PHYSICS 312. Midterm Examination. February 27, 2000

Answer all three questions.



Problem 1 :
Consider the function

$$f(x)=\sin(x)+\cos(x)$$

defined in the range $0<x<\pi$. Find the function g(x) that constitutes the
a: periodic extension of f(x).
b: odd periodic extension of f(x).
c: even periodic extension of f(x).
in the range $-\pi<x<0$

Problem 2:
Consider the following eigenvalue problem (assume $\lambda >0$)

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

$$\phi(0)+\frac{d\phi}{dx}\vert _{x=0}=0$$

$$\phi(a)+\frac{d\phi}{dx}\vert _{x=a}=0$$

Normally boundary value conditions that involve combinations of the function and its derivatives lead to transcendental equations, but in this case the equations can be solved explicitly.
Find the eigenvalues $\lambda$ and eigenfunctions $\phi$ of the problem!

Problem 3:
Heat is produced uniformly inside a long thin bar at a constant rate per unit length. Heat escapes from the ends but not from the sides of the bar.The thermal conductivity $\kappa$ can in principle depend on the temperature T. The steady state temperature inside the bar satisfies the differential equation

$$\frac{d}{dx}(\kappa(T)\frac{dT}{dx})=Q$$

The ends x=0 and x=L are kept at the temperature T=0 and

$$\kappa(T)=\kappa_0+\alpha T$$

a: When integrating the steady state differential equation you will end up with a quadratic equation for T. Solve this equation to find the steady state temperature distribution $T(x), \;0<x<L$ for
b: $\alpha>0$
c: $\alpha=0$
d: $\alpha<0$
Bonus question: In the last case (d:) you will find that if $\vert\alpha\vert$ is too large there no longer is a real solution in the middle of the bar. Can you explain physically what goes wrong then?