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 1.30R 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? Did you use the equation derived for the field on the central axis of a disk? Did you also use the equation for a ring? Beams of high-speed protons can be produced in "guns" using electric fields to accelerate the protons. What acceleration would a proton experience if the gun's electric field were 1.00 x 10^4 N/C? What speed would the proton attain if the field accelerated the proton through a distance of 0.70 cm?

Explanation / Answer

original E field is given by:

       E1= k ( 1 - z / (z2+R2)1/2)      where k is a constant.

E2/E1 = [k (1 - 1.3 R / ( (1.3R)2 + (R/2)2)1/2)]/[k (1 - 1.3 R / ( (1.3R)

E2/E1 = [(1 - 1.3/((1.3)^2 + (1/2)^2)^(1/2))]/[(1 - 1.3/((1.3)^2 + 1)^(1/2))]

E2/E1 = 32.14%

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