物理代写|PHYS350 Electromagnetism

Prove that Maxwell’s equations with magnetic charges and currents (as given in the first lecture) are invariant under the electric-magnetic duality transformation which rotates the electric and magnetic fields (as well as the corresponding charges and currents) into each other. That is, the equations have exactly the same form in terms of the rotated fields as the original ones. Use these steps. (a) Define rescaled fields and sources in the form
$$
\tilde{E}=\sqrt{\epsilon_0} \vec{E}, \tilde{B}=\vec{B} / \sqrt{\mu_0}, \tilde{\rho}=\rho / \sqrt{\epsilon_0}, \tilde{\rho}_m=\sqrt{\mu_0} \rho_m, \tilde{J}=\vec{J} / \sqrt{\epsilon_0}, \tilde{J}_m=\sqrt{\mu_0} \vec{J}_m
$$
Further define the 6-component vectors $\mathcal{E} \equiv\left(\begin{array}{c}E_B \ \bar{B}\end{array}\right)$ and $\mathcal{J} \equiv\left(\begin{array}{c}\bar{J}, \ \bar{J}_m\end{array}\right)$, the 2-component vector $\mathcal{Q}=\left(\begin{array}{c}\bar{\rho} \ \bar{\rho}_m\end{array}\right)$, the matrix $S \equiv\left(\begin{array}{cc}0 & -1 \ 1 & 0\end{array}\right)$ and $c=1 / \sqrt{\mu_0 \epsilon_0}$. Show that Maxwell’s equations can be expressed in the form
$$
\vec{\nabla} \times \mathcal{E}=\frac{1}{c} S\left(\mathcal{J}+\frac{\partial}{\partial t} \mathcal{E}\right), \quad \vec{\nabla} \cdot \mathcal{E}=\mathcal{Q}
$$
Notice that the $\vec{\nabla}$. and $\vec{\nabla} \times$ operations act only on the spatial indices of each 3 -vector inside the 6-vector, whereas the matrix $S$ acts the same way on each spatial component, and only rotates the $E$ and $B$ fields into each other.
(b) Define rotated fields and sources $\mathcal{E}=R \mathcal{E}^{\prime}, \mathcal{Q}=R \mathcal{Q}^{\prime}$ and $\mathcal{J}=R \mathcal{J}^{\prime}$, where $R$ is the standard $2 \times 2$ rotation matrix, which acts trivially on the spatial indices of the 6-component vectors. Show that Maxwell’s equations have the same form in terms of the primed quantities. It will be useful to establish that $R$ and $S$ commute, and also that $R$ commutes with the differentiation operations (see problem 3 below). Would it still work if the rotation angle was a function of space or time?

(a) Use the result of the previous problem to prove the following proposition: there is no physical difference between the statement that magnetic charges/currents do not exist in nature, versus the statement that all particles in nature have the same ratio of magnetic to electric charge. Specifically, suppose that $\rho_m / \rho=10 / \sqrt{\mu_0 \epsilon_0}$; what rotation angle in the duality tranformation would be needed to get rid of the magnetic charge? Explain why the ratio of magnetic to electric currents should be the same as the corresponding ratio of charges for a given particle.
(b) Consider a magnetic monopole at position $\vec{x}=[0,0, d(t)]$ centered above a circular wire of radius $R$ in the $x-y$ plane. Show that the magnetic flux $\Phi=\int d \vec{S} \cdot \vec{B}$ through the loop is proportional to $\operatorname{sign}(d)-d / \sqrt{R^2+d^2}$. Suppose the monopole passes through the loop at constant speed. By sketching a graph of $\Phi$ and $d \Phi / d t$, show that the EMF (voltage) in the loop is smooth as the monopole passes through it, even though $\Phi$ itself is discontinuous.

(a) Use the Levi-Civita tensor to show that $\vec{\nabla} \times f(r) \hat{r}=0$ for any spherically symmetric function $f$. (b) Use the Levi-Civita tensor to show that $R(\vec{\nabla} \times \vec{V})_i=(\vec{\nabla} \times R \vec{V})_i$ from problem 1, where $i$ (and the $\vec{\rightarrow}$ symbol) refers to the spatial index, not the twocomponent index.

(a) Show that $\delta(r) /\left(4 \pi r^2\right)$ has the right properties to be the 3D Dirac delta function expressed in spherical coordinates. (b) Show that the function $f(r)=c /\left[\left(r^2+\epsilon^2\right)^{3 / 2}\right]$ can provide a representation of the 2D Dirac delta function in the limit that $\epsilon \rightarrow 0$, and find the appropriate value of $c$.