
    @@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html index c1cf235..e1c0f6a 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:

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{\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

-
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{\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

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html index ca73c1b..8cc5467 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

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html index 828c8e0..2e24f76 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.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emdm.html b/build/emdm.html index d896018..32de66d 100644 --- a/build/emdm.html +++ b/build/emdm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emdm_Me.html b/build/emdm_Me.html index 5e7b45c..a4e9e97 100644 --- a/build/emdm_Me.html +++ b/build/emdm_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-13 Sun 21:20

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Created: 2022-02-14 Mon 20:35

diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html index c15aaeb..2ccb94c 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

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{\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:

-