Capacitance of a system of 3 concentric spheres

[Anthony]

1. 3 conducting spheres of radius a, b, c (a<b<c) are held at a potential of 0, V, 0 respectively. What is the capacitance of the system?

2. Q=CV (which I assume extends to Q_i=C_ijV_j for multiple conductors).

3. I've calculated the fields in the two interesting regions, calculated the charge on each sphere - this gives Q_a+Q_b+Q_c=0.

I can't find many references to the "total capacitance" of a system of conductors and how it would relate to my C_ij. I might be missing the totally obvious, but that'll be down to my utter incompetence when it comes to physics!

Thanks for shedding any light on the matter. It shouldn't be too difficult, because the question comes from an undergraduate example sheet on introductory EM.

 

[Nate810]

The total capacitance of the system is equal to the sum of the individual capacitances of each sphere. The capacitance of a single isolated spherical conductor is given by C = 4πε0a,where a is the radius of the sphere and ε0 is the permittivity of free space. Therefore, the total capacitance of the system is given byCtotal = 4πε0(a + b + c).

 

[m2003]

I would first like to commend you for your efforts in calculating the fields and charges in the system. It shows a good understanding of the underlying principles and equations.

To answer your question, the capacitance of a system of conductors is defined as the ratio of the charge on a conductor to the potential difference between that conductor and a reference point. In your case, the reference point is the potential of 0. So, for a single conductor, the capacitance is given by C = Q/V.

For a system of multiple conductors, the total capacitance can be calculated by considering each conductor individually and then adding them up. In your case, we have three conductors, so the total capacitance would be:

Ctotal = Ca + Cb + Cc

Now, each conductor has a capacitance that depends on its geometry and the dielectric medium surrounding it. In your system, all three conductors are concentric spheres, which simplifies the calculations. The capacitance of a single conducting sphere is given by:

C = 4πε0ab/(b-a)

Where ε0 is the permittivity of free space, a is the radius of the inner sphere, and b is the radius of the outer sphere. Using this equation, we can calculate the capacitance of each conductor in your system:

Ca = 0 (since a = 0)

Cb = 4πε0ab/(b-a)

Cc = 4πε0ac/(c-a)

Therefore, the total capacitance of your system would be:

Ctotal = 4πε0ab/(b-a) + 4πε0ab/(b-a) + 4πε0ac/(c-a)

= 4πε0ab/(b-a) + 4πε0ac/(c-a)

= 4πε0a(b+c)/(b-a)(c-a)

I hope this helps to clarify the concept of capacitance in a system of multiple conductors. Keep up the good work in your studies!