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 2.30R 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?

(  )%?

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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 2.30R 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

You need to find the electric field due to a ring. It looks like this:

E = [ k * (sigma) * (2 * pi * r * z) * dr ] / [ r^2 + z^2 ]^(3/2)

Where k is 1/( 4* pi * epsilon), z is the position along the axis that runs through the center of the disk.
Then integrate this with respect to r for r = R/4.4 to r = R.

You also want to find the electric field for a disk of radius R.
It looks like this:

E = ( (sigma)*z ) / (2 e0 ) [ 1/z - 1/sqrt[ R^2 + z^2 ] ].

Where e0 is epsilon.

Take the electric field you found for the ring and subtract it from this. This should give you your answer.

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