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 4.50R 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/4.50  (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?

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 4.50R 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/4.50 (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

Electric filed due to first disk, E1 = k [ 1 - z / ?( z 2 + R 2 )]

= k [ 1 - (4.50 R) / ?( (4.50 R)^2 + R^ 2 )]

= 0.02381 k

Electric field due to removed part of the disk, E2 = k [ 1 - z / ?( z 2 + R 2 )]

= k [ 1 - (4.50 R) / ?( (4.50 R)2 + (R/4.50) 2 )]

= 0.00122 k

Fraction of the reduced part = E2 / E1

= 0.00122 k / 0.02381 k

= 0.0512

= 5.12%

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