Question
1.32:
(a) Consider an
arbitrary electrostatic field configuration. A small test charge is placed at a
null point (i.e., where E = 0) of the
configuration. Show that the equilibrium of the test charge is necessarily
unstable.
(b) Verify this result for the simple
configuration of two charges of the same magnitude and sign placed a certain
distance apart.
Answer :
(a) Let the equilibrium
of the test charge be stable. If a test charge is in equilibrium and displaced
from its position in any direction, then it experiences a restoring force
towards a null point, where the electric field is zero. All the field lines
near the null point are directed inwards towards the null point. There is a net
inward flux of electric field through a closed surface around the null point.
According to Gauss’s law, the flux of electric field through a surface, which
is not enclosing any charge, is zero. Hence, the equilibrium of the test charge
can be stable.
(b) Two charges of same
magnitude and same sign are placed at a certain distance. The mid-point of the
joining line of the charges is the null point. When a test charged is displaced
along the line, it experiences a restoring force. If it is displaced normal to
the joining line, then the net force takes it away from the null point. Hence,
the charge is unstable because stability of equilibrium requires restoring
force in all directions.
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