Mathematical Tools for Physics

10—Partial Differential Equations 296

flowingintothis area has the reversed sign,
+κ∂T/∂y (26)

The total power flow is the integral of this over the area of the top face.
LetLbe the length of this long rectangular rod, its extent in thez-direction. The element of area along
the surface aty=bis thendA=Ldx, and the power flow into this face is

∫a

0

Ldxκ

∂T

∂y

∣

∣

∣

∣

y=b

The temperature function is the solution Eq. ( 23 ), so differentiate that equation with respect toy.

∫a

0

Ldxκ

4

π

T 0

∑∞

`=0

[(2`+ 1)π/a]

2 `+ 1

cosh

(

(2`+ 1)πy/a

)

sinh

(

(2`+ 1)πb/a

)sin

(2`+ 1)πx

a

at y=b

=

4 LκT 0

a

∫a

0

dx

∑∞

`=0

sin

(2`+ 1)πx

a

and this sum does not converge.I’m going to push ahead anyway, temporarily pretending that I didn’t notice this
minor difficulty with the series. Just go ahead and integrate the series term by term and hope for the best.

=

4 LκT 0

a

∑∞

`=0

a

π(2`+ 1)

[

−cos

(

(2`+ 1)π

)

+ 1

]

=

4 LκT 0

π

∑∞

`=0

2

2 `+ 1

=∞

This infinite series for the total power entering the top face is infinite. The series doesn’t converge (use the
integral test).
This innocuous-seeming problem is suddenly pathological because it would take an infinite power source
to maintain this temperature difference. Why should that be? Look at the corners. You’re trying to maintain
a non-zero temperature difference (T 0 − 0 ) between two walls that are touching. This can’t happen, and the