PHYSICS 312

PROBLEM SET 2:
Given January 15. Due January 22. 1999.
Problem 1:
Heat is produced uniformly at the rate H (energy per time per volume) inside a sphere of radius c. The thermal conductivity of the sphere is $\kappa$. At the surface of the sphere heat loss per unit surface area to the surroundings is

$$\lambda(u(c)-T)$$

Find the steady state temperature $u(\rho)$ at radius $\rho$ inside the sphere.

Problem 2:
A large object has a spherical hole of radius c.The thermal conductivity of the object is $\kappa$. At the surface of the hole the temperature is 0. Far away from the hole the temperature is T. Find the steady state temperature at a distance r>c from the center of the hole.

Problem 3:
The function f(x) is defined for $0<x,\pi$ as

$$f(x)=\left\{\begin{array}{ll}\frac{1}{2},&0<x<\frac{\pi}{2}\\ 1,&\frac{\pi}{2}<x<\pi\\ \end{array}\right.$$

Find the Fourier series for the

a:

periodic extension (use either the sine/cosine or complex series)

b:

even periodic extension

c:

odd periodic extension.

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