4) A long conducting solid rod of length, L , and radius a , is placed inside th

Question

4) A long conducting solid rod of length, L, and radius a, is placed inside the center of a conducting pipe of length L, inside radius b, and outside radius c, as shown in Fig.-4a.

a) A charge of +Q is placed on the centrally located, solid rod. What is the charge at r<a, r=a, a<r<b, r=b, b<r<c, r=c, and r>c?

b)What is the electric field in the regions r<a, a<r<b, b<r<

c, and r>c?

c) The charge of +2Q is left on the central solid rod, and a charge of -5Q is now placed on the outside conducting pipe. Now what is the charge at r<a, r=a, a<r<b, r=b, b<r<c, r=c, and r>c?

d) What is the electric field in the regions r < a, a < r < b, b < r < c, and r > c?

e) A very skinny wire is now connected between the central solid rod of radius a, and the outer conducting pipe as shown in Fig.-4b. Now what is the charge at r<a, r=a, a<r<b, r=b, b<r<c, r=c, and r>c?

f) With the wire still connected, what is the electric field in the regions r<a, a<r<b, b<r<c, and r>c?

Explanation / Answer

"Determine the magnitude of the electric field in the region r<a."
From Gauss' Law:
E(4?r^2) = +Q/?o
E = +Q/[(4??o)r^2]

"Determine the magnitude of the electric field in the region a<r<b"
The electric field inside of a conductor (with no current) is zero.

"Determine the magnitude of the electric field in the region r>b."
Again, from Gauss' Law:
E(4?r^2) = +3Q/?o
E = +3Q/[(4??o)r^2]

"How much charge is on the inside surface of the hollow sphere?"
For the electric field in the region a<r<b to be zero, the charge on this surface would be equal in magnitude, but opposite in sign to the point charge in the center: -Q.

"How much charge is on the exterior surface?"
If the conductor has a total charge of +2Q and the charge on the inner surface is -Q, the charge on the outside surface would be: +2Q - (-Q) = +3Q

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