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In an experiment we were given non-homogenous dielectric substances described by functions of coordinate. How can capacitance be determined from this?

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2 Answers 2

Here's a (flawed) method on doing it--which works for many cases Otherwise @Jon's method is the way to go.

Break it into infinitesimal thin rods so that you get $\infty$ parallel capacitors of area $\rm dA$. Usually one of the directions will be homogenous, so you can write $\rm dA=z\rm dy$. Otherwise, $\rm dA=dy\cdot dz$

Break each rod into infinitesimal chunks and get $\infty$ series capacitors per slice of thickness $\rm dx$.

For each rod, find the infinitesimal reciprocal of capacitance $\rm d(\frac1C_{chunk})$ (in terms of $(x,y,z)$ Integrate $\int \rm d(\frac1C_{chunk})=\frac1{dC_{rod}}$

Now integrate $\int \rm dC_{rod}=C_{eq}$

The issue here is, when you break them into a set of series capacitors, you use the capacitors-in-series formula. This assumes that charge distribution on each capacitor in series is the same--which is wrong in this case. We can't apply charge conservation on a rod here, as charge can enter and leave from various spots.

Usually, a sufficently symmetric case works.

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You need to use the following equations coming from electrostatic:

$$\nabla\cdot(\epsilon({\bf x}){\bf E})=\rho({\bf x})$$

$${\bf E}=-\nabla\phi$$

with the proper boundary conditions. Then, with the definition of capacitance you will get the result. This is not often feasible analytically and will depend on the problem at hand.

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