- #1
univox360
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According to Jackson the potential energy of a system of conductors is
[tex]W=\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n C_{ij} V_{i}V_{j}[/tex]
He calls the coefficients [tex]C_{ii}[/tex] coefficients of capacitance and [tex]C_{ij}[/tex] 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,
[tex]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})[/tex]
And if we make the substitution
[tex]V_{2}=V_{1}+\Delta V_{12}[/tex]
And then set V1 to zero,
[tex]W=\frac{1}{2}C_{22} (\Delta V_{12})^2[/tex]
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.
[tex]W=\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n C_{ij} V_{i}V_{j}[/tex]
He calls the coefficients [tex]C_{ii}[/tex] coefficients of capacitance and [tex]C_{ij}[/tex] 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,
[tex]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})[/tex]
And if we make the substitution
[tex]V_{2}=V_{1}+\Delta V_{12}[/tex]
And then set V1 to zero,
[tex]W=\frac{1}{2}C_{22} (\Delta V_{12})^2[/tex]
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.