consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point of the condfiguration. Show that the equilibrium of the test charge is necessarity unstable.
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On the contrary, assume that the equilibrium of the test charge is stable. So if the charge is displaced by a slight amount it will experience a restoring force which will bring it back to its original position, which is the null point. So all field lines near the null point must be directed inwards to the null point. This implies there is a net inward flux of electric field through a closed surface around the null point. But Gauss's Law tells that the flux through a closed surface not enclosing any charge must be zero. So here is contradiction. Thus the equilibrium of the test charge has to be unstable.
On the contrary, assume that the equilibrium of the test charge is stable. So if the charge is displaced by a slight amount it will experience a restoring force which will bring it back to its original position, which is the null point. So all field lines near the null point must be directed inwards to the null point. This implies there is a net inward flux of electric field through a closed surface around the null point. But Gauss's Law tells that the flux through a closed surface not enclosing any charge must be zero. So here is contradiction. Thus the equilibrium of the test charge has to be unstable.
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When a test charge is in equilibrium and shifted in some direction from its location, it will undergo a restore force to a null point where the electrical field is zero.
Explanation:
- Let the Test Charging balance be stable. When a test charge is in equilibrium and shifted in some direction from its location, it will undergo a restore force to a null point where the electrical field is zero.
- All field lines close to null point are pointed inward to null point. There is still a net inward flow of electrical field across the null point, via a closed wall.
- According to the law of Gauss, the flow of electric field across a surface that does not contain any charge is zero. Hence the check charge balance can be positive.
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