# Homework Help: SPHERICAL CAPACITOR - with two insulating shells

1. Nov 3, 2009

### Grommit

I could really use some help with this one, and I need to submit very soon...
Problem:
A spherical capacitor is made of two insulating spherical shells with different dielectric
constants K1 and K2 situated between two (inner and outer, shown by thick lines)
spherical metallic shells and separated by a vacuum gap. Geometrical dimensions of
the cross-section are as shown in figure 2. Outer metallic shell has charge +Q and the
inner metallic shell has charge −Q. What is the potential difference V between these
metallic shells?
Image: http://tinyurl.com/yazcvq2

If Im understanding this correctly, this can be interpreted as three capacitors connected in series. To that end, I thought I could utilize the equation for capacitors in series: 1/Ceq = 1/C1 + 1/C2 +1/C3. Then, Ceq=Q/V would indicate the potential difference. (Im thinking this may be flawed logic though).

Using K*(4*pi*Epsilon(0))*(a*b)/(b-a) for the spherical capacitances seems to be making a cumbersome mess of numbers more than anything.
Im starting to think now that maybe I should be setting up an integral along the lines of V=Q/(4*pi*Epsilon(0)) INTEGRAL (from a to b) db/b^2... hope that makes sense.
Thanks in advance for any help.

Last edited: Nov 3, 2009
2. Nov 3, 2009

### Grommit

FWIW, I found the problem partially worked out in this manner:
Basically you have 3 capacitors in series, the first has a dielectric K2 with a thickness b -a, the second has a vacuum dielectric with a thickness c-b and the third capacitor with a dielectric K3 with a thickness d -c.
C1 = K2/(b-a), C2 = 1/(c-b), and C3 = K1/(d-c)
1/Ct = 1/C1 + 1/C2 + 1/C3
1/Ct = (b-a)/K2 + (c-b) + (d-c)/K1
1/Ct = (K1(b-a) +K1(K2)(c-b) + K2(d-c))/K1(K2)
Ct = K1(K2)/(K1(b-a) +K1(K2)(c-b) + K2(d-c))