Page 369 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
Figure 22.3

where A and B are constant coefficients and are to be evaluated. To evaluate these coefficients,
n n
we know that the gravitational potential is symmetric about the z-axis and therefore any point
P on the same distance (a 2 r 2 ) from all the points of the ring, where a is the radius of the ring
and distance OP = r.

Let M denote the total mass of the ring, then the gravitational potential at P due to the ring will
be

mass M
V = ...(iv)
distance (a 2 r 2 )

M 2 2 1/2 M r 2 1/2
but = M (a r ) 1 2
(a 2 r 2 ) a a

M r 2 1 3 r 4
or V = 1 2 . . 4 ... ...(v)
a 2a 2 4 a
by Binomial theorem for r < a

However in case r > a, we can write

1/2
M 2 2 1/2 M a 2
= M (a r ) 1 2
(a 2 r 2 ) r r

M 1 a 2 1 3 a 4
= 1 2 . 4 ...
r 2 r 2 4 r

M a 1 a 3 1 3 a 5
or V = 3 . 5 ... ...(vi)
a r 2 r 2 4 r
Now, for point situated on the z-axis, = 0 and the general solution as contained in equation (iii)
must reduce either to equation (v) or equation (vi). Now the Legendre polynomials P (cos ) for
n
a point on the z-axis (cos 0°) become
P (cos0 ) = P (1) 1
n n

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