Update 2022-03-02 15:47

2022-03-02 15:47:54 +01:00
parent ac1e628013
commit 21bf9fdba5
194 changed files with 1653 additions and 1216 deletions
@@ -1,7 +1,7 @@
 <!DOCTYPE html>
 <html lang="en">
 <head>
-<!-- 2022-03-01 Tue 08:14 -->
+<!-- 2022-03-02 Wed 15:45 -->
 <meta charset="utf-8">
 <meta name="viewport" content="width=device-width, initial-scale=1">
 <title>Pre-Quantum Electrodynamics</title>
@@ -1628,7 +1628,7 @@ Two sources of electric fields:  electric charges, and changing magnetic fields.
 <p>
 Electric fields induced by a changing magnetic field are determined in an exactly
 parallel way as magnetostatic fields from the current:  exploit parallel
-between Ampère and Faraday!
+between Ampère and Faraday
 \[
 {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}
 \hspace{3cm}
@@ -1645,42 +1645,60 @@ law in integral form:



-<div class="example div" id="org3b5285b">
+<div class="example div" id="org045463b">
 <p>
-{\bf Example 7.7:}
-\({\bf B}(t)\) points up in circular region of radius \(R\).  What is the induced \({\bf E}(t)\) ?
-\paragraph{Solution:}
+<b>Example: loop with time-dependent flux</b>
+</p>
+
+<p>
+Consider a time-dependent magnetic field \({\bf B}(t)\) directed vertically
+through a horizontal circular region of radius \(R\).
+</p>
+
+<p>
+<b>Task</b>: find the induced \({\bf E}(t)\).
+</p>
+
+<p>
+<b>Solution</b>:
 amperian loop of radius \(s\), apply Faraday:
 \[
-    \oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt}
-    \Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
-  \]
+\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 \varphi}.
+\]
 Increasing \({\bf B}\):  clockwise (viewed from above) \({\bf E}\) from Lenz.
 </p>

 </div>


-<div class="example div" id="org68f8620">
+<div class="example div" id="org047ea10">
 <p>
-{\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 \(a &lt; b\),
-pointing up.  Field turned off.  What happens ?
-\paragraph{Solution:}  the wheel starts spinning to compensate the reduction of field.
+<b>Example: wheel with charged rim traversed by flux</b>
+</p>
+
+<p>
+Consider a wheel of radius \(b\) with line charge \(\lambda\) on the rim.
+A uniform magnetic field \({\bf B}_0\) pointing up is traversing the central region
+up to radius \(a &lt; b\). The field is then turned off.  What happens?
+</p>
+
+<p>
+<b>Solution</b>:  the wheel starts spinning to compensate the reduction of field.
 Faraday:
 \[
-    \oint {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt} = - \pi a^2 \frac{dB}{dt}
-    \Rightarrow {\bf E} = -\frac{a^2}{2b} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
-  \]
+\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 \varphi}.
+\]
 Torque on segment \(d{\bf l}\):  \(|{\bf r} \times {\bf F}| = b \lambda E dl\).
 Total torque:
 \[
-    N = b\lambda \oint E dl = -b \lambda \pi a^2 \frac{dB}{dt}
-  \]
-so total angular momentum imparted is
+N = b\lambda \oint E dl = -b \lambda \pi a^2 \frac{dB}{dt}
+\]
+so total angular momentum imparted to the wheel is
 \[
-    \int N dt = -\lambda \pi a^2 b \int_{B_0}^0 dB = \lambda \pi a^2 b B_0.
-  \]
+\int N dt = -\lambda \pi a^2 b \int_{B_0}^0 dB = \lambda \pi a^2 b B_0.
+\]
 </p>

 </div>
@@ -1690,33 +1708,44 @@ The precise way the field is turned off doesn't matter.  Only electric field doe
 </p>

 <p>
-{\bf N.B.:}  we use magnetostatic formulas for changing fields.  This is
-called the {\bf quasistatic} approximation, and works provided we deal with
-'slow enough' phenomena.
+<b>N.B.</b>:  we use magnetostatic formulas for changing fields.  This is
+called the <b>quasistatic</b> approximation, and works provided we deal with
+<i>slow enough</i> phenomena.
 </p>

-<div class="example div" id="org5b14490">
+<div class="example div" id="org733cbdd">
 <p>
-{\bf Example 7.9:}  infinitely long straight wire carries \(I(t)\).  Find
-induced \({\bf E}\) field as a function of distance \(s\) from wire.
-\paragraph{Solution:}  quasistatic:  magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\)
+<b>Example: field from wire with time-dependent current</b>
+</p>
+
+<p>
+Consider an infinitely long straight wire which carries current \(I(t)\).
+</p>
+
+<p>
+<b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(s\) from wire.
+</p>
+
+<p>
+<b>Solution</b>:  assuming we can use the quasistatic approximation, the
+magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\)
 and circles the wire.  Like \({\bf B}\) field of solenoid, \({\bf E}\) runs parallel
 to wire.  Amperian loop with sides at distances \(s_0\) and \(s\):
 \[
-    \oint {\bf E} \cdot d{\bf l} = E(s_0)l - E(s)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
-    = -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{s_0}^s \frac{ds'}{s'}
-    = -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(s/s_0).
-  \]
+\oint {\bf E} \cdot d{\bf l} = E(s_0)l - E(s)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
+= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{s_0}^s \frac{ds'}{s'}
+= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(s/s_0).
+\]
 So:
 \[
-    {\bf E} (s) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln s + K \right] \hat{\bf x}
-    \label{Gr(7.19)}
-  \]
+{\bf E} (s) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln s + K \right] \hat{\bf x}
+\label{Gr(7.19)}
+\]
 where \(K\) is a constant (depends on the history of \(I(t)\)).
 </p>

 <p>
-{\bf N.B.:}  this can't be true always, since it blows up as \(s \rightarrow \infty\).
+<b>N.B.</b>: this can't be true always, since it blows up as \(s \rightarrow \infty\).
 Reason:  in this case, we've overstepped the quasistatic limit.  We need
 \(s \ll c\tau\) where \(\tau\) is a typical time scale for change of \(I(t)\).
 </p>
@@ -1743,7 +1772,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
 <p class="validation"></p>
 </div>