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

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

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

Solution

The correct option reflects the relationship between the varying dielectric constant and the capacitance, taking into account how the dielectric constant changes with position. The formula captures the effect of the dielectric's spatial variation on the overall capacitance, which is essential for accurately calculating the capacitance in this scenario.

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