A thick spherical shell with inner radius R i and outer radius R o has uniform c

Question

A thick spherical shell with inner radius Ri and outer radius Ro has uniform charge density p. What is the total charge on the shell Q? What is the electric field magnitude E in terms of Ri, Ro, p, and distance r from the center of shell? Graph the electric field magnitude versus distance from the center of the shell for Ri = 1.0 cm, Ro = 2.0 cm, and p = 10-3 C/m3. The graph may be drawn, but use a ruler and correctly measure the distance between points along both the E and r axes. Include the points r = 0.0 cm, 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, 2.5 cm, and 3.0 cm on the graph.

Explanation / Answer

Let a = inner radius, b = outer radius, p = volume charge density, Q = total charge.

Consider three regions:

I - 0 < r < a
II - a <= r < b
III - r >= b

In region I, there is no charge so by Gauss's Law the E field is 0 ---> E = 0

In region II, the field is dependent on how large the raidus is and we need to do some math. For a <= r< b, the charge enclosed in a sphere of radius r is:

q(r) = p*4*pi/3*(r^3 - a^3)

Define Q = p*4*pi/3*(b^3 - a^3) ---> p = Q/{4*pi/3*(b^3 - a^3)}

So q(r) = Q*(r^3 - a^3)/(b^3 - a^3)

Now apply Gauss's Law :

Integral (E*dA) = q(r)/e0 ---> E*4*pi*r^2 = Q*(r^3 - a^3)/{e0*(b^3 - a^3)}

E = Q*(r^3 - a^3)/{4*pi*e0*r^2*(b^3 - a^3)} for a<= r < b

In region III, a Gaussian sphere encloses the entire charge distribution so:

Integral(E*da) = Q/e0 ---> E = Q/(4*pi*e0*r^2) r >= b

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