A capacitor is formed from two concentric spherical...

A capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has radius 12.5 cm, and the outer sphere has radius 14.8 cm. A potential difference of 120 V is applied to the capacitor. (a) What is the energy density at $r$ = 12.6 cm, just outside the inner sphere? (b) What is the energy density at $r$ = 14.7 cm, just inside the outer sphere? (c) For a parallel-plate capacitor the energy density is uniform in the region between the plates, except near the edges of the plates. Is this also true for a spherical capacitor?

Key Concepts

Gauss's Law

Gauss's law is a fundamental principle of electrostatics used to relate the electric field to the charge enclosed by a closed surface. For spherically symmetric charge distributions, this law greatly simplifies the calculation of the electric field at any point. This concept is foundational for understanding and solving problems involving spherical capacitors, where the electric field varies with radius.

Energy Density

Energy density in an electric field quantifies the energy stored per unit volume and is given by the formula u = 1/2 ?0 E^2 in vacuum. This concept is key in electromagnetism because it allows the analysis of how energy is distributed in space, and how modifications in geometry or field strength directly affect the energy stored in capacitive systems.

Comparison with Parallel-Plate Capacitors

Unlike parallel-plate capacitors, which typically exhibit a uniform electric field and energy density between the plates (ignoring edge effects), the energy density in a spherical capacitor is non-uniform due to the radial variation of the electric field. This concept highlights the importance of geometry in determining field distributions and energy storage characteristics in different capacitor configurations.

Spherical Capacitor

A spherical capacitor is a device consisting of two concentric spherical conductive shells separated by a dielectric (in this case, vacuum). This setup creates a radial electric field that varies with distance from the center. Understanding spherical capacitors is essential in electrostatics because their geometry leads to a distinct distribution of electric fields and potential differences compared to other capacitor geometries.

Electric Field in Spherical Geometry

In spherical systems, the electric field emanating from a charged conductor decreases with the square of the distance from the center, a result that can be derived using Gauss’s law. The radial symmetry simplifies the analysis, as the field at any point depends solely on the distance from the center. This concept is crucial for determining quantities such as potential differences and energy densities at various radial positions within the capacitor.

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