Gravitational Field of an
Infinitely Long Rod
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Consider
an infinitely long rod lying on the x-axis. Find the gravitational potential F(r)
at a distance r perpendicular to the rod as shown below in Fig. 1
Solution:
The approach used will be to find the gravitational potential for a rod of
finite length off the midpoint of the rod and then let the rod become infinitely
long. The gravitational potential is defined as
Implicit
in Eq. (1) is that the potential goes to zero as r goes to infinity.
However when the source is infinite this is no longer a valid assumption. A
straightforward calculation shows that the integral in Eq. (1) diverges. Since
the gravitational potential is defined up to an arbitrary constant we can choose
that constant, C, as we see fit. We therefore choose C such that
the potential vanishes at a distance r0 off axis in
the middle of the rod, as shown above in Fig. 1. We therefore add a constant C
such that F(r0)
= 0. C will at most be a function of the length of the rod and the value r0 and
therefore a function of both r0
and a. The value F
along the r axis is then given by
The “a” subscript indicates that the potential is off the middle of a rod of, each end of which is a distance “a” away from the origin as shown in Fig. 1. The gravitational field of the infinitely long rod is then given by
Assuming a linear mass density l we can replace rdV with ldx. R is the distance from source to field point and from the diagram above can readily be seen to have the value
Eq. (2)
now becomes
The
integral in Eq. (2) has the value
We may
now evaluate Fa(r).
We find
Now
determine C by imposing the boundary condition
Substitute
C into Eq. (7) to obtain
Since
we want to take the limit as the length of the rod goes to infinity we can
assume that r is much smaller than a and thus make the
approximation
With
this Eq. (10) becomes
We
may now take the limit in Eq. (3) to obtain
which is
the desired result.