Suppose you design an apparatus in which a uniformly chargeddisc of a radius R i

Question

Suppose you design an apparatus in which a uniformly chargeddisc of a radius R is to produce an electric field. The fieldmagnitude is most important along the central perpendicular axis ofthe disc at a point P at distance 2R from the disc. Cost analysissuggests you switch to a ring of the same radius R but with aninner radius R/2. Assume that the ring will have the same surfacecharge density as the original disc. If you switch to the ring, bywhat percentage will you decrease the electric field magnitude atP?

Explanation / Answer

If it were depended on area then replacing the disk withunlimited infinite plate will result in infinite electricfield, instead of 2 pi k rho.

Spherical angle covered by the disk of radius r:
A(r) = 2 pi (1 - cos(atan(r/2R))) = 2 pi (1 - ((r/2R)^2 +1)^-1/2).

The decrease is electric field is therefore equal to
A(R/2) / A(R) = (1- sqrt(16/17))/(1 - sqrt(4/5)) = 28.28%

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