---
title: "Two identical air-filled parallel-plate capacitors, each with plate area \\(A\\) and plate separation \\(d\\), are connected in series with an ideal battery of potential difference \\(V_0\\). A slab of linear dielectric material with dielectric constant \\(\\kappa\\) is inserted between the plates of one of the capacitors, completely filling the space between its plates, while the battery remains connected. In terms of \\(\\kappa\\), \\(\\varepsilon_0\\), \\(V_0\\), and \\(d\\), what is the magnitude of the bound surface charge density \\(\\sigma_b\\) induced on the faces of the dielectric slab?"
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url: "https://nerd-notes.com/ubq/124830/"
date_modified: "2026-09-28T14:11:36+00:00"
---

# Two identical air-filled parallel-plate capacitors, each with plate area \(A\) and plate separation \(d\), are connected in series with an ideal battery of potential difference \(V_0\). A slab of linear dielectric material with dielectric constant \(\kappa\) is inserted between the plates of one of the capacitors, completely filling the space between its plates, while the battery remains connected. In terms of \(\kappa\), \(\varepsilon_0\), \(V_0\), and \(d\), what is the magnitude of the bound surface charge density \(\sigma_b\) induced on the faces of the dielectric slab?

Two identical air-filled parallel-plate capacitors, each with plate area \(A\) and plate separation \(d\), are connected in series with an ideal battery of potential difference \(V_0\). A slab of linear dielectric material with dielectric constant \(\kappa\) is inserted between the plates of one of the capacitors, completely filling the space between its plates, while the battery remains connected. In terms of \(\kappa\), \(\varepsilon_0\), \(V_0\), and \(d\), what is the magnitude of the bound surface charge density \(\sigma_b\) induced on the faces of the dielectric slab?

![A single-loop rectangular circuit diagram oriented horizontally. On the left vertical wire segment is a DC battery symbol labeled \(V_0\), with the longer horizontal line at the top indicating the positive terminal. The top horizontal wire connects to a parallel-plate capacitor with two vertical parallel lines of equal length, between which the space is filled with uniform light gray shading labeled \(\kappa\); double-ended horizontal arrows indicate plate separation \(d\). Continuing to the right along the top wire, a second identical parallel-plate capacitor is shown with two vertical parallel lines and empty white space between them, also labeled with plate separation \(d\). The right vertical wire and bottom horizontal wire return to the negative terminal of the battery. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604696-NXKftB.jpg)

- **A.** \(\left(\dfrac{1}{\kappa + 1}\right)\dfrac{\varepsilon_0 V_0}{d}\)
- **B.** \(\left(\dfrac{\kappa - 1}{\kappa + 1}\right)\dfrac{\varepsilon_0 V_0}{d}\)
- **C.** \(\left(\dfrac{\kappa}{\kappa + 1}\right)\dfrac{\varepsilon_0 V_0}{d}\)
- **D.** \(\left(\dfrac{\kappa(\kappa - 1)}{\kappa + 1}\right)\dfrac{\varepsilon_0 V_0}{d}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124830/*