Conductive and grounded shells

[Kosta1234]

Homework Statement

conductive and grounded shells

Relevant Equations

$E\cdot dS = \frac {q}{\epsilon_0}$

Problem Statement: conductive and grounded shells
Relevant Equations: $E\cdot dS = \frac {q}{\epsilon_0}$

Hi.
I'll be glad if you can help me with this question.I've two conductive and grounded shells with radius 'a' and radius 'b' with their center on the same point.
And another conductive (But not grounded) shell with radius R (a<R<b) and charge density $\sigma$. I'm asked to figure out what is the potential in space and the charge distribution on the grounded shells.So my way to solution was first to figure out what is the potential in space is using gaus law.
so where $r < a$ the Electric field is zero therefore the potential in $r < a$ is constant and because of the potential on the edge is zero so $V_{r<a} = 0$.

When I'm moving to $R < r < a$, (between the grounded conductive shell and the conductive shell) . I'm in a little bit in a trouble here.
grounding the shell will bring the potential to be to zero, and will make not trivial charge distribution on it, but does that mean that in TOTAL all the charge is zero on the conductive grounded shell? So that I could use gaus law here again, and in this way I get that $E\vec = 0$ in $R < r < a$ as well?

thanks.

 

[TSny]

grounding the shell will bring the potential to be to zero, and will make not trivial charge distribution on it, but does that mean that in TOTAL all the charge is zero on the conductive grounded shell?

No.

Using Gauss' law, you should be able to show that the fields in the two regions (1) $a < r < R$ and (2) $R < r < b$ are inverse square fields. That is, $E_1 = A/r^2$ and $E_2 = B/r^2$, where A and B are unknown constants. Try to think of a way to set up two equations to solve for $A$ and $B$.

Hint: The statement that the inner and outer conductors are grounded means that those two conductors are at the same potential. Also, can you use Gauss' law to determine the jump in E as you cross the middle conductor at r = R?

 

[Kosta1234]

No.

Using Gauss' law, you should be able to show that the fields in the two regions (1) $a < r < R$ and (2) $R < r < b$ are inverse square fields. That is, $E_1 = A/r^2$ and $E_2 = B/r^2$, where A and B are unknown constants. Try to think of a way to set up two equations to solve for $A$

Why are those inverse?
Is thia because the amount of charge that coming to the inner grounded conductive shell?

 

[haruspex]

Why are those inverse?

Inverse square, i.e. each field is proportional to the inverse square of the distance, i.e. r^-2.

 

[Sriram Akella]

The flaw in your argument is that you've assumed the Potential, V(r) is continuous (i.e if it is a constant inside the volume of the shell and zero on the surface, then it must be zero inside) which in general need not be the case. ( This problem is a clear counter-example to that).
The way you go about solving this is by assuming some total charge, say q1 and q2 is deposited by the ground on the shells with radii 'a' and 'b' respectively to maintain equilibrium/minimize potential energy. Now you can apply Gauss' Law to find the electric field in the region bounded by the conductors and outside. Now think of a way to solve for q1 and q2.

 

[haruspex]

The flaw in your argument is that you've assumed the Potential, V(r) is continuous

Seems like a fair assumption to me. Each shell produces a continuous potential, therefore the sum is continuous.

find the electric field

There is no need to find any fields. It can all be done with potentials.

 

[Sriram Akella]

Yeah, you're right. I'm sorry. The Potential is continuous, I stand corrected. And yes, it can all be done with potentials. I'm really sorry about the previous post...

 

[Kosta1234]

Thank you I hope I solved it right.

On one of the next questions I've been asked to figure out what is the area charge density on the edges of the grounded shells if the shells were with width $\Delta a$ and $\Delta b$.

How can I figure out this?
I know that the charge density with jump of electric field is:
$\Delta E = \frac {\sigma}{\epsilon _0}$

But what about the edges?

Edit:
I can figure out easly the electric field in the space, and the Electric field inside the conductor is $\vec E = 0$ (even if it's grounded?), and I can know the charge density on the edges using

$\Delta E = \frac {\sigma}{\epsilon _0}$​

 

[haruspex]

I missed this before...

does that mean that in TOTAL all the charge is zero on the conductive grounded shell?

No.

I hope I solved it right.

If you post your solution someone will check it.

 

[haruspex]

what is the area charge density on the edges of the grounded shells if the shells were with width Δa and Δb .

How can I figure out this?

Consider a Gaussian surface lying between the inner and outer surfaces of the shell.