PHYSICS 312. Midterm Examination. February, 2002
Allowed aids formula sheet (2 pages). Calculator.

Answer all three questions.

Problem 1 :
a: Heat is produced inside a sphere of radius $R$ at the rate $H$ ($H=$ energy per unit time and volume). The surface is kept at the temperature $T_0=0$, the thermal conductivity is $\kappa$. Find the steady state temperarure distribution.
b: Answer the same question in the case of a cylinder of radius $R$ length $L$. The ends of the cylinder are thermally insulated from the surroundings.
Problem 2:
a: Find the solution to the wave equation

$$\frac{\partial ^2u}{\partial x^2}=\frac{1}{c^2}\frac{\partial ^2u}{\partial t^2},\;-\infty<x<\infty;\; t>0$$

satisfying

$$u(x,0)=\sin(x)$$

$$\frac{\partial u(x,t)}{\partial t}\vert _{t=0}=0$$

b: Solve the same problem if

$$u(x,0)=\sin(x)$$

$$\frac{\partial u(x,t)}{\partial t}\vert _{t=0}=\cos(x)$$

Problem 3:
Find the three lowest eigenvalues and sketch the eigenfunctions of the problem

$$\phi^{\prime\prime}+\lambda^2\phi=0;\;0<x<a$$

a:

$$\phi^\prime(a)=0;\phi(0)=0$$

b:For which values of $\lambda>1$ will

$$\frac{d^2\phi}{dx^2}+2\frac{d\phi}{dx}+\lambda^2\phi=0$$

have solutions (other than $\phi=0$ satisfying

$$\phi(0)=\phi(a)=0$$