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 3.90R 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/3.90 (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

Okay, I will give you an idea of how to approach this. If I typed out all of the steps, it would become quite messy here. And plus it benefits you more if you can work through this on your own =). 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. Hope this helps.

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