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(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.
(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart.
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(a) Proving it by contradiction. Suppose the equilibrium is stable; then the test charge displaced slightly in any direction will experience a restoring force towards the null-point. That is, all field lines near the null point should be directed inwards towards the null point. That is, there is a net inward flux of electric field through a closed surface around the null-point. But by Gauss's law, the flux of electric field through a surface, not enclosing any charge, must be zero. Hence, the equilibrium cannot be stable.
(b) The mid-point of the line joining the two charges is a null-point. Displace a test charge from the nullpoint slightly along the line. There is a restoring
force. But displace it, say, normal to the line. You will see that the net force takes it away from the null-point. Remember, stability of equilibrium needs restoring force in all directions.
(b) The mid-point of the line joining the two charges is a null-point. Displace a test charge from the nullpoint slightly along the line. There is a restoring
force. But displace it, say, normal to the line. You will see that the net force takes it away from the null-point. Remember, stability of equilibrium needs restoring force in all directions.
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