Question

A point charge `q` is located at the centre fo a thin ring of radius `R` with uniformly distributed charge `-q`, find the magnitude of the electric field strength vectro at the point lying on the axis of the ring at a distance `x` from its centre, if `x gt gt R`.

Answer 1

Electric field at `P` :
Due to `q , E_(1) = (1)/(4 pi in_(0)) .(q)/(x^(2))` , towards right
Due to ring , `E_(2) = (1)/(4 pi in_(0)) .(qx)/((R^(2) + x^(2))^((3)/(2)))` , towards left
`E_(1) gt E_(2) , (x gt gt R)`
`E_(P) = E_(1) - E_(2) = (q)/(4 pi in_(0)) [(1)/(x^(2)) - (x)/((R^(2) +x^(2))^((3)/(2)))]`
`= (q)/(4 pi in_(0)) [(1)/(x^(2)) - (x)/(x^(3) (1 + (R^(2))/(x^(2)))^((3)/(2)))]`
`= (q)/(4 pi in_(0) x^(2)) [ 1 - (1 + (R^(2))/(x^(2)))^((3)/(2))]`
`= (q)/(4 pi in_(0)x^(2)) [ 1 - (1 - (3)/(2) (R^(2))/(x^(2)))]`
`= (3qR^(2))/(8 pi in_(0) x^(4))`, towards right