Update 2022-03-02 15:47
This commit is contained in:
Jean-Sébastien
+95
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@@ -1,7 +1,7 @@
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-03-01 Tue 08:14 -->
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<!-- 2022-03-02 Wed 15:45 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@@ -1622,15 +1622,11 @@ Table of contents
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</svg></a><span class="headline-id">emd.Fl.Fl</span></h4>
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<div class="outline-text-4" id="text-emd_Fl_Fl">
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<p>
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1831: 3 experiments by Faraday (according to Griffiths! but it's historically incorrect)
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\paragraph{1)} Pull a loop of wire through a magnetic field.
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\paragraph{2)} Move magnet around a still loop.
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\paragraph{3)} Change strength of field, holding magnet and loop still.
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Around 1831, Faraday performed a number of experiments pertaining to
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the effects of time-dependent fields.
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</p>
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<p>
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Actually, historically, things didn't happen like that.
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The first experiment that Faraday performed (1831) involved two metal coils wound
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on opposite sides of a metal ring. When a current was turned on through the first
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coil, it generated a transient current in the second coil (as measured by a
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@@ -1647,27 +1643,39 @@ on this idea. Faraday observed transient current in a circuit when:
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<p>
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Faraday's big insight was to summarize these effects by noticing that
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</p>
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<p>
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\[
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\boxed{
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\mbox{\bf A changing magnetic field induces an electric field}
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}
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\boxed{
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\mbox{A changing magnetic field induces an electric field.}
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}
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\]
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</p>
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<p>
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Empirically: the changing magnetic field induces an electric current around
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the circuit. This current is really driven by an electric field having a component
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along the wire. The line integral of this field is called the
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</p>
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<div class="core div" id="org03c55ba">
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<div class="core div" id="orgdfb0aad">
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<p>
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<b>Electromotive force (or electromotance)</b>,
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\[
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{\cal E} \equiv \oint_{\cal P} {\bf E} \cdot d{\bf l}.
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\]
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</p>
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<div class="eqlabel" id="org64afdaf">
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<p>
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<a id="elmofo"></a><a href="./emd_Fl_Fl.html#elmofo"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgb3cda90">
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<ul class="org-ul">
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<li>Gr (7.9)</li>
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</ul>
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</div>
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</div>
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<p>
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\[
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{\cal E} \equiv \oint_{\cal P} {\bf E} \cdot d{\bf l}.
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\tag{elmofo}\label{elmofo}
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\]
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</p>
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</div>
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@@ -1676,21 +1684,56 @@ You can think of the emf in different ways. It's the energy accumulated as a uni
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</p>
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<p>
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The precise statement is that the electromotive force is proportional
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The precise statement associated to Faraday's observations
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is that the electromotive force is proportional
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to the rate of change of the magnetic flux,
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</p>
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<div class="eqlabel" id="orgec7b520">
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<p>
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<a id="Fl_flux"></a><a href="./emd_Fl_Fl.html#Fl_flux"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgf70f495">
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<ul class="org-ul">
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<li>Gr (7.14)</li>
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</ul>
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</div>
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</div>
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<p>
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\[
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{\cal E} = \oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt}
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\label{Gr(7.14)}
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\tag{Fl_flux}\label{Fl_flux}
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\]
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so we obtain
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</p>
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<div class="core div" id="org93f9990">
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<div class="core div" id="orgdfedc05">
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<p>
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<b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
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\[
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\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\int_{\cal S} \frac{\partial {\bf B}}{\partial t} \cdot d{\bf a}
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\label{Gr(7.15)}
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\]
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</p>
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<div class="eqlabel" id="orge87df83">
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<p>
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<a id="Fl_int"></a><a href="./emd_Fl_Fl.html#Fl_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org7b32fe7">
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<ul class="org-ul">
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<li>Gr (7.15)</li>
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</ul>
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</div>
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</div>
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<p>
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\[
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\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\int_{\cal S} \frac{\partial {\bf B}}{\partial t} \cdot d{\bf a}
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\tag{Fl_int}\label{Fl_int}
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\]
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</p>
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</div>
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@@ -1702,20 +1745,36 @@ for any loop (on a wire or not). Using Stokes' theorem,
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\]
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we obtain
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</p>
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<div class="core div" id="org6046a76">
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<div class="core div" id="orgafa0d15">
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<div class="eqlabel" id="org13d3c14">
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<p>
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<a id="Fl"></a><a href="./emd_Fl_Fl.html#Fl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgd662a28">
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<ul class="org-ul">
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<li>Gr (7.16)</li>
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</ul>
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</div>
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</div>
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<p>
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<b>Faraday's law</b> (differential form)
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\[
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{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}
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\label{Gr(7.16)}
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\]
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\[
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{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}
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\tag{Fl}\label{Fl}
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\]
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</p>
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</div>
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<p>
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Right-hand rule always sorts signs out. Easier rule: {\bf Lenz's law}, which
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states that {\bf nature resists a change in flux}. This is in fact just
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{\bf Le Ch\^atelier's principle} of any action at an equilibrium point leading
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Right-hand rule always sorts signs out. Easier rule: <b>Lenz's law</b>, which
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states that physical systems naturally resist a change in flux.
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This is in fact just
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<b>Le Châtelier's principle</b> of any action at an equilibrium point leading
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to an opposing counter-reaction.
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</p>
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</div>
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@@ -1739,7 +1798,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-01 Tue 08:14</p>
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<p class="date">Created: 2022-03-02 Wed 15:45</p>
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<p class="validation"></p>
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</div>
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