Page 370 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 22: Solution of Laplace Differential Equation

Therefore for all points situated on the z-axis, the general form of the potential as contained in Notes
(iii), reduces to

B
n
V = A r n n 1 ...(vii)
n
n 0 r
Comparing this equation with equation (vi) we see that for r > a, the coefficients A = 0 and B are
n n
the coefficients of equation (vi).
Again comparing equation (vii) with (v), we see that for r < a, the coefficients B = 0 and A are
n n
the coefficients of equation (v).
Hence the solution for the case r > a may be written as

M a 1 a 3 1 3 a 5
V = P 0 (cos ) 3 P 2 (cos ) . . 5 P 4 (cos )... ...(viii)
a r 2 r 2 4 r
and that for r < a is

M 1 r 2 1 3
V = P 0 (cos ) P 2 (cos ) . . (cos )... ...(ix)
P
4
a 2 a 2 2 4
Example 2: Electrical Potential about a Spherical Surface
Let us consider a spherical surface which is being kept at a fixed distribution of the electrical
potential of the form
V = f( ) ...(i)
On the surface of the sphere.
Figure 22.4

Let us assume that the space both inside and outside the surface is free of electrical charge and we
will determine the potential at points within and outside the spherical surface under consideration.

Obviously, the potential V is quite symmetric around the z-axis and as such it shall be independent
of angle .

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