PHYSICS 312
Problem set 5:

Given March 8 1999
Due March 17 1999
Problem 1:
A thin metal ring is thermally insulated from its surroundings. The radius of the ring is L meter and its thermal diffusivity is k meter²sec^-1. Assume that the temperature of the ring is $T(\theta,t)$.

a:

Assume the initial temperature is $T(\theta,0)=f(\theta)$. Find the subsequent temperature distribution.

b:

Assume that

$$f(\theta)=T_0\cos\theta$$

How long will it take for the temperature difference between the hottest and coldest spot on the ring to halve?

Problem 2:
Solve the 2-dimensional Laplace equation

$$\nabla^2u(r,\theta)=0$$

in polar coordinates in the region a<r<2a with boundary conditions

$$u(a,\theta)=\cos\theta;\;u(2a,\theta)=\cos\theta$$

Problem 3:
Solve

$$\nabla^2u=1$$

inside a sphere of radius 1. The boundary condition is

$$u(1,\theta,\phi)=0$$

Hint: Assume that u is independent of $\theta$ and $\phi$. It is possible to find a particular solution proportional to r². The solution should be finite at the origin.

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