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title: "A parallel-plate capacitor with plate area \\(A\\) is charged to a net free charge \\(+Q\\) on its top plate and \\(-Q\\) on its bottom plate, then disconnected from the voltage source. The space between the plates is filled by two uniform dielectric slabs stacked on top of one another. Slab 1 (dielectric constant \\(\\kappa_1\\)) rests against the top plate, and Slab 2 (dielectric constant \\(\\kappa_2\\)) rests against the bottom plate, meeting Slab 1 at a planar interface parallel to the plates. Which of the following gives the correct ratio of the electric field magnitudes \\(E_1 / E_2\\) inside the two slabs, and the net bound surface charge density \\(\\sigma_b\\) accumulated at the interface between Slab 1 and Slab 2?"
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url: "https://nerd-notes.com/ubq/118266/"
date_modified: "2026-08-04T08:08:29+00:00"
---

# A parallel-plate capacitor with plate area \(A\) is charged to a net free charge \(+Q\) on its top plate and \(-Q\) on its bottom plate, then disconnected from the voltage source. The space between the plates is filled by two uniform dielectric slabs stacked on top of one another. Slab 1 (dielectric constant \(\kappa_1\)) rests against the top plate, and Slab 2 (dielectric constant \(\kappa_2\)) rests against the bottom plate, meeting Slab 1 at a planar interface parallel to the plates. Which of the following gives the correct ratio of the electric field magnitudes \(E_1 / E_2\) inside the two slabs, and the net bound surface charge density \(\sigma_b\) accumulated at the interface between Slab 1 and Slab 2?

A parallel-plate capacitor with plate area \(A\) is charged to a net free charge \(+Q\) on its top plate and \(-Q\) on its bottom plate, then disconnected from the voltage source. The space between the plates is filled by two uniform dielectric slabs stacked on top of one another. Slab 1 (dielectric constant \(\kappa_1\)) rests against the top plate, and Slab 2 (dielectric constant \(\kappa_2\)) rests against the bottom plate, meeting Slab 1 at a planar interface parallel to the plates. Which of the following gives the correct ratio of the electric field magnitudes \(E_1 / E_2\) inside the two slabs, and the net bound surface charge density \(\sigma_b\) accumulated at the interface between Slab 1 and Slab 2?

![A cross-sectional schematic of a parallel-plate capacitor oriented horizontally. Two thick horizontal lines represent the conducting plates of area A: the top plate is labeled +Q and the bottom plate is labeled -Q. The region between the plates is divided into two horizontal rectangular layers of equal width. The upper layer is shaded light gray and labeled Slab 1, \kappa_1. The lower layer is shaded medium gray and labeled Slab 2, \kappa_2. A dashed horizontal line marks the boundary interface between Slab 1 and Slab 2. A vertical downward arrow next to the capacitor indicates the direction of the electric field from the top plate to the bottom plate. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830909-DSg5jo.jpg)

- **A.** \( \dfrac{E_1}{E_2} = \dfrac{\kappa_1}{\kappa_2} \) and \( \sigma_b = \dfrac{Q}{A}\left(\dfrac{1}{\kappa_1} - \dfrac{1}{\kappa_2}\right) \)
- **B.** \( \dfrac{E_1}{E_2} = \dfrac{\kappa_2}{\kappa_1} \) and \( \sigma_b = \dfrac{Q}{A}\left(\dfrac{1}{\kappa_2} - \dfrac{1}{\kappa_1}\right) \)
- **C.** \( \dfrac{E_1}{E_2} = \dfrac{\kappa_2}{\kappa_1} \) and \( \sigma_b = \dfrac{Q}{A}\left(1 - \dfrac{1}{\kappa_1 \kappa_2}\right) \)
- **D.** \( \dfrac{E_1}{E_2} = \dfrac{\kappa_1}{\kappa_2} \) and \( \sigma_b = \dfrac{Q}{A}\left(\dfrac{\kappa_1 - \kappa_2}{\kappa_1 + \kappa_2}\right) \)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118266/*