How Does Charge Distribution Affect Electric Fields in Nested Conductors?

[theunbeatable]

Homework Statement

A solid conducting sphere of radius r₁ has a total charge of +3Q. It is placed inside (and concentric with) a conducting spherical shell of inner radius r₂ and outer radius r₃. Find the electric field in these regions:
r < r₁
r₁ < r < r₂
r₂ < r < r₃
r > r₃

Homework Equations

E = F / q
E = kq / R (for a point charge)

The Attempt at a Solution

Because it's a conductor, I know that the electric field for r < r₁ should be 0 (I also forgot the reasoning for this. I think it's because the charges are moving and somehow they cancel each other out?)

I'm confused on the other parts, though. For r₁ < r < r₂, I said that the electric field is 3kQ/r because the electric field from the sphere could be felt from that point. For r₂ < r < r₃, I said that the electric field was 0 for the same reasoning as r < r₁, and I'm not even sure if that makes sense. Finally, for r > r₃, I said it was 3kQ / r.

 

[Doc Al]

E = kq / R (for a point charge)

Careful. That formula is not quite right.

Except for that, your answers are fine. For electrostatic equilibrium, the field inside a conductor must be zero, otherwise charges would move until they canceled the field.

 

[theunbeatable]

Oops, that should be R² haha. So my answers were correct then?

And also, if both the sphere and the shell were insulators, would the answers be the opposite? Like for r < r₁, the answer would be 3kQ/(r₁²)?

 

[Doc Al]

Oops, that should be R² haha. So my answers were correct then?

Yes, except as noted.

And also, if both the sphere and the shell were insulators, would the answers be the opposite? Like for r < r₁, the answer would be 3kQ/(r₁²)?

Not exactly. If the sphere were a uniformly charged insulator, then the field at any point within the sphere (say at r = r_a, where r_a < r₁) would only depend on the charge within the region 0 < r < r_a.

 

[barryj]

Are you sure that E = kq / R (for a point charge) is correct?

 

[theunbeatable]

So it would have to be 3kQ/(r²) then.

 

[Doc Al]

So it would have to be 3kQ/(r²) then.

No. 3Q is the total charge of the sphere. All you want is the charge within r.

 

[theunbeatable]

No. 3Q is the total charge of the sphere. All you want is the charge within r.

Since r is an arbitrary length that is less than r₁, how would we determine what the charge would be?

 

[Doc Al]

Since r is an arbitrary length that is less than r₁, how would we determine what the charge would be?

If the sphere were uniformly charged, then the charge would be proportional to the volume.

 

[CAF123]

Since r is an arbitrary length that is less than r₁, how would we determine what the charge would be?

To find the charge contained within a sphere with radius less than $r_1$, you will need to use the volume charge density of the sphere, where $dq = \rho dV$

 

[theunbeatable]

To find the charge contained within a sphere with radius less than $r_1$, you will need to use the volume charge density of the sphere, where $dq = \rho dV$

That makes sense. Thanks for the help everybody!