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title: "A parallel-plate capacitor with vacuum between its plates has plate area \\(A\\), plate separation \\(d\\), and free surface charge density of magnitude \\(\\sigma_0\\) on each plate. The capacitor is disconnected from the charging source, and an uncharged dielectric slab of dielectric constant \\(\\kappa > 1\\) is inserted to completely fill the space between the plates.  | | Net Electric Field \\(E\\) | Induced Surface Charge Density \\(\\sigma_{\\text{ind}}\\) | Stored Electrostatic Energy \\(U\\) | |—|—|—|—| | (A) | \\(\\dfrac{E_0}{\\kappa}\\) | \\(\\sigma_0\\left(1 – \\dfrac{1}{\\kappa}\\right)\\) | \\(\\dfrac{U_0}{\\kappa}\\) | | (B) | \\(\\dfrac{E_0}{\\kappa}\\) | \\(\\dfrac{\\sigma_0}{\\kappa}\\) | \\(\\dfrac{U_0}{\\kappa}\\) | | (C) | \\(\\dfrac{E_0}{\\kappa}\\) | \\(\\sigma_0\\left(1 – \\dfrac{1}{\\kappa}\\right)\\) | \\(\\kappa U_0\\) | | (D) | \\(E_0\\) | \\(\\sigma_0(\\kappa – 1)\\) | \\(U_0\\) |  Which row in the table correctly compares the resulting net electric field magnitude \\(E\\), the induced surface charge density magnitude \\(\\sigma_{\\text{ind}}\\) on the dielectric faces, and the stored electrostatic energy \\(U\\) to their initial values \\(E_0\\), \\(\\sigma_0\\), and \\(U_0\\)?"
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url: "https://nerd-notes.com/ubq/121177/"
date_modified: "2026-08-23T04:58:24+00:00"
---

# A parallel-plate capacitor with vacuum between its plates has plate area \(A\), plate separation \(d\), and free surface charge density of magnitude \(\sigma_0\) on each plate. The capacitor is disconnected from the charging source, and an uncharged dielectric slab of dielectric constant \(\kappa > 1\) is inserted to completely fill the space between the plates.

| | Net Electric Field \(E\) | Induced Surface Charge Density \(\sigma_{\text{ind}}\) | Stored Electrostatic Energy \(U\) |
|—|—|—|—|
| (A) | \(\dfrac{E_0}{\kappa}\) | \(\sigma_0\left(1 – \dfrac{1}{\kappa}\right)\) | \(\dfrac{U_0}{\kappa}\) |
| (B) | \(\dfrac{E_0}{\kappa}\) | \(\dfrac{\sigma_0}{\kappa}\) | \(\dfrac{U_0}{\kappa}\) |
| (C) | \(\dfrac{E_0}{\kappa}\) | \(\sigma_0\left(1 – \dfrac{1}{\kappa}\right)\) | \(\kappa U_0\) |
| (D) | \(E_0\) | \(\sigma_0(\kappa – 1)\) | \(U_0\) |

Which row in the table correctly compares the resulting net electric field magnitude \(E\), the induced surface charge density magnitude \(\sigma_{\text{ind}}\) on the dielectric faces, and the stored electrostatic energy \(U\) to their initial values \(E_0\), \(\sigma_0\), and \(U_0\)?

A parallel-plate capacitor with vacuum between its plates has plate area \(A\), plate separation \(d\), and free surface charge density of magnitude \(\sigma_0\) on each plate. The capacitor is disconnected from the charging source, and an uncharged dielectric slab of dielectric constant \(\kappa > 1\) is inserted to completely fill the space between the plates.

| | Net Electric Field \(E\) | Induced Surface Charge Density \(\sigma_{\text{ind}}\) | Stored Electrostatic Energy \(U\) |
|---|---|---|---|
| (A) | \(\dfrac{E_0}{\kappa}\) | \(\sigma_0\left(1 - \dfrac{1}{\kappa}\right)\) | \(\dfrac{U_0}{\kappa}\) |
| (B) | \(\dfrac{E_0}{\kappa}\) | \(\dfrac{\sigma_0}{\kappa}\) | \(\dfrac{U_0}{\kappa}\) |
| (C) | \(\dfrac{E_0}{\kappa}\) | \(\sigma_0\left(1 - \dfrac{1}{\kappa}\right)\) | \(\kappa U_0\) |
| (D) | \(E_0\) | \(\sigma_0(\kappa - 1)\) | \(U_0\) |

Which row in the table correctly compares the resulting net electric field magnitude \(E\), the induced surface charge density magnitude \(\sigma_{\text{ind}}\) on the dielectric faces, and the stored electrostatic energy \(U\) to their initial values \(E_0\), \(\sigma_0\), and \(U_0\)?

![A schematic cross section of a parallel-plate capacitor. Two horizontal parallel gray rectangular plates of length L are separated vertically by distance d. The top plate is labeled with charge density +\sigma_0 and the bottom plate is labeled -\sigma_0. Between the two plates, a solid light gray slab completely fills the entire gap and is labeled with dielectric constant \kappa. On the top interior surface of the slab, four evenly spaced minus signs represent induced surface charge. On the bottom interior surface of the slab, four evenly spaced plus signs represent induced surface charge. A vertical double-headed arrow between the plates is labeled d. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461103-uyN4sk.jpg)

- **A.** Row A
- **B.** Row B
- **C.** Row C
- **D.** Row D

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121177/*