Suppose you design an apparatus in which a uniformly charged disk of radius R is

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 2.10R from the disk (see Figure (a)). Cost analysis suggests that you switch to a ring of the same outer radius R but with inner radius R/2.10 (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?

Explanation / Answer

original E field is given by:

       E1 = k (1 - [z / (z2 +R2)1/2])      where k is a constant.

So     E1 = k (1 - [2.1R / ((2.1R)2 + R2)1/2])

         E1 = k ( 1 - [2.1R / 2.33R])

         E1 = k (1 - 0.9012) = 0.0987k

Now... consider the contribution only from the part thatis removed. I'll call this   E2. Use the same expression, but now the radius of the disk is R/2 so

   E2 = k(1 - 2.1R / ( (2.1R)2 + (R/2)2)1/2 )   = k (1 - 2.1R/2.1587R)

           k (1 - 0.9728) = 0.0272 k

So the E field would decrease by this much. The percentage would be...

[ E2 ] / E1

= [0.0272 / 0.0987] * 100

= 27.56%

Hope this helps

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