Equivalent Capacitance of Concentric Spheres

[TheSoftAttack]

Hey guys! This problem has been perplexing me for ages, and everything I do to try to solve it doesn't seem to work. So I was wondering if you guys could help point me in the right direction. Anyway:

Homework Statement

The figure below shows six concentric conducting spheres, a, b, c, d, e, and f having radii R, 2 R, 3 R, 5 R, 6 R, and 8 R, respectively. Spheres b and c are connected by a conducting wire, as are spheres d and e. Determine the equivalent capacitance of this system.

Homework Equations

I believe the equation needed for this problem is the equation for a spherical capacitor (with just two concentric spheres). This equation is C = ab/k(b-a), where a and b are the radii of the two spheres, k = 1/(4piε₀) and C is the capacitance.

The Attempt at a Solution

My idea for this problem was that the spheres would essentially act like a set of conductors in series. Since spheres B and C; and spheres D and E are connected, they will be equipotential, and so we are left with only three conductors. I thought the capacitance should be given by the equation 1/C = 1/Cab + 1/Ccd + 1/Cef.

Thus, 1/C = k(b-a)/ab + k(d-c)/cd + k(f-e)/ef

And finally, C = k[(b-a)/ab + (d-c)/cd + (f-e)/ef]

Substituting in a = R, b = 2R, c = 3R, d = 5R, e = 6R, and f = 8R, I got:

C = k[(2R-R)/(2R*R) + (5R-3R)/(5R*3R) + (8R-6R)/(8R*6R)]
C = k[R/(2R^2) + 2R/(15R^2) + 2R/(48R^2)]
C = k[1/2R + 2/15R + 1/24R]
C = 27k/40R

This isn't right though, so I have no idea what to do from here. Any help would be greatly appreciated.

EDIT: Oh wow, I figured out what I had done wrong as soon as I posted this. The answer I originally got was actually 1/C. C would be 40R/27k. Well, that's embarrassing.

 

[anathema]

Hello! It looks like you are on the right track with your solution. However, there are a few things that you could adjust to make it more accurate.

First, when finding the equivalent capacitance of a system, it is important to remember that the capacitors are connected in parallel, not in series. This means that the equation should be 1/C = 1/Cab + 1/Ccd + 1/Cef. This will give you the correct answer of 40R/27k.

Secondly, when substituting in the values for the radii, it is important to remember that the radii are not all equal. In the equation, you have used a = R, b = 2R, c = 3R, d = 5R, e = 6R, and f = 8R. However, this is not the case for all of the spheres. For example, sphere a has a radius of R, but sphere b has a radius of 2R, not R + 1. This may be where you got the incorrect answer of 27k/40R.

Lastly, make sure to double check your calculations and units. It looks like you may have mixed up the order of operations in your final calculation, as the units should be in Farads (F), not Farads multiplied by meters (F*m). If you make these adjustments, your final answer should be more accurate.

I hope this helps and good luck with your future problem solving!