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title: "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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url: "https://nerd-notes.com/ubq/121155/"
date_modified: "2026-08-23T04:58:18+00:00"
---

# 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?

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?

![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.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461098-MXbumn.jpg)

- **A.** \(C = \dfrac{\varepsilon_0 A}{d^2} \displaystyle\int_0^d (1 + \alpha x)\,dx\)
- **B.** \(C = \dfrac{\varepsilon_0 A}{\displaystyle\int_0^d (1 + \alpha x)\,dx}\)
- **C.** \(C = \dfrac{\varepsilon_0 A}{\displaystyle\int_0^d \dfrac{1}{1 + \alpha x}\,dx}\)
- **D.** \(C = \dfrac{\varepsilon_0 A}{\displaystyle\int_0^d \dfrac{x}{d(1 + \alpha x)}\,dx}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121155/*