---
title: "A parallel-plate capacitor with plate dimensions of length \\(L\\) and width \\(w\\), separated by distance \\(d\\), is connected across an ideal battery that maintains a constant potential difference \\(V_0\\). An uncharged dielectric slab of dielectric constant \\(\\kappa\\), length \\(L\\), width \\(w\\), and thickness \\(d\\) is initially outside the capacitor. An external agent slowly inserts the slab at constant velocity into the space between the plates until it completely fills the capacitor. In terms of the given quantities and fundamental constants, what is the total work \\(W_{\\text{ext}}\\rangle done by the external agent on the dielectric slab during this process?"
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url: "https://nerd-notes.com/ubq/118279/"
date_modified: "2026-08-04T08:08:37+00:00"
---

# A parallel-plate capacitor with plate dimensions of length \(L\) and width \(w\), separated by distance \(d\), is connected across an ideal battery that maintains a constant potential difference \(V_0\). An uncharged dielectric slab of dielectric constant \(\kappa\), length \(L\), width \(w\), and thickness \(d\) is initially outside the capacitor. An external agent slowly inserts the slab at constant velocity into the space between the plates until it completely fills the capacitor. In terms of the given quantities and fundamental constants, what is the total work \(W_{\text{ext}}\rangle done by the external agent on the dielectric slab during this process?

A parallel-plate capacitor with plate dimensions of length \(L\) and width \(w\), separated by distance \(d\), is connected across an ideal battery that maintains a constant potential difference \(V_0\). An uncharged dielectric slab of dielectric constant \(\kappa\), length \(L\), width \(w\), and thickness \(d\) is initially outside the capacitor. An external agent slowly inserts the slab at constant velocity into the space between the plates until it completely fills the capacitor. In terms of the given quantities and fundamental constants, what is the total work \(W_{\text{ext}}\rangle done by the external agent on the dielectric slab during this process?

![A parallel-plate capacitor shown in cross-section with two horizontal rectangular plates of length L and separation d. The top plate is connected by a thin wire to the positive terminal of a battery of potential difference V_0, and the bottom plate is connected by a thin wire to the negative terminal. A rectangular dielectric slab of length L, thickness d, and dielectric constant \kappa is partially inserted from the left between the plates along the horizontal x-axis, entering a distance x into the region between the plates. A single horizontal arrow labeled x points to the right to show the direction of insertion. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830916-GxO5DX.jpg)

- **A.** \(+\dfrac{(\kappa - 1)\varepsilon_0 w L V_0^2}{2d}\)
- **B.** \(+\dfrac{(\kappa - 1)\varepsilon_0 w L V_0^2}{d}\)
- **C.** -\dfrac{(\kappa - 1)\varepsilon_0 w L V_0^2}{d}
- **D.** -\dfrac{(\kappa - 1)\varepsilon_0 w L V_0^2}{2d}

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118279/*