How to treat the "ideal" plate capacitor more rigorously?

[greypilgrim]

Hi.

The derivation of the capacity of an ideal parallel-plate capacitor is inconsistent: On the one hand, the plates are assumed to be infinitely large to exploit symmetries to compute an expression for the electric field, on the other the area is finite to get a finite expression for the charge. Usually this is justified by the fact that the boundary only increases with the square root of the area and hence boundary effects can be neglected.

However, this can be tricky: An ideal, finite capacitor doesn't even agree with Maxwell's equation since the line integral of the electric field from one plate to the other is path-dependent ($E\cdot d$ for paths between the plates, zero for paths outside). When it comes to voltage or work, boundary effects clearly cannot be neglected.

How can this subject be treated more rigorously than just what seems to me to pick and choose when to neglect boundary effects and when not to?

 

[MisterX]

The fields outside are not zero for a finite capacitor, and the work is path independent. We say that there is a fringing effect at the boundary. Numerical techniques are used for realistic geometries for electrostatics and magnetostatics.
For example this person has used the finite difference method to produce the fields of a finite capacitor.
http://www.drjamesnagel.com/EM_Beauty.htm

 

[greypilgrim]

I know all this (except for the exact term fringing effect, due to English not being my first language). Nevertheless, the derivation of the widely used formula $C=\varepsilon\frac{A}{d}$ is inconsistent due to the reasons I stated and I'd like to get a better understanding why it's still acceptable for most applications.

 

[Dale]

I'd like to get a better understanding why it's still acceptable for most applications.

Because generally the errors produced by neglecting the fringing effect are less than the errors in manufacturing the specified A and d.

 

[greypilgrim]

Can you make a quantitative estimation about the errors produced by the fringing effect?

 

[Dale]

Sure. The usual way is to do a FEM analysis like the one mentioned by @MisterX above, then make small changes to the parameters

 

[greypilgrim]

And how did people justify this derivation back in the early days of electromagnetism when those techniques weren't available?

 

[Dale]

That I don’t know. I am not aware of any analytical techniques for it. Perhaps they approximated it as a dipole as a worst case?

 

[greypilgrim]

A similar example is using Ampere's law to find an expression of the magnetic field inside a long solenoid.

 

[marcusl]

For a 2D capacitor (long parallel strips) you can compute the fringe fields analytically using conformal mapping. For spacing small compared to width, the fringing fields are negligible.

 

[Nik_2213]

Guard rings ?
I've vague memories of a Uni PhysLab session where there were co-planar guard rings just outside the circular 'standard' electrodes...
I've equally vague memories of later seeing circuit for an 'active' guard-ring set-up using op-amps...
Ideas ??

 

[marcusl]

Guard rings eliminate (or greatly reduce) the fringe fields, so are useful for standard capacitors as used by the old National Bureau of Standards. This skirts, rather than addresses, the OP question of mathematically treating a capacitor of finite area.