Gravitational
Field of an Infinite Sheet of Mass
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Problem: Find the gravitational potential due to an infinite plane. Let the plane be the z = plane and let the surface density of mass be s.
Solution:
To find the gravitational potential of an infinite plane first find the
gravitational potential above a disk and then let the radius of the disk become
infinite. See Fig. 1 below.
The gravitational potential for a mass distribution defined by r is given by
The
volume integral reduces to a surface integral by substituting rdV
= sds.
Thus Eq. (1) becomes
As
can be seen from Fig. 1 R has the value
Also
from Fig. 1 it is readily seen that ds = r dr dq. Therefore
for the region z > 0 Eq. (2) becomes
The subscript denotes that the potential applies to a disc or radius a. Change variables by making the substitution a = r/z to obtain
Substitute
the integral
into
Eq. (5) to obtain
Eq.
(7) holds only along the z-axis and does not apply to points off the z-axis.
However this will make no difference in the end result since the sheet will be
infinite. Eq. (7) cannot be used to obtain the potential for an infinite sheet
because the assumption in Eq. (1) is that the gravitational potential at
infinity is zero. For an infinite sheet this assumption is no longer valid.
However we can always add a constant, C, to Eq. (7) without affecting the
physical meaning of the equation. Eq. (7) then becomes
We choose C such that F(0) = 0. Therefore
Solving
for C gives
Eq.
(8) then becomes
We are now ready to obtain the potential F(z) for an infinite sheet. Simple let a go to infinity, i.e.
First let a be much larger than z. F(z) may be approximated then as
Canceling
terms gives an expression that does not depend on the radius of the disk.
Therefore in the limit we have the exact relation
