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AP Physics C: E&M
10.4 Dielectrics
10.3 Capacitors
AdvancedMCQMathematical21.3k
A schematic cross-section of a parallel-plate capacitor with two vertical parallel conducting plates representing area A, separated horizontally by a distance d. The left plate is located at x = 0 and the right plate is at x = d. A horizontal coordinate axis labeled x points from left to right along the bottom. The space between the plates is filled with shading whose density increases continuously from x = 0 to x = d, labeled with \kappa(x) = 1 + \alpha x. A double-headed dimension arrow indicates the plate separation d. No other labels, lines, text, or axes appear.
Cross-section of parallel-plate capacitor filled with a non-uniform dielectric.
A parallel-plate capacitor consists of two parallel conducting plates, each of area \(A\), separated by a distance \(d\). The region between the plates is filled with an inhomogeneous dielectric material whose dielectric constant varies continuously with distance \(x\) from the left plate (at \(x = 0\)) to the right plate (at \(x = d\)) according to the function \(\kappa(x) = 1 + \alpha x\), where \(\alpha\) is a positive constant. Fringing fields are negligible. Which of the following expressions represents the correct integral setup to determine the total capacitance \(C\) of this capacitor?

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