Two charges each are separated by distance 2d. A third charge
is kept at midpoint O. Find potential energy of
as a function of small distance x from O due to
charges. Sketch PE versus x and convince yourself that the charge at O is in an unstable equilibrium.
In the above figure, charge is at a point away from O towards
.
This can be written as
At x=0;
If we differentiate both sides of the equation with respect to x, we get
when
and when
With the help of this two, we can assume that the charge on the particle to be
Hence,
When
a) = positive, equilibrium is stable
b) = negative, equilibrium is unstable
c) , equilibrium is neutral
Therefore, when
Which shows that the system is an unstable equilibrium.