**MY-ASSIGNMENTEXPERT™**可以为您提供 wgtn.ac PHYS415 Electromagnetism电磁学的代写代考和辅导服务！

**MY-ASSIGNMENTEXPERT™**

#### 这是惠靈頓維多利亞大學的电磁学课程的代写成功案例。

**PHYS415**课程简介

This course is available to take both fully online and in person (dual mode).Lectures for this course will be delivered in person and recordings of these will be available for students who need to attend remotely. Recordings of lectures will be available for preparation for assessment.This is an advanced course in classical electromagnetism following from PHYS 304. The course covers a selection of topics from

- Interactions of fields and charged particles
- Kinetic description of plasmas
- Deduction of fluid models from kinetic models
- Implications of field-particle dynamics for space physics
- Maxwell’s equations in media
- Conservation laws
- Electromagnetic waves in media, polarization, reflection, absorption and dispersion
- Guided waves
- Anisotropic and nonlinear constitutive relations

## Prerequisites

The class representative provides a useful way to communicate feedback to the teaching staff during the course. They also work with the VUWSA Education Office on any academic issues that arise in their course. Reps are elected by students by the third week of classes every trimester. Being a rep requires a weekly commitment.

**PHYS415** **Electromagnetism** HELP（EXAM HELP， ONLINE TUTOR）

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.

** MY-ASSIGNMENTEXPERT™**可以为您提供 wgtn.ac PHYS415 Electromagnetism电磁学的代写代考和辅导服务