Question

Suppose you design an apparatus in which a uniformly charged disk of radius R is to produce an electric field. The field magnitude is most important along the central perpendicular axis of the disk, at a point P at distance 2.00R from the dis (see Figure (a)). Cost analysis suggests that you switch to a ring of the same outer radius R but with inner radius R/2.00 (see Figure (b)). Assume that the ring will have the same surface charge density as the original disk. If you switch to the ring, by what part will you decrease the electric field magnitude at P?

Answer 1

Let the surface chare density of both ring and disk be .

Distance of the point P from the centre is .

Now we know that for a disk, Electric Field at a point is given by

Using the given value of "x",

  

  

  ................................(1)

Also for the ring with different inner and outer radius, Electric Field at a point "x" distance away from centre is

  

  

  

  

using equation 1 in above,

  

  

Then decrease in the magnitude of electric field will be

  

(Negative sign denotes decrease in the field)