Electrostatics and current carrying loop in magnetic field
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Problem 22.58
(please see the attached file)
Charge density ρ(r) = ρ0 (1 - 4r/3R) r < R
= 0 r > R
a) Let us consider a spherical shell of radius r and thickness dr co-centric with the spherical charge distribution. Volume of the dr thick shell is given by : dV = 4Πr2dr
Total charge in volume dV = dQ = ρ(r)dV = ρ0 (1 - 4r/3R) 4Πr2dr
R
Total charge in the whole sphere = Q = ρ0 4Π ∫(r2 - 4r3/3R) dr
R 0
Or Q = ρ0 4Π[r3/3 - (4/3R)(r4/4)] = ρ0 4Π[R3/3 - R3/3] = 0
0
Q = 0 ..........(1)
b) Considering a Gaussian surface with radius r > R, and applying Gauss theorem to the same we get :
∫E.ds = E4Πr2 = Q/Є0 = 0 [where Q = Total charge enclosed by the Gaussian surface = 0]
Or E = 0
c) Considering a Gaussian surface with radius r < R, and applying Gauss theorem to the same we get :
∫E.ds = E4Πr2 = ρ0 4Π[r3/3 - (4/3R)(r4/4)] /Є0
Or E = (ρ0/ Є0)[r/3 - r2/3R] ............(2)
d)
(please see the attached ...
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