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A parallel-plate capacitor has its entire gap filled with a dielectric whose dielectric constant changes with position x according to K(x) = K0 + λx, where λ is a constant. How is its capacitance C related to the vacuum capacitance C0?

1. C = (λd / ln(1 + K0/λd)) C0
2. C = (λ / d ln(1 + K0/λd)) C0
3. C = (λd / ln(1 + λd/K0)) C0
4. C = (λ / d ln(1 + K0/λ)) C0

Correct answer: C = (λd / ln(1 + λd/K0)) C0

Solution

1/C = (1/(eps0*A)) * integral dx/(K0+lambda*x) over 0..d = (1/(eps0*A*lambda)) ln(1+lambda*d/K0). Hence C = (lambda*d/ln(1+lambda*d/K0)) * C0, with C0 = eps0*A/d.

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