---
title: "A group of students designs an experiment to determine the dielectric constant \\(\\kappa\\) of an unknown liquid. They use a parallel-plate capacitor with large rectangular plates of width \\(W = 0.50 \\text{ m}\\) and height \\(H = 0.30 \\text{ m}\\), separated by a small gap of \\(d = 2.0 \\times 10^{-3} \\text{ m}\\). The capacitor is placed upright in an empty container, as shown in Figure 1, and connected to a meter that measures capacitance \\(C\\). The students slowly pour the liquid into the container, measuring the capacitance \\(C\\) as a function of the depth \\(x\\) of the liquid."
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url: "https://nerd-notes.com/ubq/117847/"
date_modified: "2026-08-04T07:59:23+00:00"
---

# A group of students designs an experiment to determine the dielectric constant \(\kappa\) of an unknown liquid. They use a parallel-plate capacitor with large rectangular plates of width \(W = 0.50 \text{ m}\) and height \(H = 0.30 \text{ m}\), separated by a small gap of \(d = 2.0 \times 10^{-3} \text{ m}\). The capacitor is placed upright in an empty container, as shown in Figure 1, and connected to a meter that measures capacitance \(C\). The students slowly pour the liquid into the container, measuring the capacitance \(C\) as a function of the depth \(x\) of the liquid.

A group of students designs an experiment to determine the dielectric constant \(\kappa\) of an unknown liquid. They use a parallel-plate capacitor with large rectangular plates of width \(W = 0.50 \text{ m}\) and height \(H = 0.30 \text{ m}\), separated by a small gap of \(d = 2.0 \times 10^{-3} \text{ m}\). The capacitor is placed upright in an empty container, as shown in Figure 1, and connected to a meter that measures capacitance \(C\). The students slowly pour the liquid into the container, measuring the capacitance \(C\) as a function of the depth \(x\) of the liquid.

![A side-view cross-section of an open rectangular container. Inside the container, two tall vertical parallel plates are positioned side-by-side, separated by a small gap labeled \( d \). The plates have a total height labeled \( H \). A liquid is pooled in the bottom of the container, submerging the lower portion of both plates up to a depth labeled \( x \). Wires extend from the top edges of both plates upward to a rectangular box labeled 'Capacitance Meter'. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830362-6OgexI.jpg)

**Part a)** The capacitor can be modeled as two capacitors in parallel: one with air between the plates and one with the liquid between the plates. **Derive** an expression for the total capacitance \(C\) as a function of the liquid depth \(x\). Express your answer in terms of \(W\), \(H\), \(d\), \(x\), \(\kappa\), and physical constants, as appropriate. *(3 points)*

**Part b)** The students collect the following data. | \( x \) (m) | \( C \) (nF) | |-------------|--------------| | 0.00        | 0.66         | | 0.05        | 0.89         | | 0.10        | 1.12         | | 0.15        | 1.31         | | 0.20        | 1.56         | | 0.25        | 1.75         | **Plot** the data points on the axes provided. **Draw** a best-fit line for the data. *(2 points)*

**Part c)** The following analysis uses the best-fit line from your graph. *(4 points)*

**Part d)** The container is now filled so that the capacitor is fully submerged in the liquid (\(x = H\)). The capacitor is connected to a battery that maintains a constant potential difference \(V_0\) across the plates. While the battery remains connected, a valve at the bottom of the container is opened, and the liquid drains out until the container is empty. *(3 points)*

**Part e)** The experiment is repeated, but this time the fully submerged capacitor is first charged to a potential difference \(V_0\) and then **disconnected** from the battery before the liquid begins to drain. As the liquid drains, does the potential difference across the plates increase, decrease, or remain the same? - [ ] Increases - [ ] Decreases - [ ] Remains the same **Justify** your answer. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117847/*