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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.

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.

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For equilibrium to be stable, there must be a restoring force and hence all the field lines should be directed inwards towards the null point. This implies that there is a net inward flux of the electric field. But this violates Gauss's law, which states that the flux of electric field through a surface not enclosing any charge must be zero. Hence, the equilibrium of the test charge cannot be stable.

Therefore, the equilibrium is necessarily unstable.

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