- #1

godtripp

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I have given myself the situation where I have a capacitor of area "A" and separation "d"

Between them is a non-uniform but continuous dielectric, so that the dielectric on the right plate is significantly more dense than that of the left.

I have also given the dielectric the function: [tex]\epsilon = \epsilon_{0}+\alpha x[/tex]

where x is the distance from the left plate.

Now I can find my answer by taking an infinite sum of infinitesimal capacitors in series.

I'm just confused now on how to properly build the formula. Should I take the derivative of

my capacitance formula with respect to d, plug in for epsilon as a function of distance and integrate? Or how?

I've listed a few of my ideas on formulas below. If anyone can point out the right one and why.

[tex] C = \frac{\epsilon_{ave} A}{d} [/tex]

Where

[tex]\epsilon_{ave}=\frac{1}{d}\int \epsilon_{0}+\alpha x dx[/tex]

[tex] C_{eq} = \int \frac{-(\epsilon_{0}+\alpha x)A dx}{x^2} [/tex]

[tex] \frac{1}{C_{eq}}= \int \frac{dx}{(\epsilon_{0}+\alpha x)A} [/tex]

[tex] \frac{1}{C_{eq}}= \int \frac{x dx}{(\epsilon_{0}+\alpha x)A} [/tex]