Page 372 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 22: Solution of Laplace Differential Equation

Potential in Region within the Spherical Surface Notes

The potential within the spherical surface cannot be infinite and therefore negative powers of r
are inadmissible in the general solution as contained in equation (iv). This means that potential
inside spherical surface will be

n
V = A r P n (cos ) for r a ...(x)
n
n 0
Again the coefficients A are determined by the boundary condition at the surface, viz., V = f( )
n
at r = a
V = F ( ) f (cos )

n
= A a P (cos ) ...(xi)
n n
n 0
Let u cos , then

n
u
u
V = F ( ) A a P n ( ) ...(xii)
n
n 0
multiplying both sides by P n ( ) and integrating within the limits 1 to +1, we get
u
1 1
2
u
u
u
F ( ) P n ( )du = A a n [P n ( )] du
n
1 1
u
All other coefficients vanish on account of the orthogonal property of P n ( )
1 n 2
F ( )P n ( )du = A a (2n 1)
u
u
n
1
1
(2n 1)
or A n = n F ( ) P n ( )du
u
u
2a
1
1
(2n 1)
or A n = 2a n F ( ) P n (cos )sin d ...(xiii)
1
So the potential within the spherical surface is given by equation (xi) or (xii) with values of A
n
given by the equation (xiii).
Self Assessment

2. Solve

2 u
r 0
r r
subject to the boundary conditions

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