A positive charge distribution exists within a nonconducting spherical region of

Question

A positive charge distribution exists within a nonconducting spherical region of radius a. The volume charge density B is not uniform but varies with the distance r from the center of the spherical charge distribution, according to the relationship p=Br for 0<=0<=a, where ? is a positive constant, and p=0, and r>a.

a. In terms of B, a, and fundamental constants, determine the total charge Q in the spherical region.

b. In terms of B, r, a, and fundamental constants, determine the magnitude of the electric field at a point a distance r from the center of the spherical charge distribution for each of the following cases.
i. r > a
ii. r = a
iii. 0 < r < a

c. Quantitatively graph the electric field vs. r for 0 < r , 3a. Include labels and values on the axes.

d. In terms of B, a, and fundamental constants, determine the electric potential at a point a distance r from the center of the spherical charge distribution for each of the following cases:
i. r > a
ii. 0 < r < a

Explanation / Answer

dq = p 4r^2 dr

dq = Br 4r^2 dr = B4r^3 dr with limits r= 0 to a => q = Ba^4

Beyond r>a , E = Ba^4/ (4 r^2) = Ba^4/4r^2

at r=a , E = Ba^2/4

r<a, E.4r^2 = q/ gauss law

so E = Br^4/4r^2 = Br^2/4

graph r<a parabolic and then it will decrease with a concave shape(hyperbolic shape ie; y proportional to x^-2)

potential for r>a , V=Ba^4/4r

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