Questions

1)

Prove the generalisation of Green's theorem:

$$\int_{\cal V} (v {\cal L} u - u{\cal L} v) d\tau = \oint_{\cal S} p(v\vec{\nabla} u - u\vec{\nabla }v)\cdot d{\bf a} ~.$$

Here ${\cal L}$ is the operator

$${\cal L} = \vec{\nabla} \cdot [ p({\bf r})\vec{\nabla} ] +q({\bf r})$$

and $p,q,u$, and $v$ are scalar functions of position, $p$ and $q$ having continuous first derivatives, and $u$ and $v$ having continuous second derivatives. The RHS integral over ${\cal S}$ denotes the surface that encloses the volume ${\cal V}$, and $d{\bf a}$ is the surface element vector pointing in the normal direction. (25 points)

2)

Consider a uniformly charged circular cylindrical shell having total charge $Q$, radius $R$, and height $h$. Determine the electric field and the electric potential at a point on the axis of the cylinder a distance $d$ from the right side of the cylinder, as shown in Fig. 1. (25 points)

Figure 1: Prob 2

3)

A rectangular pipe, running parallel to the $z$-axis (from $-\infty$ to $+\infty$), has three grounded metal sides, at $y=0$, $y=a$, and $x=0$. The fourth side, at $x=b$, is maintained at a specified potential $V_0$ (constant). Find the electric potential explicitly. (25 points)

4)

A dielectric sphere with permitivity $\epsilon_1$ and radius $R$ is embedded in another dielectric material (of infinite size) with permitivity $\epsilon_2$. Suppose that a point charge of $q$ is embedded at the centre of the sphere and that there is no other embedded (free) charge. Find the electric displacement, the electric field, and the electric potential as a function of radius $r$ from the centre of the sphere, and calculate the bound surface charge density at $r=R$. Assume that both dielectric media have linear response. See Fig. 2. (25 points)

Figure 2: Prob 4