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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 an insulating material whose dielectric constant varies continuously with perpendicular distance \\(y\\) from the bottom plate (\\(y = 0\\)) to the top plate (\\(y = d\\)) according to \\(\\kappa(y) = \\kappa_1 + (\\kappa_2 – \\kappa_1)\\dfrac{y}{d}\\), where \\(\\kappa_1\\) and \\(\\kappa_2\\) are positive constants with \\(\\kappa_2 > \\kappa_1\\). Which of the following expressions represents the equivalent capacitance of this capacitor?"
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url: "https://nerd-notes.com/ubq/121160/"
date_modified: "2026-08-23T04:58:20+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 an insulating material whose dielectric constant varies continuously with perpendicular distance \(y\) from the bottom plate (\(y = 0\)) to the top plate (\(y = d\)) according to \(\kappa(y) = \kappa_1 + (\kappa_2 – \kappa_1)\dfrac{y}{d}\), where \(\kappa_1\) and \(\kappa_2\) are positive constants with \(\kappa_2 > \kappa_1\). Which of the following expressions represents the equivalent capacitance 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 an insulating material whose dielectric constant varies continuously with perpendicular distance \(y\) from the bottom plate (\(y = 0\)) to the top plate (\(y = d\)) according to \(\kappa(y) = \kappa_1 + (\kappa_2 - \kappa_1)\dfrac{y}{d}\), where \(\kappa_1\) and \(\kappa_2\) are positive constants with \(\kappa_2 > \kappa_1\). Which of the following expressions represents the equivalent capacitance of this capacitor?

![A parallel-plate capacitor shown in cross section with two horizontal conducting plates of width representing area \(A\) separated vertically by distance \(d\). The bottom plate is at \(y = 0\) and the top plate is at \(y = d\). A vertical dashed coordinate axis labeled \(y\) points upward from the bottom plate to the top plate, with tick marks at \(0\) and \(d\). The region between the plates is filled with a shaded gradient representing the dielectric, varying from lighter shading at the bottom labeled \(\kappa_1\) to darker shading at the top labeled \(\kappa_2\). A thin horizontal rectangular slice of thickness \(dy\) is outlined at an arbitrary height \(y\) between the plates. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461099-md2l6f.jpg)

- **A.** \(C = \dfrac{\varepsilon_0 A}{d}\left(\dfrac{\kappa_1 + \kappa_2}{2}\right)\)
- **B.** \(C = \dfrac{\varepsilon_0 A (\kappa_2 - \kappa_1)}{d \ln\left(\dfrac{\kappa_2}{\kappa_1}\right)}\)
- **C.** \(C = \dfrac{\varepsilon_0 A}{d}\left(\dfrac{\ln\left(\dfrac{\kappa_2}{\kappa_1}\right)}{\kappa_2 - \kappa_1}\right)\)
- **D.** \(C = \dfrac{\varepsilon_0 A}{d}\left(\dfrac{2\kappa_1 \kappa_2}{\kappa_1 + \kappa_2}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121160/*