Potential energy of a system of conductors

[univox360]

According to Jackson the potential energy of a system of conductors is

$$W=\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n C_{ij} V_{i}V_{j}$$

He calls the coefficients $$C_{ii}$$ coefficients of capacitance and $$C_{ij}$$ coefficients of induction.

I want to derive from this formula the well known result for a parallel plate capacitor, top plate of potential V1 and bottom plate of potential V2.

Certainly,

$$W=\frac{1}{2}(C_{11} V_{1}V_{1}+C_{12} V_{1}V_{2}+C_{21} V_{2}V_{1}+C_{22} V_{2}V_{2})$$

And if we make the substitution

$$V_{2}=V_{1}+\Delta V_{12}$$

And then set V1 to zero,

$$W=\frac{1}{2}C_{22} (\Delta V_{12})^2$$

Which is the familiar result. However, my question has to do with what C22 actually is. Since I am only familiar with capacitance of a system the term C22 doesn't make sense since the indices only involve the top plate, unless it is some sort of self capacitance. But that doesn't make sense either since we know that the energy between the plates is dependent upon the plate-plate geometry.

I need to solve this problem with more conductors but without a sense of what the coefficients actually are I cannot continue.

 

[praharmitra]

Capacitance is always defined w.r.t. a potential difference as follows -

If we charge a conductor to a value Q, what would be its potential V w.r.t. a reference potential (in effect a potential difference)? Capacitance is then given by C = Q/V.

Now, The formula in jackson assumes an arbitrary reference potential w.r.t. which all the potentials $V_i$ are calculated. What is meant by $C_{ii}$ then is the capacitance b/w the conductor i and the reference conductor which is at a potential zero.

In the formula you derived you set the reference potential to be $V_1=0$ In effect $C_{22}$ defines the capacitance b/w 1 and 2. You could alternatively have set $V_2=0$, then $C_{11}$ would define the capacitance b/w 1 and 2.

Usually the reference potential is set to be 0 at infinity. One could then measure the capacitance of a single conductor, e.g. sphere, or spherical shell, etc.

 

[univox360]

That seems reasonable.

What if we had a system of three conductors?

If I were to set, say conductor one to potential zero, would the coefficient C22 still correspond to the capacitance between conductor two and the conductor held at zero?

Also, what would the term C23 correspond to? Is that the inductance of conductors two and three with respect to the conductor held at zero?

Thanks!