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Question
Suppose the gravitational potential due to a small system is k/r2 at a distance r from it. What will be the gravitational field? Can you think of any such system? What happens if there were negative masses?
Solution
The gravitational potential due to the system is given as \[V = \frac{k}{r^2}\]
Gravitational field due to the system :
\[E = - \frac{dV}{dr}\]
\[ \Rightarrow E = - \frac{d}{dr}\left( \frac{k}{r^2} \right) = - \left( - \frac{2k}{r^3} \right)\]
\[ \Rightarrow E = \frac{2k}{r^3}\]
We can see that for this system , \[E \propto \frac{1}{r^3}\]
This type of system is not possible because \[F_g\] is always proportional to inverse of square of distance(experimental fact).
If there were negative masses, then this type of system is possible.
This system is a dipole of two masses, i.e., two masses, one positive and the other negative, separated by a small distance.
In this case, the gradational field due to the dipole is proportional to \[\frac{1}{r^3}\]
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