# Capacitance of a spherical capacitor?

## Main Question or Discussion Point

1-capacitance of sphere with radius r : 4πεr
2-capacitance of 2 concentric spheres with inner radius a and outer radius b : 4πε(a.b/b-a)
3-when the outer the outer sphere is earthed it gives the same capacitance as in number 2
4-when the inner sphere is earthed it gives the sum of capacitances in number 1 and 2
my question : why does number 2 and 3 give the same capacitance while they are different capacitors (the inner sphere is earthed in one and not earthed in the other one) ? and when the inner sphere is earthed : why does it give 2 capacitances (sum of number 1 and 2) ?

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mfb
Mentor
(2) is actually composed of two different capacitances, one between the shells (with the formula you have there) and one between the outer shell and "ground" (infinite distance).
If you connect the outer shell to ground the outer capacitance does not do anything (between ground and ground).
If you connect the inner shell to ground (but not the outer one) you have both capacitances in parallel.

(2) is actually composed of two different capacitances, one between the shells (with the formula you have there) and one between the outer shell and "ground" (infinite distance).
If you connect the outer shell to ground the outer capacitance does not do anything (between ground and ground).
If you connect the inner shell to ground (but not the outer one) you have both capacitances in parallel.
yeah, but why connecting the inner one to the ground makes you have both capacitances exist ? while when you earth nothing the outer capacitance is zero

mfb
Mentor
"Infinity" is like ground. If you also ground the inner one, you have:

GND--| |--plate--| |--GND
Two capacitors in parallel between plate and GND.

If you ground nothing, you have:

GND--| |--plate1--| |--plate2
If you measure capacitance between 1 and 2 (!), you get just the right capacitor.

"Infinity" is like ground. If you also ground the inner one, you have:

GND--| |--plate--| |--GND
Two capacitors in parallel between plate and GND.

If you ground nothing, you have:

GND--| |--plate1--| |--plate2
If you measure capacitance between 1 and 2 (!), you get just the right capacitor.
i will write what i understand and tell me if i am right or wrong
for 2 concentric spheres with inner diameter a and outer diameter b :
if we earthed the outer sphere and gave the inner sphere charge +Q, it will induce -Q on the inner surface of the outer sphere and +Q on the outer surface of the outer sphere, the +Q on the outer surface of the outer sphere will go to earth, so the system will have the capacitance : 4πε(ab/b-a)
if we earthed the inner sphere and gave the outer sphere charge +Q, it will distribute itself over the inner and outer surfaces of outer sphere, Some charge +Q2 will remain on the outer surface because it is surrounded by earth all around. Also some charge +Q1 will shift to its inner side because there is an earthed sphere inside (+Q2 +Q1 = +Q), therefore +Q1 will induce -Q1 on the outer surface of the inner sphere and +Q1 on the inner surface of the inner sphere, the +Q1 on the inner surface of the inner sphere will go to earth, so the system will have 2 capacitances : 4πε(ab/b-a) + 4πεb
if we didn't earth anything the battery +ve terminal will be connected to inner sphere and the battery -ve terminal will be connected to the outer sphere, so inner sphere will have +Q and the outer sphere will have -Q therefore the capacitance will be : 4πε(ab/b-a)

mfb
Mentor
Capacitance is not a property of a system, it is a property of a pair of conductors. It does not make sense to talk about the capacitance of everything. You can consider the capacitance between a sphere and ground, or between the two spheres.

If you ground the outer sphere, there is no electric field outside of it, so the total charge of inner and outer sphere is zero. If you have +Q on the inner sphere, the inner surface of the outer sphere gets -Q, the outer surface of the outer sphere does not get a charge.

if we earthed the inner sphere and gave the outer sphere charge +Q, it will distribute itself over the inner and outer surfaces of outer sphere, Some charge +Q2 will remain on the outer surface because it is surrounded by earth all around. Also some charge +Q1 will shift to its inner side because there is an earthed sphere inside (+Q2 +Q1 = +Q), therefore +Q1 will induce -Q1 on the outer surface of the inner sphere and +Q1 on the inner surface of the inner sphere, the +Q1 on the inner surface of the inner sphere will go to earth
Right.
so the system will have 2 capacitances : 4πε(ab/b-a) + 4πεb
That is just a single capacitance.
if we didn't earth anything the battery +ve terminal will be connected to inner sphere and the battery -ve terminal will be connected to the outer sphere, so inner sphere will have +Q and the outer sphere will have -Q therefore the capacitance will be : 4πε(ab/b-a)
Right.

If you have +Q on the inner sphere, the inner surface of the outer sphere gets -Q, the outer surface of the outer sphere does not get a charge.
yes, i said the +Q induced on the outer surface of outer sphere will go to earth
That is just a single capacitance.
yeah i was unclear in this, i mean capacitors connected in parallel produce this equivalent capacitance

thanks !!