A thin wire ring of radius `a` carrying a charge `q` approcahes the observation point `P` so that its centre moves rectilinearly with a constant velocity `v`. The plane of the ring remians perpendicular to the motion direction. At what distance `x_(m)` from the point `P` will the ring be located at the moment when the displacement current density at the point `P` becomes maximum? What is the magnitude of this maximum density ?

Explanation

We have, `E_(P) = (q x)/(4pi epsilon_(0) (a^(2) + x^(2))^(3//2))`
then `j_(4) = (del D)/(del t) = epsilon_(0) (del E)/(del t) = (qv)/(4pi (a^(2) + x^(2))^(5//2)) (a^(2) - 2x^(2))`
This is maximum, when `x = x_(m) = 0`, and minimum at some other value. The maximum displacement current density is
`(j_(d))_(max) = (qv)/(4pi a^(3))`
To check this we calculate `(del j_(d))/(del x)`,
`(del j_(4))/(del x) = (q v)/(4pi) [(-4x (a^(2) + x^(2)) -5x(a^(2) - 2x^(2))]`
This vanishes for `x = 0` and for `x = sqrt((3)/(2)) a`. The latter is easily shown to be a smaller local minimum (negative maximum).