# Coefficients of Capacitance

1. Nov 8, 2007

### psholtz

I have a question about coefficients of capacitance..

Specifically I'm reviewing the treatment on the subject in Chap 3.6 in Purcell's classic book on E&M. He starts out by considering a system of four conductors (actually 3 main conductors, and an "infinte" boundary condition surrounding the other three at potential zero).

We have three conductors, C1, C2 and C3 at potentials V1, V2 and V3 respectively. He goes on to consider a State I, where:

V1 = V
V2 = 0
V3 = 0

He then states that by holding the potentials V2 and V3 at zero, all charges in the system will be determined by the voltage (i.e., charge) on conductor 1, and goes on to define a linear relation for State I:

Q1 = C11 * V1
Q2 = C21 * V1
Q3 = C31 * V1

My question is: how can we produce/create a charge on conductors 2 or 3, when they are still being held at potential zero? Certainly I can see how adding charge (i.e., raising potential) of conductor 1 could *induce* a charge on part of conductors 2 or 3, but that induced charge (on the "close" side of the conductor, the side closest to C1) would have to be counterbalanced by an equal and opposite charge on the far side of the conductor, no?

I don't see how the expressions Q2 or Q3 could be anything other than zero, if we are to take Q2 and Q3 to be the "net" total charge on these conductors.

If, on the other hand, Q2 and Q3 are supposed to represent the amount of charge "induced" on those conductors by the charge/potential on C1, then I can understand that, but (a) I question what the usefulness of that information is, since it must be balanced by an equal and opposite charge on that same conductor that cancels it out; and (b) this interpretation doesn't seem totally consistent w/ the gist of the treatment Purcell seems to be trying to give in this section..

2. Nov 8, 2007

### clem

"My question is: how can we produce/create a charge on conductors 2 or 3, when they are still being held at potential zero?"

To keep 2 and 3 at zero potential, they have to be connected the ground of an electrical circuit. The circuit adds or removes the charge needed to keep the electrode at V=0.

3. Nov 8, 2007

### psholtz

Well yes, sure.

But then why won't C21 and C31 in the above equation be equal to zero?

How is it that by adding charge to Conductor1, you can induce/create charge on nearby conductors? Surely there's not a "net" increase in charge on Conductors 2 and 3, just b/c charge was added to Conductor 1?

4. Nov 9, 2007

### Meir Achuz

The C_ij are determined only by the geometry.
They are not affected by the charges on the electrodes.

This is called charging by induction.
The electrical source connected to electrodes 2 and 3 produces the charge necessary to keep tham at zero potential.
Surely there IS a "net" increase in charge on Conductors 2 and 3, just b/c charge was added to Conductor 1.

Read that chapter in Purcell again, carefully.

5. Nov 11, 2007

### psholtz

Ah yes... that "outer" conducting shell in Purcell's treatment serves the role of "ground"; as the source of the charges which are moved to the other conductors (C2 and C2, in the case of State I (following Purcell's diagram)) in order to effect the "charging by induction."

Purcell even explains this in book: "we have kept it in the picture because it makes the process of charge transfer easier to follow".. Indeed, there is charge transfer.

Thanks also for the tip about "charging by induction".. Google has much more to say about this search term than it does under "coefficients of capacitance"..