# Gauss's Law:Metal sphere of radius 'a' surrounded by a shell

## Homework Statement

A metal sphere of radius a is surrounded by a thick concentric metal shell (inner radius b, outer radius c). Neither the shell nor the sphere carries any charge, but there is a point charge +Q located inside an irregularly shaped cavity in the otherwise solid sphere as shown in the figure. The irregular cavity is not concentric with the sphere.

a) Sketch the induced charges on all the relevant surfaces.
b) What is the surface density of the charge on the outer surface of the sphere r=c?
c) What is the electric field where a<r<b?
c) What is the electric field where b<r<c?

## The Attempt at a Solution

a) The positive charge within the irregular cavity will induce a negative charge on the surface of the cavity. This will in turn induce a positive charge on the surface of the sphere. A negative charge will be induced on the surface at b and a positive charge will be induced on the surface at c. This is illustrated in this image.

b) Q=σA where A=area of surface.
σ=Q/A
σ=Q/(4πc^2)

c) E=1/(4πε0) * Q/r^2
The total enclosed charge is Q.

d) This point is inside the metal shell so the E-field should be zero.
The enclosed charges are +Q from the sphere and -Q from the 'b' surface. The net charge is zero, so E= 1/(4πε0) * 0/r^2 == 0.

I'm using Griffith's E+M text.

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collinsmark
Homework Helper
Gold Member
'Looks correct to me. [Just out of curiosity though, part b) uses the word "sphere" but then has "r=c" which would imply outer radius of the "shell." Maybe it's nothing, sometimes shells are called spheres. But it seems a little inconsistent for Griffiths. Are you sure that part b) isn't asking you for charge density at r = a?]

Last edited:
'Looks correct to me.  Awesome, thanks!

collinsmark
Homework Helper
Gold Member Awesome, thanks!
In case you missed my edit (which I didn't update until after your last post), I'll repeat it again here:

Just out of curiosity though, part b) uses the word "sphere" but then has "r=c" which would imply outer radius of the "shell." Maybe it's nothing, sometimes shells are called spheres. But it seems a little inconsistent for Griffiths. Are you sure that part b) isn't asking you for charge density at r = a?

In case you missed my edit (which I didn't update until after your last post), I'll repeat it again here:

Just out of curiosity though, part b) uses the word "sphere" but then has "r=c" which would imply outer radius of the "shell." Maybe it's nothing, sometimes shells are called spheres. But it seems a little inconsistent for Griffiths. Are you sure that part b) isn't asking you for charge density at r = a?
My homework problem is based off a similar problem in griffith's, so its the teacher's language and not Griffith's. Part-b says (verbatim) What is the surface density of charge on the outer surface of the sphere (r=c)?