A non-conducting sphere of uniform charge density p = +6 mue C/m^3 and outer rad

Question

A non-conducting sphere of uniform charge density p = +6 mue C/m^3 and outer radius R = 15 m has a smaller spherical cavity of radius a = 5 m cut out of its interior. This cavity is not centered-the center of the larger sphere and the center of the cavity are a distance b = 6 m apart, as shown below. Find the electric field inside the cavity at the following locations: A distance of a/2 to the left of the cavity's center. A distance of a/2 above the cavity's center. Make sure to compare/contrast your answers. (Super challenge: Can you find a general expression for the electric field within the cavity?) Repeat part a, if this object were instead a conductor which has the same total charge.

Explanation / Answer

Based on superposition principles, we can say that the larger sphere with the cavity is a superposition of a large sphere with radius R=15 and charge density + and a smaller sphere of radius a=5 with charge density -. Finding the E of a uniform charge density of a sphere without a cavity we get:

EA=(V)/
E(4r^2)=(4/3r^3)/
E=r/(3)

So to find the E anywhere within the cavity we can simply take
E1=+r/(3) (with r<R)
E2=-(r-b)/(3).

So there for...

E=E1+E2
E=+R/(3)-(r-b)/(3)
E=+R/(3)-R/(3)+b/(3)
E=+b/(3)
E=(+6C/m^3)(6m)/(3)

E= 1355932.20 i N/C +0 j N/C +0 k N/C=answer to both a and b.

For c, all three components will be 0 because there is no charge inside a conductor.

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