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title: "A spherical capacitor is formed by a solid conducting sphere of radius \\(a\\) concentric with a thin conducting spherical shell of radius \\(b\\), where \\(a < b\\). The inner sphere carries charge \\(+Q\\), and the outer shell carries charge \\(-Q\\). The region between the conductors is filled with an inhomogeneous dielectric whose dielectric constant depends on the radial distance \\(r\\) according to \\(\\kappa(r) = \\kappa_0 \\dfrac{r}{a}\\), where \\(\\kappa_0\\) is a positive dimensionless constant. Which of the following expressions correctly represents the potential difference \\(\\Delta V = V(a) – V(b)\\) between the two conductors?"
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url: "https://nerd-notes.com/ubq/121152/"
date_modified: "2026-08-23T04:58:18+00:00"
---

# A spherical capacitor is formed by a solid conducting sphere of radius \(a\) concentric with a thin conducting spherical shell of radius \(b\), where \(a < b\). The inner sphere carries charge \(+Q\), and the outer shell carries charge \(-Q\). The region between the conductors is filled with an inhomogeneous dielectric whose dielectric constant depends on the radial distance \(r\) according to \(\kappa(r) = \kappa_0 \dfrac{r}{a}\), where \(\kappa_0\) is a positive dimensionless constant. Which of the following expressions correctly represents the potential difference \(\Delta V = V(a) – V(b)\) between the two conductors?

A spherical capacitor is formed by a solid conducting sphere of radius \(a\) concentric with a thin conducting spherical shell of radius \(b\), where \(a < b\). The inner sphere carries charge \(+Q\), and the outer shell carries charge \(-Q\). The region between the conductors is filled with an inhomogeneous dielectric whose dielectric constant depends on the radial distance \(r\) according to \(\kappa(r) = \kappa_0 \dfrac{r}{a}\), where \(\kappa_0\) is a positive dimensionless constant. Which of the following expressions correctly represents the potential difference \(\Delta V = V(a) - V(b)\) between the two conductors?

![A cross-sectional view of two concentric circles centered at a common origin point. The inner circle of radius \(a\) is solid gray, with a thin solid radial arrow from the center to its outer edge labeled \(a\), and text labeled \(+Q\) positioned in its interior. The outer circle has a larger radius \(b\), with a thin solid radial arrow from the center to its circumference labeled \(b\), and text labeled \(-Q\) just outside the outer boundary. The annular region between radius \(a\) and radius \(b\) has a light gray fill with a radial gradient, annotated by text \(\kappa(r) = \kappa_0 \dfrac{r}{a}\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461098-8u0oBL.jpg)

- **A.** \(\Delta V = \dfrac{Q \kappa_0}{4\pi \varepsilon_0 a} \displaystyle\int_a^b \dfrac{1}{r} \, dr\)
- **B.** \(\Delta V = \dfrac{Q a}{4\pi \varepsilon_0 \kappa_0} \displaystyle\int_a^b \dfrac{1}{r^3} \, dr\)
- **C.** \(\Delta V = \dfrac{Q}{4\pi \varepsilon_0 \kappa_0} \displaystyle\int_a^b \dfrac{1}{r^2} \, dr\)
- **D.** \(\Delta V = \dfrac{Q}{4\pi \varepsilon_0 \kappa_0 a} \displaystyle\int_a^b \dfrac{1}{r} \, dr\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121152/*