物理代写|PHYS350 Electromagnetism

Show that the Green’s function for the 1-D Laplacian operator $\frac{d^2}{d x^2}$ is proportional to $\left|x-x^{\prime}\right|$. Find the proportionality constant. This could be useful in electrodynamics in a case where the charge density depends only on $x$ and not $y$ or $z$.

(a) Solve the $2 \mathrm{D}$ Laplace equation for the region $x \in[0, a], y \in[0, a]$ with boundary conditions that $V=0$ on all the sides except at $y=a$; there $V(x)=-E x$ for $x \leq a / 2$ and $-E(a-x)$ for $x \geq a / 2$. You may use Maple to do the integrals for the Fourier coefficients if you wish (and it would be a good idea to check your result with Maple even if you do it another way). (b) Use “plot3d” in Maple to plot the potential (use the “axes=boxed” option so that axes will be drawn), and verify that your solution obeys the boundary condition. How many terms in the Fourier series seem to be needed to get good convergence? How does the potential look when extended to the region $x \in[0,2 a]$ ? (c) Use “with(plots)” and “contourplot” to plot the equipotential surfaces. Sketch the result, and show also the electric field lines.

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.

Suppose you have solved the above problem for an abitrary potential $V_0(x)$ on the boundary at $y=a$, to get the potential $V(x, y)$. Show how the solution for the following problems can be expressed in terms of the above solution $V(x, y)$ : (a) $V=0$ on all boundaries except at $y=0$, where the value is $V_0(x)$; (b) $V=0$ at all boundaries except at $x=a$, where $V=V_0(y)$; (c) $V=V_0(y)$ at $x=0$ and vanishes at the other boundaries. (d) Using these results and the principle of superposition, find the solution for problem 1 when the function $V_0$ describes all four of the boundaries. Use Maple to plot $V$ again.

Do problem 5.15 , except assume there are 4 conductors, alternating between $V_0$ and $-V_0$, with the positive ones for $\phi \in[0, \pi / 2]$ and $[\pi, 3 \pi / 2]$, while the negative ones are at $\phi \in[\pi / 2, \pi],[3 \pi / 2,2 \pi]$. Make sure any coefficients of the Fourier series which vanish are so recognized. Hint: note that $\cos (n \pi)=(-1)^n$; show that $\sin (3 n \pi / 2)=(-1)^n \sin (n \pi / 2)$ and $\cos (3 n \pi / 2)=(-1)^n \cos (n \pi / 2)$.

Find the potential (for both $rR$ ) for the case where $V=V_0 \cos (\theta)^4$ on a spherical surface of radius $R$. Hint: $\cos (\theta)^4 \operatorname{can}$ be expressed as a linear combination of Legendre polynomials.