Charged Spherical Shell and Solid Sphere

[andyrk]

A spherical shell and a conducting sphere each of radius R are charged to maximum potential. Which of the two has more charge?

My attempt:
Since in a conductor, no charge can reside inside the conductor so all charge is on the surface of the conductor just like the spherical shell. Now : Potential(V)=KQ/R

K and R are same for both. They are charged to maximum potential. Means there is a limit to the amount of charge the spherical shell and the solid conducting sphere can possess. My question is how to determine who can possesses more charge? I think we need to calculate capacitance of a solid sphere and a spherical shell in this. We know the capacitance of an isolated spherical capacitor.But how to calculate the capacitance of a isolated spherical shell?? Please reply soon!

 

[andyrk]

Anybody there?

 

[voko]

What is the maximum potential you are talking about?

 

[andyrk]

That is not mentioned in the question.

 

[voko]

Well, this is rather strange. However, your observation that all the charge on a conducting sphere is at its surface is correct. Then, can the potential of the sphere be different from the potential of the shell, charges being equal?

 

[andyrk]

Well, this is rather strange. However, your observation that all the charge on a conducting sphere is at its surface is correct. Then, can the potential of the sphere be different from the potential of the shell, charges being equal?

No it can't be. But in the problem it says, which has greater charge. For this I think we need capacitance C for both the bodies. Like for the solid sphere C=4∏ε_oR

I don't think I have ever heard of capacitance of an isolated spherical shell. If we somehow get that and compare both of them, the one which is greater can be said to have the capacity to have more charge on it. But this is where I got stuck. There isn't any formula for the capacitance of a spherical shell. So if I can't do it this way then is there any other possible way to do this problem?

 

[voko]

Capacitance is defined via charge and potential. If charges are equal, and potentials are equal, too, there is no other option for capacitance.

 

[andyrk]

Capacitance is defined via charge and potential. If charges are equal, and potentials are equal, too, there is no other option for capacitance.

No! Its not necessary that capacitance always involves charge and potential!
http://ta.ramk.fi/~jouko.teeriaho/capac.pdf

 

[voko]

No! Its not necessary that capacitance always involves charge and potential!
http://ta.ramk.fi/~jouko.teeriaho/capac.pdf

Capacitance is defined via charge and potential, just like the second slide at your link indicates.

 

[BruceW]

No it can't be. But in the problem it says, which has greater charge. For this I think we need capacitance C for both the bodies. Like for the solid sphere C=4∏ε_oR

I don't think I have ever heard of capacitance of an isolated spherical shell. If we somehow get that and compare both of them, the one which is greater can be said to have the capacity to have more charge on it. But this is where I got stuck. There isn't any formula for the capacitance of a spherical shell. So if I can't do it this way then is there any other possible way to do this problem?

OK, so they are both kept at the same potential. And they ask 'which has greater capacitance?' So this is equivalent to 'which has stored greater charge?' right? (since we know C=Q/V). So now, you can either calculate the charge stored, or if you know the capacitance, then just use that.

hint: you can sort of figure out the answer logically. But you could also go through the calculation.

 

[andyrk]

OK, so they are both kept at the same potential. And they ask 'which has greater capacitance?' So this is equivalent to 'which has stored greater charge?' right? (since we know C=Q/V). So now, you can either calculate the charge stored, or if you know the capacitance, then just use that.

hint: you can sort of figure out the answer logically. But you could also go through the calculation.

I am not able to sort it out. Tried to think alot. Could you help?

 

[BruceW]

start by thinking the isolated sphere has charge Q1 and the isolated shell has charge Q2. (which may or may not be the same). Now in both situations, where are the charges, and what is the form of the potential in both situations? From this, what would you say about Q1 and Q2?

 

[andyrk]

See, they are charged to maximum potential, which we do not know the value of. So we can't say which is greater, Q₁ or Q₂.?

 

[rude man]

As voko pointed out, the term 'maximum potential' is weird. If you don't know what it means you should inquire of your lecturer or whoever as to its meaning. We at PF are telling you it's meaningless in the context of the remainder of the question.

We shall assume it means the potential of the sphere and the shell are the same. It doesn't matter how high the potential is.

One can visualize the meaning of 'capacitance' of an isolated sphere or shell as a capacitor formed by the sphere or inner shell of radius a and an outer shell of infinite radius b. So, as you know, the capacitance of such a system is (1/k)(1/a - 1/b) with k = 1/(4 pi epsilon-sub-zero). For b = infinity we get C = a/k.

So if V is the same for both capacitors (sphere plus infinite outer shell vs. inner shell plus infinite outer shell), then the capacitance is the same and since Q = CV it follows immediately that Q is the same for both capacitors also.

 

[andyrk]

No..it is not 2 concentric spherical shells. Its an isolated solid sphere and an isolated spherical shell of the same radius R which are charged to the maximum potential. THe formula of capacitance you wrote is of two concentric spherical shells of radii a and b (a>b)..

 

[rude man]

No..it is not 2 concentric spherical shells. Its an isolated solid sphere and an isolated spherical shell of the same radius R which are charged to the maximum potential. THe formula of capacitance you wrote is of two concentric spherical shells of radii a and b (a>b)..

Ah, thank you, I know. What I said was one can visualize a capacitor formed by a sphere of radius a and an outer concentric shell of radius b = infinity.

Or, an inner shell of radius a and again an outer shell of radius b = infinity.

Believe me, it works.

 

[andyrk]

Ah, thank you, I know. What I said was one can visualize a capacitor formed by a sphere of radius a and an outer concentric shell of radius b = infinity.

Or, an inner shell of radius a and again an outer shell of radius b = infinity.

Believe me, it works.

Oh yes. Thanks. It really works :)