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title: "A parallel-plate capacitor has plates of area \\(A\\) separated by a distance \\(d\\) carrying uniform surface charges \\(+\\sigma\\) at \\(x = 0\\) and \\(-\\sigma\\) at \\(x = d\\). Two solid dielectric slabs of equal thickness \\(\\dfrac{d}{2}\\) completely fill the gap: Slab 1 with dielectric constant \\(\\kappa_1\\) occupies \\(0 < x < \\dfrac{d}{2}\\), and Slab 2 with dielectric constant \\(\\kappa_2\\) occupies \\(\\dfrac{d}{2} < x < d\\). The graph shows the electric field magnitude \\(E(x)\\) as a function of position \\(x\\) between the plates.  Which of the following claims about the system is correct based on the graph?"
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url: "https://nerd-notes.com/ubq/121143/"
date_modified: "2026-08-23T04:58:18+00:00"
---

# A parallel-plate capacitor has plates of area \(A\) separated by a distance \(d\) carrying uniform surface charges \(+\sigma\) at \(x = 0\) and \(-\sigma\) at \(x = d\). Two solid dielectric slabs of equal thickness \(\dfrac{d}{2}\) completely fill the gap: Slab 1 with dielectric constant \(\kappa_1\) occupies \(0 < x < \dfrac{d}{2}\), and Slab 2 with dielectric constant \(\kappa_2\) occupies \(\dfrac{d}{2} < x < d\). The graph shows the electric field magnitude \(E(x)\) as a function of position \(x\) between the plates.

Which of the following claims about the system is correct based on the graph?

A parallel-plate capacitor has plates of area \(A\) separated by a distance \(d\) carrying uniform surface charges \(+\sigma\) at \(x = 0\) and \(-\sigma\) at \(x = d\). Two solid dielectric slabs of equal thickness \(\dfrac{d}{2}\) completely fill the gap: Slab 1 with dielectric constant \(\kappa_1\) occupies \(0 < x < \dfrac{d}{2}\), and Slab 2 with dielectric constant \(\kappa_2\) occupies \(\dfrac{d}{2} < x < d\). The graph shows the electric field magnitude \(E(x)\) as a function of position \(x\) between the plates.

Which of the following claims about the system is correct based on the graph?

![A Cartesian graph with a vertical axis labeled \(E(x)\) and a horizontal axis labeled \(x\). The horizontal axis has tick marks labeled \(0\), \(\dfrac{d}{2}\), and \(d\). The vertical axis has tick marks labeled \(0\), \(E_0\), and \(2E_0\). A solid horizontal line segment is plotted at vertical height \(2E_0\) spanning horizontally from \(x = 0\) to \(x = \dfrac{d}{2}\). A vertical dashed line extends downward at \(x = \dfrac{d}{2}\) from \(2E_0\) to \(E_0\). A second solid horizontal line segment is plotted at vertical height \(E_0\) spanning horizontally from \(x = \dfrac{d}{2}\) to \(x = d\). Light horizontal dashed gridlines extend from the vertical axis ticks at \(E_0\) and \(2E_0\) to the respective line segments. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461097-oq74vv.jpg)

- **A.** The dielectric constant of Slab 1 is twice that of Slab 2 (\(\kappa_1 = 2\kappa_2\)) because the electric field magnitude is directly proportional to the dielectric constant.
- **B.** The potential difference across Slab 1 is twice the potential difference across Slab 2 because the area under the \(E(x)\) curve from \(x = 0\) to \(x = \dfrac{d}{2}\) is twice the area from \(x = \dfrac{d}{2}\) to \(x = d\).
- **C.** The electrostatic energy density in Slab 1 is four times that in Slab 2 because energy density depends solely on the square of the electric field magnitude.
- **D.** The electric potential is discontinuous at the interface \(x = \dfrac{d}{2}\) because the electric field graph exhibits a step discontinuity at that boundary.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121143/*