物理代写|PHYS350 Electromagnetism

A hemispherical shell of radius $R$ and charge $Q$ subtends polar angles $0 \leq \cos \theta \leq 1$. Assume the charge is uniformly distributed on the shell. (a) Find the electric field at a position $P$ along the positive $z$ axis by integrating appropriately over the shell. (Hint: show that all charge elements within a concentric ring at fixed $\theta$ contribute equally to $E_z$. Don’t forget to project the field from one of these charge elements onto the $z$ axis.) Show that the result has the form $\vec{E}=\hat{z} f(\epsilon) Q /\left(4 \pi \epsilon_0 P^2\right)$ where $\epsilon=R / P$, and that $f \rightarrow 1$ as $\epsilon \rightarrow 0$. You can use Maple or a table of integrals to evaluate the integral over $\cos \theta$.
(b) Find the electric potential at the same point $P$ using a similar technique. Show that $V=g(\epsilon) Q /\left(4 \pi \epsilon_0 P\right)$ where $g$ has the same property as $f$ in part (b).
(c) Verify that $\vec{E}=-\vec{\nabla} V$ at the point $P$.

A hemispherical shell of radius $R$ and charge $Q$ subtends polar angles $0 \leq \cos \theta \leq 1$. Assume the charge is uniformly distributed on the shell. Calculate the dipole and quadrupole moments. (b) A point charge $Q$ is located at the position $(0,0, a)$. Find its multipole moments up to the octopole. Express your answers using Kronecker delta symbols. Write the octopole moment in such a way that it is completely symmetric under interchange of any of its three indices.

For a tutorial about solving this problem using Maple, see the other side. An electric dipole $\vec{p}$ with mass $m$ and moment of inertia $I$ starts out a distance $z_0$ along the $z$-axis above a point charge $Q$, with initial angle $\theta_0$ between $\vec{p}$ and $\vec{E}$. Assume it starts from rest. (a) Find the coupled equations of motion for $\theta$ and $z$. (b) Use Maple to numerically solve these equations and plot the solutions both $\theta(t)$ and $z(t))$ assuming that $p Q /\left4 \pi \epsilon_0 I\right=p Q /\left(2 \pi \epsilon_0 m\right=z_0=1$ in some units. Plot the solutions for $\theta_0$ close to 0 and to $\pi$ to show the different kinds of qualitative behavior. (c) By trial and error, find the critical value $\theta_c$ of $\theta_0$ such that the dipole just avoids crashing into the charge, to three significant figures. What curious behavior do the solutions display near this value? (d) Notice that for $\theta_0>\theta_c$, the ratio of rotational to translational energy approaches a constant at late times. Plot this ratio as a function of $\theta_0$, and find approximately the value of $\theta_0$ and the value of the ratio such that the latter is maximized. Although you could use Maple to find the rotational to translational energy as a function of $\theta_0$ and plot it directly, it may be faster for you to simply calculate it for a series of $\theta_0$ values and sketch the graph by hand.

The atomic bomb dropped on Nagasaki had a yield of $8.8 \times 10^{13} \mathrm{~J}$ and an approximate radius of $1 \mathrm{~m}$. (a) Find the amount of charge needed on the surface of a sphere of the same size in order for the stored electrostatic energy to be the same. (b) Repeat the calculation when the charge is uniformly distributed in the spherical volume. (Think about building up the distribution one shell at a time.)

Complete the derivation started in lecture to find the torque on an electric quadrupole in a nonuniform electric field. Express it in component notation. Notice that this would have been difficult to do without the Levi-Civita tensor!