---
title: "Spherical Capacitor | Principles of Physics II"
description: "A spherical capacitor is two concentric conducting spheres that store charge in the electric field between them, with capacitance set by geometry and dielectric."
canonical: "https://fiveable.me/principles-physics-ii/key-terms/spherical-capacitor"
type: "key-term"
subject: "Principles of Physics II"
unit: "Unit 3"
---

# Spherical Capacitor | Principles of Physics II

## Definition

A spherical capacitor in Principles of Physics II is two concentric conducting shells separated by an insulator. It stores charge in the electric field between the shells, and its capacitance depends on the radii and the dielectric in between.

## What It Is

A spherical capacitor is a capacitor made from two concentric conducting spheres, one inside the other, with an insulating gap between them. In Principles of Physics II, you study it as a clean electrostatics model where geometry controls how charge, electric field, and energy fit together.

When the inner sphere carries charge +Q, the outer sphere carries -Q, and the field exists only in the region between the shells. Because the shells are spherical, the electric field in that region points radially and has the same magnitude everywhere on a sphere of radius r centered on the shells. That symmetry is what makes the math work out neatly.

Using Gauss’s law, the field between the shells comes from the enclosed charge on the inner conductor. The potential difference is found by integrating that radial field from the inner radius r1 to the outer radius r2. Once you know V, capacitance follows from C = Q/V, which gives the standard result C = 4πϵ0(r1r2)/(r2 - r1) for air or vacuum.

That formula tells you something physical, not just mathematical. If the spheres are closer together, the capacitance gets larger because the same charge produces a smaller voltage difference. If you insert a dielectric, the capacitance increases again because the material reduces the effective electric field and lets the system hold more charge for the same V.

A nice way to picture it is to compare it to a parallel-plate capacitor. A spherical capacitor is not built for flat, uniform plates, so the field pattern is different, but the same core idea still applies: separated charge creates an electric field, and that field stores energy. The spherical version is especially useful as a symmetry example, since it shows how conductor shape changes the field while the capacitor idea stays the same.

## Why It Matters

A spherical capacitor shows you how electrostatics connects geometry to measurable electrical quantities. In Principles of Physics II, that matters because many later topics rely on the same workflow: identify symmetry, find the electric field, relate field to potential, then use potential to get capacitance or energy.

This term also gives you practice with Gauss’s law in a setting where the symmetry is strong enough to make the calculation manageable. If you can handle a spherical capacitor, you are in good shape for other radial-field problems, like isolated charged spheres or charge distributions with spherical symmetry.

It also sharpens your understanding of what capacitance really means. Capacitance is not just a property of a material or a device label, it is a ratio, Q/V, that depends on shape, separation, and dielectric. That idea shows up again when you compare capacitor designs or explain why one geometry stores more charge than another at the same voltage.

For labs, homework, and quizzes, this term often appears in setups where you identify which conductor is inner, which is outer, and how the potential difference is built from the field in between. It is a good check on whether you can move from a picture of charge on shells to an equation for stored energy or capacitance without mixing up the roles of Q, V, and E.

## Connections

### Capacitance

Spherical capacitors are one specific place where capacitance shows up as C = Q/V. The geometry changes the value of C, but the definition stays the same. In problems, you often start with the capacitor shape, find the potential difference, and then compute capacitance from the charge stored on the shells.

### Dielectric

The material between the spheres can raise the capacitance by reducing the electric field for a given charge. In a spherical capacitor problem, a dielectric changes the relationship between charge and voltage, so you may need to use the dielectric constant or permittivity instead of vacuum values.

### Electric Field

The field between the concentric shells is the reason the capacitor stores energy. Because the geometry is spherical, the field points radially and depends on distance from the center. That makes spherical capacitors a strong example of how field direction and symmetry control the whole calculation.

### [cylindrical capacitor](/principles-physics-ii/key-terms/cylindrical-capacitor)

A cylindrical capacitor uses a different geometry, but the same core idea: two conductors separated by an insulator store energy in the field between them. Comparing spherical and cylindrical capacitors helps you see how symmetry changes the field equation and the final capacitance formula.

## On the AP Exam

A quiz or problem-set question usually gives you the inner and outer radii, a charge, or a potential difference and asks you to find capacitance, voltage, or stored energy. The move is to recognize spherical symmetry, use the radial electric field between the shells, and connect that field to the potential difference with an integral or the provided formula.

If the problem includes a dielectric, you replace the vacuum permittivity with the appropriate permittivity for the region. If the shells are connected to a battery or isolated after charging, pay attention to what stays fixed, charge or voltage, because that changes the answer. On diagram questions, you should be able to label the inner conductor, outer conductor, and the direction of the field lines in the space between them.

## spherical capacitor vs cylindrical capacitor

These are easy to mix up because both use curved conductors and both store charge in the field between them. The difference is symmetry: a spherical capacitor has concentric spheres, while a cylindrical capacitor has coaxial cylinders. That changes the electric field pattern and the capacitance formula.

## Key Takeaways

- A spherical capacitor is two concentric conducting spheres separated by an insulator.
- Its capacitance depends on the radii of the spheres and on the dielectric between them.
- The electric field between the shells is radial and comes from the charge on the inner conductor.
- Capacitance is still defined by C = Q/V, so geometry changes voltage for a given charge.
- You can use spherical capacitors to practice Gauss’s law, potential difference, and energy storage.

## FAQs

### What is a spherical capacitor in Principles of Physics II?

It is a capacitor made of two concentric spherical conductors separated by an insulating gap. Charge sits on the shells, and the electric field between them stores energy. The symmetry makes it a standard electrostatics model for using Gauss’s law.

### How do you find the capacitance of a spherical capacitor?

For air or vacuum, use C = 4πϵ0(r1r2)/(r2 - r1), where r1 is the inner radius and r2 is the outer radius. The formula comes from finding the radial electric field between the spheres and integrating it to get the voltage difference.

### Is a spherical capacitor the same as a parallel-plate capacitor?

No. They both store charge and energy, but the field geometry is different. A parallel-plate capacitor has nearly uniform field between flat plates, while a spherical capacitor has a radial field between curved shells.

### What happens if you put a dielectric in a spherical capacitor?

The capacitance increases because the dielectric lowers the effective electric field for the same free charge. In problems, that usually means using the material’s permittivity instead of the vacuum value. It can also change the voltage required to hold a given charge.

## Related Study Guides

- [3.1 Capacitors](/principles-physics-ii/unit-3/capacitors/study-guide/4GgdhS3PGMTxTMsX)
- [3.2 Capacitance](/principles-physics-ii/unit-3/capacitance/study-guide/zRXmTODTx0RWE8cu)

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