---
title: "A long cylindrical capacitor of length \\(L\\) consists of a solid inner conducting cylinder of radius \\(a\\) and a concentric outer conducting cylindrical shell of inner radius \\(c\\) and outer radius \\(d\\). The space between the conductors is filled with two concentric layers of dielectric material. Layer 1 fills the region \\(a < r < b\\) and has a dielectric constant \\(\\kappa_1\\). Layer 2 fills the region \\(b < r  \\kappa_2\\). The total length \\(L\\) is much greater than \\(d\\). The inner cylinder has a net positive charge \\(+Q\\) and the outer shell has a net negative charge \\(-Q\\)."
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url: "https://nerd-notes.com/ubq/117858/"
date_modified: "2026-08-04T07:59:41+00:00"
---

# A long cylindrical capacitor of length \(L\) consists of a solid inner conducting cylinder of radius \(a\) and a concentric outer conducting cylindrical shell of inner radius \(c\) and outer radius \(d\). The space between the conductors is filled with two concentric layers of dielectric material. Layer 1 fills the region \(a < r < b\) and has a dielectric constant \(\kappa_1\). Layer 2 fills the region \(b < r  \kappa_2\). The total length \(L\) is much greater than \(d\). The inner cylinder has a net positive charge \(+Q\) and the outer shell has a net negative charge \(-Q\).

A long cylindrical capacitor of length \(L\) consists of a solid inner conducting cylinder of radius \(a\) and a concentric outer conducting cylindrical shell of inner radius \(c\) and outer radius \(d\). The space between the conductors is filled with two concentric layers of dielectric material. Layer 1 fills the region \(a < r < b\) and has a dielectric constant \(\kappa_1\). Layer 2 fills the region \(b < r < c\) and has a dielectric constant \(\kappa_2\). It is given that \(\kappa_1 > \kappa_2\). The total length \(L\) is much greater than \(d\). The inner cylinder has a net positive charge \(+Q\) and the outer shell has a net negative charge \(-Q\).

![A flat cross-sectional view of a concentric cylindrical system. A central solid circle has a radius labeled \(a\). A larger concentric dashed circle has a radius labeled \(b\). A larger concentric solid circle has a radius labeled \(c\). An outermost concentric solid circle has a radius labeled \(d\). The annular region between \(a\) and \(b\) is shaded light gray and labeled 'Layer 1 \(\kappa_1\)'. The annular region between \(b\) and \(c\) is hatched with diagonal lines and labeled 'Layer 2 \(\kappa_2\)'. The annular region between \(c\) and \(d\) is shaded dark gray. A label \(+Q\) is placed on the inner circle. A label \(-Q\) is placed on the dark gray region. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830380-lsyeKn.jpg)

**Part a)** On the axes provided, **sketch** a graph of the magnitude of the electric field \(E\) as a function of the radial distance \(r\) from the center of the inner cylinder, from \(r = 0\) to a point outside the capacitor (\(r > d\)).

**Part b)** **Derive** an expression for the capacitance \(C\) of this cylindrical capacitor. Express your answer in terms of \(L\), \(a\), \(b\), \(c\), \(\kappa_1\), \(\kappa_2\), and fundamental constants, as appropriate.

**Part c)** Consider the boundary between the two dielectric layers at \(r=b\).

**Part d)** Suppose Layer 1 is removed from the capacitor, leaving a vacuum (\(\kappa = 1\)) in the region \(a < r < b\). The capacitor remains isolated, so the total charges \(+Q\) and \(-Q\) on the conductors remain unchanged.


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117858/*