For parts A) and B), consider a sphere of radius R, centered on the origin, with
ID: 1900306 • Letter: F
Question
For parts A) and B), consider a sphere of radius R, centered on the origin, with a radially symmetric charge distribution p(r). What rho (r) is required for the E-field in the sphere to have the power-law form E(r) = c rn , where c and n are constants? The case n = -2 is special. How so? Some values of n are unphysical since these would lead to an infinite amount of charge in the sphere. Which values of n are physically allowed? What kind of charge distribution is required for the radial E-field inside the sphere to be of constant magnitude; that is, what rho (r) produces E(r) = constant (inside only)?Explanation / Answer
(A) The differential form of Gauss Law says = 0 div E. If we let E = crn times the unit vector in the radial direction, then using the formula for div in spherical coordinates we know
= (n+2)c0rn-1
which is the first answer. n = -2 is special because of the constant (n+2) in front, which makes =0 for that n. And for any n < -2 the charge is infinite thus not allowed, because the integral that you use to calculate total charge from will have 0n+2 in the evaluation, which is undefined when n < -2. So only n > -2 is allowed.
(B) Just evaluate 0 div E for E = (c,0,0) in spherical coordinates, and you get = 2c0/r
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