4.4 Potential in Unbounded Regions 273
EXERCISES
- Find a formula for the constantsanin Eq. (7).
- Ve r i f y t h a tu 1 (x,y)in the form given in Eq. (7) satisfies the potential equa-
tion and the homogeneous boundary conditions. - Find formulas forA(μ)andB(μ)of Eq. (9).
- Solve the potential equation in the slot, 0<x<a,0<y, for each of these
sets of boundary conditions.
a.u( 0 ,y)=0, u(a,y)=0, 0 <y; u(x, 0 )=1, 0 <x<a;
b.u( 0 ,y)=0, u(a,y)=e−y,0<y; u(x, 0 )=0, 0 <x<a;
c. u( 0 ,y)=f(y)={
1 , 0 <y<b,
0 , b<y, u(a,y)=0,^0 <y;
u(x, 0 )=0, 0 <x<a.- Solve the potential equation in the slot, 0<x<a,0<y, for each of these
sets of boundary conditions.
a. ∂u
∂x( 0 ,y)=0, u(a,y)=0, 0 <y; u(x, 0 )=1, 0 <x<a;b. ∂u
∂x( 0 ,y)=0, u(a,y)=e−y,0<y;
u(x, 0 )=0, 0 <x<a;c. u( 0 ,y)=0, u(a,y)=f(y)={
1 , 0 <y<b,
0 , b<y;
∂u
∂y(x, 0 )=0, 0 <x<a.- Show that if the separation constant had been chosen as−μ^2 instead of
μ^2 in solving foru 2 (leading toY′′−μ^2 Y=0), thenY(y)≡0isthe
only function that satisfies the differential equation, satisfies the condition
Y( 0 )=0, and remains bounded asy→∞. - Solve the problem of potential in a slot under the boundary conditions
u(x, 0 )= 1 , u( 0 ,y)=u(a,y)=e−y. - Show that the functionv(x,y)given here satisfies the potential equation
and the boundary conditions on the “long” sides in Exercise 7, provided