Exercises — Maxwell's equations — integral form, all four

(a) Gauss's Law for Electricity — closed surface, enclosed charge. See Gauss's Law. (b) Gauss's Law for Magnetism ($\oint \vec{B}\cdot d\vec{A}=0$, no monopoles). (c) Faraday's Law — changing magnetic flux makes circulating $\vec{E}$. See Faraday's Law and Lenz's Law. (d) Ampère–Maxwell Law — the current term is zero in the gap, so the changing-electric-flux term does the work (the Displacement Current).

Gauss: $\Phi =\frac{Q_{enc}}{{\epsilon }_0}=\frac{3.0\times 10^{-6}}{8.85\times 10^{-12}}=3.39\times 10^5\text{N}\cdot {\text{m}}^2/\text{C}$. The outside charge changes nothing: its lines enter and exit the closed surface, net contribution zero. Only enclosed charge counts.

Why a cylinder? By symmetry $\vec{E}$ points straight out (radially) and has the same size everywhere on the curved wall; on the flat caps $\vec{E}$ is parallel to the surface so its flux is zero. So all flux comes through the curved side of area $2\pi rL$. $\oint \vec{E}\cdot d\vec{A}=E(2\pi rL)=\frac{Q_{enc}}{{\epsilon }_0}=\frac{\lambda L}{{\epsilon }_0}.$ The length $L$ cancels — good, an infinite line has no special length: $E=\frac{\lambda }{2\pi {\epsilon }_{0}r}=\frac{2.0\times 10^{-8}}{2\pi (8.85\times 10^{-12})(0.050)}=7.19\times 10^3\text{N/C}.$

Flux ${\Phi }_B=B\cdot (\pi R^2)$ (field $\perp$ loop, so full area counts). $|\text{EMF}|=\left|\frac{d{\Phi }_B}{dt}\right|=\pi R^2\frac{dB}{dt}=\pi (0.10{)}^2(0.30)=9.42\times 10^{-3}\text{V}.$ Only $B$ changes here, area is fixed — that's why $\pi R^2$ comes out front.

With plate area $A$, field $E=Q/({\epsilon }_{0}A)$, so ${\Phi }_E=EA=Q/{\epsilon }_0$. Then $I_d={\epsilon }_0\frac{d{\Phi }_E}{dt}={\epsilon }_0\cdot \frac{1}{{\epsilon }_0}\frac{dQ}{dt}=\frac{dQ}{dt}=4.0\text{A}.$ The displacement current equals the wire current exactly — current is continuous through the gap.

Between the plates there is no conduction current, so use the displacement-current term. The displacement current is spread uniformly over the plate area, so the fraction enclosed by a loop of radius $r$ is $\frac{\pi r^2}{\pi R^2}=\frac{r^2}{R^2}$. Ampère–Maxwell around the circular loop (where $\vec{B}$ is tangential and constant): $B(2\pi r)={\mu }_0{I}_{d,\text{enc}}={\mu }_{0}I\frac{r^2}{R^2}.$ $B=\frac{{\mu }_{0}Ir}{2\pi R^2}=\frac{(4\pi \times 10^{-7})(4.0)(0.010)}{2\pi (0.020{)}^2}=2.0\times 10^{-5}\text{T}.$

${\Phi }_B=B{s}^2$. Only the area changes, so by the chain rule $\text{EMF}=-\frac{d}{dt}(B{s}^2)=-2Bs\frac{ds}{dt}=-2(0.50)(0.20)(-0.10)=+0.020\text{V}.$ Magnitude $=0.020\text{V}$. Because $ds/dt<0$ (shrinking), the two minus signs make the EMF positive, meaning the induced current fights the decreasing out-of-page flux by trying to add out-of-page flux (counterclockwise current, by Lenz).

(a) Total displacement current uses the whole field and whole area: $I_d={\epsilon }_0\frac{d{\Phi }_E}{dt}={\epsilon }_0(\pi R^2)\frac{dE}{dt}=(8.85\times 10^{-12})\pi (0.020{)}^2(1.0\times 10^{12})=1.11\times 10^{-2}\text{A}.$ (b) At $r=R$ the loop encloses all of $I_d$: $B=\frac{{\mu }_0{I}_d}{2\pi R}=\frac{(4\pi \times 10^{-7})(1.11\times 10^{-2})}{2\pi (0.020)}=1.11\times 10^{-7}\text{T}.$ This tiny $\vec{B}$, made purely by a changing $\vec{E}$ with zero moving charge, is exactly the ingredient that lets Electromagnetic Waves exist.

Ampère–Maxwell: a changing $\vec{E}$ (along $y$) curls $\vec{B}$ around it. For the wave to carry energy in $+x$, the field pattern requires $\vec{B}$ along $\pm z$. Faraday: that changing $\vec{B}$ (along $z$) must in turn curl an $\vec{E}$ back along $y$. The two laws close a self-consistent loop only if $\vec{E}\times \vec{B}$ points along $+x$ (the travel direction) — so $\vec{E}$ along $+y$ forces $\vec{B}$ along $+z$. Consistency check via the wave condition: both laws are satisfied simultaneously when $E=cB$ and $\vec{E}\perp \vec{B}\perp \hat{x}$. No contradiction — they are the two halves of the same wave.

Faraday says a wiggling $\vec{B}$ makes a circulating $\vec{E}$; Ampère–Maxwell says a wiggling $\vec{E}$ makes a circulating $\vec{B}$. Feeding one into the other for a plane wave produces a wave equation whose speed is set entirely by the coupling constants: $c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}.$ Numerically: $c=\frac{1}{\sqrt{(4\pi \times 10^{-7})(8.85\times 10^{-12})}}=3.00\times 10^8\text{m/s}.$ This matches the measured speed of light — the punchline that told Maxwell light is an electromagnetic wave. See Electromagnetic Waves.

Interior field: $B={\mu }_{0}nI$, so $\frac{dB}{dt}={\mu }_{0}n\frac{dI}{dt}$. Flux through the loop: ${\Phi }_B=B\pi r^2$ (uniform field inside). $|\text{EMF}|=\pi r^2\frac{dB}{dt}=\pi r^2{\mu }_{0}n\frac{dI}{dt}.$ $=\pi (0.030{)}^2(4\pi \times 10^{-7})(1000)(50)=1.78\times 10^{-4}\text{V}.$

$B=\frac{E}{c}=\frac{300}{3.00\times 10^8}=1.0\times 10^{-6}\text{T}.$ Note $\vec{B}$ is tiny compared to $\vec{E}$ in SI units — that's just the factor $c$, not a sign that magnetism is "weaker." Both halves carry equal energy in the wave.

Which law's derivation gives $c=1/\sqrt{{\mu }_0{\epsilon }_0}$? ::: The combination of Faraday and Ampère–Maxwell — the two "changing-field" laws feeding each other.