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 2.80R from the disk (Fig. a). Cost analysis suggests that you switch to a ring of the same outer radius R but with inner radius R/2.00 (Fig. b). Assume that the ring will have the same surface charge density as the original disk. If you switch to the ring, by what percentage will you decrease the electric field magnitude at P?

Explanation / Answer

Dimensional analysis would suggest the field strength along
the axis is proportional to the sperical angle covered by disk,
not the area (otherwise replacing the disk with unlimited infinite
plate will result in infinite electric field, 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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