物理代写|PHYS415 Electromagnetism

Prove that the torque on an arbitrarily shaped loop of current in a constant external magnetic field can be written as
$$
\vec{\tau}=I \oint d \vec{l}(\vec{r} \cdot \vec{B})
$$
You will first need to reduce the double cross product in $\vec{r} \times d \vec{F}=\vec{r} \times(I d \vec{l} \times \vec{B})$ using the epsilon tensor (see eq. (2.27) of textbook). Use Stoke’s theorem to prove that the extra term you get in addition to the one above vanishes.
(b) By considering $\oint d \vec{l} \cdot \vec{\nabla}\left(x_i x_j\right)$, prove that $\oint d l_i x_j=-\oint d l_j x_i$. Therefore the vector $\epsilon_{k i j} \oint d l_i x_j$ contains the same information as the tensor $\oint d l_i x_j$.
(c) Show that $\frac{1}{2} \epsilon_{k i j} d l_i x_j$ is the element of area of a triangle in the plane of the loop, bounded by $\overrightarrow{d l}$ and the radial lines which are projections onto the plane of the loop of the vectors $\vec{x}$ which go from the origin to the two endpoints of $\overrightarrow{d l}$, as shown in the figure. Show that the direction of $d A_k=\frac{1}{2} \epsilon_{k i j} d l_i x_j$ is normal to the element of area.

(a) Find the vector potential for two oppositely oriented point dipoles, $\vec{m}$ located at $\vec{r}=\vec{a}$, and $-\vec{m}$ located at $\vec{r}=-\vec{a}$. Expand your result to the leading order in powers of $1 / r$ to get just the quadrupole contribution to $\vec{A}$.
(b) Show that your result can be expressed in the form $\epsilon_{i j k} m_j Q_{k l}(r) a_l$, where $Q_{k l}(r)$ has the same tensor structure as for an electric quadrupole.
(c) Prove that $\vec{\nabla} \cdot \vec{A}=0$, i.e., $\vec{A}$ is in the Coulomb gauge. Use the Levi-Civita tensor.
(d) Calculate $\vec{B}$ for the quadrupole field, as a linear combination of $\vec{m}, \vec{r}$ and $\vec{a}$, again using the Levi-Civita tensor. An exercise in keeping track of indices!

(a) Estimate the magnetic susceptibility of antimony (Sb), a diamagnetic material, using the atomic radius $0.14 \mathrm{~nm}$, atomic weight 122 , and mass density $6.7 \mathrm{~g} / \mathrm{cm}^3$.
(b) Calculate $|\Delta \vec{m}| /|\vec{m}|$ for a $1 \mathrm{~T}$ applied field, assuming $|\vec{m}|=1 \mu_B$. (Note the estimate given in lecture was smaller than this, due to a calculational error.)
(c) Consider a $1 \mathrm{~g}$ needle of $\mathrm{Sb}$ which is $1 \mathrm{~cm}$ in length, in a nonuniform magnetic field $B \hat{z}$ with $d \vec{B} / d z= \pm 1 \hat{z} \mathrm{~T} / \mathrm{cm}$ at positions near the top and the bottom of the needle, respectively. Estimate the torque on the needle based on this information. Take it to be oriented at a $45^{\circ}$ angle relative to $\vec{B}$, and pretend the mass is concentrated near the ends of the needle.

Find the vector potential $\vec{A}$ in the $x-y$ plane for a wire carrying current $I$ along the $z$ axis. Assume the wire goes from $z=-L$ to $+L$. As $L \rightarrow \infty$, the answer blows up, but the divergent part is just a constant. Show that the part which depends upon $x$ and $y$ is finite as $L \rightarrow \infty$, and that $\vec{\nabla} \times \vec{A}$ gives the known result for $\vec{B}$ around a long straight wire.