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Supplementary Problems

(b) two long conducting concentric cylinders on each of which the voltage distri-
bution is specified;

(c) two long conducting concentric cylinders on each of which the charge distri-
bution is specified;
(d) a semi-infinite string the end of which is made to move in a prescribed way.

Exercise 102. Suppose that heat is flowing in a uniform rod of cross section a
and perimeter p, and that it is assumed that the temperature T does not vary over
a cross section,and hence is a function only of time t and distance x measured along

the rod. Assume also that heat escapes from the lateral boundary by radiation, in
such away that the rate of heat loss per unit of area is µK(T −T ), where K is the
0
conductivity of the rod material, T the temperature of the surrounding medium,
0
and µ is a constant.
(a) By considering differential thermal equilibrium in an element (x, x+dx) of the

rod, show that there must followo

∂ ∂T ∂T
(K )adx − µK(T − T )pdx = sρa dx,
0
∂x |partialx ∂t

and hence deduce that T then must satisfy the equation

∂ ∂T p
(K ) = sρ + µK (T − T ).
0
∂x ∂x a
(b) For a rod of circular cross section, of diameter d and of uniform conductivity
K, show that T(x, t) must satisfy the equation

2
∂ T 1 ∂T 4µ
= + (T − T ),
0
2
∂x 2 α ∂t d
2
where α = K/sρ.
(c) Verify that, if T is assumed to remain constant, the substitution T(x, t) =
0
2
2
1 ∂U
T + U(x, t)e −4µα t/d leads to the normal heat flow equation ∂ U = α ∂t .
0
2
2
∂x

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