Irodov Solutions : Problem 3.5

A thin wire ring of radius R has an electric charge q. What will be the increment of the force stretching the wire when a point charge qₒ is placed at the centre of the ring?

In this question, we have been asked to find the increment of tension in the charged ring, when an additional point charge qₒ is placed at the ring's centre. Let the distribution of charge q on the ring is uniform.

When the charge qₒ is placed at the ring's centre, the wire gets stretched due to the mutual electrostatic repulsions between the positive charge qₒ and each infinitesimal charge element dq on the wire. This stretching causes internal restoring forces to arise that results in additional tension in the wire of ring. Let T be this additional tension which is to be calculated. We can adopt either of the methods discussed below to get T.

Method 1

Consider an infinitesimally small section of the ring subtending an angle dθ at the centre. The FBD will look like :

As this portion of ring is in equilibrium, resultant force on it along any direction must be zero. 

Method 2

Consider only the half portion of the ring say the portion lying above the x-axis where x-axis is taken to be along one of the diameters of the ring. 

Resultant force on this portion should be zero. Therefore we can simply write :

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