The figure below shows a spherical shell with uniform volume charge density ? =

Question

The figure below shows a spherical shell with uniform volume charge density ? = 1.53 nC/m3, inner radius a = 12.0 cm, and outer radius b = 3.00a.

What is the magnitude of the electric field at the following radial distances?

(a)    r = 0
  N/C

(b)    r = a/2.00
N/C

(c)    r = a
  N/C

(d)    r = 1.50a
  N/C

(e)    r = b
N/C

(f)    r = 3.00b
N/C

I know a, b, and c are 0 I just cant seem to be able to answer d-f.

*Did you construct a Gaussian sphere with a radius matching any of the given radial distances? Did you find the charge enclosed by the Gaussian sphere? Did you then use Gauss' law to find an expression for the field on the surface of the Gaussian sphere?

Explanation / Answer

draw a sphere with radius r, Gauss Law says E 4 pi r^2 = (total charge enclosed) / epsilon.

for questions (a), (b), (c), the spheres enclose no charges, so E = 0.

for (d), (e)
E = [rho * (4/3) * pi (r^3-a^3)] / (4 pi epsilon r^2)

particularly, for (d)
E = 9e9*1.97e-9*(4/3)*pi*(1.5^3-1)*0.126/1.5...
I got E = 9.9 N/C

for (e)
E = 9e9*1.97e-9*(4/3)*pi*(3.4^3-1)*0.126/3.4...
I got E = 31 N/C

for (f)
E = [rho * (4/3) * pi (b^3-a^3)] / (4 pi epsilon r^2)
=9e9 [1.97e-9 * (4/3) * pi * (3.4^3-1)* 0.126^3] / (3.4*0.126*3)^2
=3.44 N/C

A shortcut for (f) is that the answer should be (1/3)^2 of that of (e), E=31/9 = 3.44 N/C, because at the outside of the shell, it acts as a point charge, and E falls as 1/r^2

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