---
title: "A parallel-plate capacitor with plate area \\( A \\) and separation distance \\( d \\) is connected in a circuit to a battery that maintains a constant potential difference \\( V_0 \\). Initially, there is a vacuum between the plates, and the capacitor is fully charged. A solid dielectric slab with dielectric constant \\( \\kappa \\) (where \\( \\kappa > 1 \\)) is placed just outside the plates. An external agent slowly pushes the slab into the capacitor until it completely fills the space between the plates. The capacitor remains connected to the battery during this entire process."
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url: "https://nerd-notes.com/ubq/117706/"
date_modified: "2026-08-04T07:53:35+00:00"
---

# A parallel-plate capacitor with plate area \( A \) and separation distance \( d \) is connected in a circuit to a battery that maintains a constant potential difference \( V_0 \). Initially, there is a vacuum between the plates, and the capacitor is fully charged. A solid dielectric slab with dielectric constant \( \kappa \) (where \( \kappa > 1 \)) is placed just outside the plates. An external agent slowly pushes the slab into the capacitor until it completely fills the space between the plates. The capacitor remains connected to the battery during this entire process.

A parallel-plate capacitor with plate area \( A \) and separation distance \( d \) is connected in a circuit to a battery that maintains a constant potential difference \( V_0 \). Initially, there is a vacuum between the plates, and the capacitor is fully charged. A solid dielectric slab with dielectric constant \( \kappa \) (where \( \kappa > 1 \)) is placed just outside the plates. An external agent slowly pushes the slab into the capacitor until it completely fills the space between the plates. The capacitor remains connected to the battery during this entire process.

![A schematic showing a DC battery with a voltage label \( V_0 \) connected by wires to a parallel-plate capacitor. The capacitor consists of two horizontal, parallel plates. To the right of the plates, a rectangular shaded block labeled 'Dielectric, \( \kappa \)' is shown partially inserted between the plates. A horizontal arrow points to the left from the right edge of the dielectric, indicating its direction of movement into the capacitor gap. Vertical dashed lines with a double-headed arrow indicate the plate separation distance \( d \). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830015-QVSlBG.jpg)

**Part a)** **Indicate** whether the electric potential energy stored in the capacitor increases, decreases, or remains the same after the dielectric is fully inserted. - [ ] Increases - [ ] Decreases - [ ] Remains the same **Justify** your answer qualitatively using physical principles, without manipulating equations. *(2 points)*

**Part b)** For the following parts, express your answers in terms of \( A \), \( d \), \( V_0 \), \( \kappa \), and fundamental constants, as appropriate. *(4 points)*

**Part c)** A student looks at the derived expressions for \( \Delta U \) and \( W_{batt} \) and states, "The work done by the battery is greater than the energy gained by the capacitor. To satisfy the law of conservation of energy, the external agent must have done negative work to keep the dielectric moving slowly, meaning the electric field of the capacitor exerted an inward attractive force on the dielectric." **Evaluate** the student's claim. **Explain** how the expressions you derived in part (b) support or refute the student's reasoning about the conservation of energy and the forces involved. *(3 points)*

**Part d)** Now consider a modified experiment: The identical capacitor is connected to the battery and allowed to fully charge. It is then completely *disconnected* from the battery so that it is isolated. The identical dielectric slab is then slowly inserted by the external agent until it fills the space between the plates. Does the electric potential energy of the isolated capacitor increase, decrease, or remain the same when the dielectric is inserted? **Support your claim** with evidence using physical principles, and **explain** how this qualitative claim is consistent with the idea that the electric field of the capacitor exerts an inward attractive force on the dielectric. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117706/*