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title: "A parallel-plate capacitor consists of two conducting plates, each of area \\(A\\), separated by a distance \\(d\\). The region between the plates is filled with a dielectric material whose dielectric constant varies continuously with distance \\(y\\) from the bottom plate (\\(y = 0\\)) to the top plate (\\(y = d\\)) according to \\(\\kappa(y) = \\kappa_0 \\left(1 + \\dfrac{y}{d}\\right)\\), where \\(\\kappa_0\\) is a positive constant. Which of the following is a correct expression for the total capacitance \\(C\\) of this capacitor?"
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url: "https://nerd-notes.com/ubq/118251/"
date_modified: "2026-08-04T08:08:26+00:00"
---

# A parallel-plate capacitor consists of two conducting plates, each of area \(A\), separated by a distance \(d\). The region between the plates is filled with a dielectric material whose dielectric constant varies continuously with distance \(y\) from the bottom plate (\(y = 0\)) to the top plate (\(y = d\)) according to \(\kappa(y) = \kappa_0 \left(1 + \dfrac{y}{d}\right)\), where \(\kappa_0\) is a positive constant. Which of the following is a correct expression for the total capacitance \(C\) of this capacitor?

A parallel-plate capacitor consists of two conducting plates, each of area \(A\), separated by a distance \(d\). The region between the plates is filled with a dielectric material whose dielectric constant varies continuously with distance \(y\) from the bottom plate (\(y = 0\)) to the top plate (\(y = d\)) according to \(\kappa(y) = \kappa_0 \left(1 + \dfrac{y}{d}\right)\), where \(\kappa_0\) is a positive constant. Which of the following is a correct expression for the total capacitance \(C\) of this capacitor?

![A side-view diagram of a parallel-plate capacitor. Two thick horizontal lines represent parallel conducting plates of length L separated vertically by distance d along a vertical y-axis. The lower plate is located at y = 0 and the upper plate is located at y = d. A vertical axis line with an upward arrow is drawn on the left side, labeled y at the top, with tick marks labeled y = 0 at the lower plate level and y = d at the upper plate level. The region between the two plates is filled with a shaded rectangular region featuring a smooth vertical color gradient, transitioning from light gray near y = 0 to darker gray near y = d. The plates are labeled with capital letter A to indicate plate area. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830905-csReag.jpg)

- **A.** \(\dfrac{\kappa_0 \varepsilon_0 A}{d \ln 2}\)
- **B.** \(\dfrac{\kappa_0 \varepsilon_0 A \ln 2}{d}\)
- **C.** \(\dfrac{2 \kappa_0 \varepsilon_0 A}{3 d}\)
- **D.** \(\dfrac{3 \kappa_0 \varepsilon_0 A}{2 d}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118251/*