24.12. A spherical capacitor is formed from two concentric,...

24.12. A spherical capacitor is formed from two concentric, spherical, conducting shells separated by vacuum. The inner sphere has radius 15.0 $\mathrm{cm}$ and the capacitance is 116 $\mathrm{pF}$ . (a) What is the radius of the outer sphere? (b) If the potential difference between the two spheres is 220 $\mathrm{V}$ , what is the magnitude of charge on each sphere?

Key Concepts

Spherical Capacitors

A spherical capacitor consists of two concentric spherical conductors separated by a dielectric, often vacuum. Its behavior and storage capacity depend on the physical dimensions of the spheres. This concept is fundamental in understanding how geometry affects the storage of electrical energy in an electrostatic system.

Capacitance Formula for Spherical Capacitors

The capacitance of a spherical capacitor is determined by the formula C = 4???(ab)/(b - a), where 'a' and 'b' are the radii of the inner and outer spheres, respectively, and ?? is the permittivity of free space. This formula illustrates how the capacitance increases with the size of the spheres and decreases with the distance between them.

Relationship Between Charge, Potential Difference, and Capacitance

In any capacitor, the charge stored (Q) is directly proportional to the capacitance (C) and the potential difference (V) across its plates, given by the relation Q = CV. This principle connects the geometric and material properties of a capacitor with its electrical performance.

Permittivity of Free Space

The permittivity of free space, denoted by ??, is a fundamental constant that measures the ability of the vacuum to support an electric field. Its value plays a critical role in the calculation of capacitance for devices like spherical capacitors, affecting how electric field lines propagate in a vacuum.

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