Irodov Solutions : Problem 3.11

The problem is stated as :
A system consists of a thin charged wire ring of radius R and a very long uniformly charged thread lying along the ring's axis, with one of its ends coinciding with the centre of the ring. The total charge of the ring is q and the charge of the thread per unit length is λ. Find out the interaction force between the ring and the thread. 

The given electrostatic system is a combination of a uniformly charged ring and a semi-infinite uniformly charged thread oriented along the ring's axis. The mutual electrostatic forces of interaction can be found by using either of the following two ways described below.

Method 1

The electric field strength due to a uniformly charged ring at a point on its axis at a distance x from the ring's centre is given by

Also the field vector E at any point on the axis is directed along the axis (From symmetry). 

Consider a small segment dx on the thread which is located at a distance x from the centre of the ring (see the below fig.)

This small selection carries a charge dq = λdx. The force experienced by it can be written as

Since force on each such small section of the thread is directed along the axis, we can directly integrate the above expression of infinitesimal force to get the resultant electric force on the charged thread due to the charged ring. 

Method 2

We can also find the force applied by semi-infinite charged thread on the charged ring. The field strength due to a semi-infinite line charge (uniform) is given by