What Is the Capacitance of Three Concentric Spherical Shells?

[MarkovMarakov]

Homework Statement

Hi, I would really appreciate clarification on the definition of capacitance in this context: Suppose I have 3 concentric metal spherical shells and they have charges Q1,Q2,Q3 and potentials 0,V,0 and radii a,b,c respectively, what is the capacitance of the configuration?

Homework Equations

C=Q/V

The Attempt at a Solution

I know what the capacitance is with only 2 spheres but I am not sure what it means with 3.

 

[MarkovMarakov]

Don't worry, I have figured it out! :)

 

[gash789]

I am inclined to answer although I am not 100% this solution is correct so please tell me if this is wrong.

As I'm sure your aware we calculate the capacitance for two concentric circles by
[itex]C=Q/V \; , [/itex]
where from Gauss's law we draw a sphere around the smallest shell.
[itex]V=\int_{a}^{b}E dr = \frac{Q_{enc}}{4 \pi \epsilon} \int_{a}^{b} r^{-2} dr =\frac{Q_{enc}}{4 \pi \epsilon} \left(\frac{1}{a}-\frac{1}{b} \right)[/itex]
Since we are explicitly given the charges on each sphere, I believe the enclosed charge should be that of the innermost sphere. Now here is where I am stuck, the charge described in the capacitance equation refers normally to have a charge q and -q on each side, I am therefore going to assume that Q=abs(Q1-Q2)/2. This is probably wrong but I can't see any other way.
[itex]C_{a \rightarrow b}=\frac{2 \pi \epsilon |Q_{1}-Q_{2}|}{Q_{1}\left(\frac{1}{a}-\frac{1}{b} \right)} [/itex]

This result holds in the three sphere setup since we only considered the enclosed charge (from Gauss). Therefore this could be easily extended for the third shell giving a second capacitance seen between b and c.

Hope that helps, and isn't completely wrong!

 

[rude man]

Homework Statement

Hi, I would really appreciate clarification on the definition of capacitance in this context: Suppose I have 3 concentric metal spherical shells and they have charges Q1,Q2,Q3 and potentials 0,V,0 and radii a,b,c respectively, what is the capacitance of the configuration?

Homework Equations

C=Q/V

The Attempt at a Solution

I know what the capacitance is with only 2 spheres but I am not sure what it means with 3.

What is the implication if the inner and outer shells have the same ("zero") potential?

 

[paranormal]

Hi there,

Capacitance is a measure of the ability of a system to store electrical charge. In this context, it refers to the ability of the 3 concentric metal spherical shells to hold their respective charges (Q1, Q2, and Q3) at their given potentials (0, V, and 0) and radii (a, b, and c).

In this scenario, the capacitance of the configuration can be calculated using the formula C = Q/V, where Q is the total charge stored in the system and V is the potential difference between the shells. However, since there are 3 shells involved, it is important to consider the capacitance of each individual shell as well as the overall capacitance of the entire configuration.

To calculate the individual capacitance of each shell, you can use the formula C = 4πε₀ab, where ε₀ is the permittivity of free space, a is the inner radius, and b is the outer radius of each shell.

To calculate the overall capacitance of the configuration, you can use the formula 1/C = 1/C1 + 1/C2 + 1/C3, where C1, C2, and C3 are the individual capacitances of each shell. This accounts for the fact that the capacitance of the entire system is dependent on the capacitance of each individual shell.

I hope this helps clarify the concept of capacitance in this context. Let me know if you have any further questions. Good luck with your homework!