A system consits fo a thin charged wire ring of radius `R` and a very long uniformly charged thread oriented along the axis of the ring, with one of its ends coinciding with the centre of the ring. The total charge of the ring, with one of the ring so equal to `q`. The charge of the thread (per unit length) is equal to `lambda`. Find the interaction froce between the ring and the thread.

Explanation

From the solution of 3.9, the electric field strength due to ring at a point on its axis (say x-axis) at distance `x` from the centre of the ring is given by :
`E(x) = (qx)/(8pi epsilon_(0) (R^(2) + x^(2))^(3//2))`
And from symmetry `vec(E)` at every point on the axis is directed along the x-axis (Fig.) Let us condider an element `(dx)` on thread whcih carries the charge `(lambda, dx)` The electric force experienced by the element in the field or ring.
`dF = (lambda, dx) E(x) = (lambda qx dx)/(4pi epsilon _(0) (R^(2) + x^(2))^(3//2))`
Thus the sought interaction
`F = int_(0) ^(oo) (lambda q x dx)/(4pi epsilon_(0) (R^(2) + x^(2))^(3//2))`
On intergating we get, `F = (lambda q)/(4pi epsilon_(0) R)`