A 20.0-μF spherical capacitor is composed of two concentric...

A $20.0-\mu \mathrm{F}$ spherical capacitor is composed of two concentric metal spheres, one having a radius twice as large as the other. The region between the spheres is a vacuum. Determine the volume of this region.

Key Concepts

Spherical Capacitor Geometry

Spherical capacitors consist of two concentric conducting spheres creating a radial symmetry where the electric field is defined solely by the distance from the center. The geometry involves an inner sphere and an outer sphere, and the relation between their radii fundamentally determines the electrical and spatial properties of the device.

Volume of a Spherical Shell

The space between two concentric spheres, known as a spherical shell, has a volume calculated by subtracting the volume of the inner sphere from that of the outer sphere. This is given by the formula V = (4/3)?(R_outer^3 - R_inner^3), which is central to determining the amount of space where no conductive material exists.

Capacitance of a Spherical Capacitor

The capacitance of a spherical capacitor in a vacuum is derived from its geometry, specifically the radii of the inner and outer spheres. It is calculated using the formula C = 4??? (R_inner * R_outer) / (R_outer - R_inner), a relationship that ties the electrical storage capability to the physical dimensions of the capacitor.

Vacuum Permittivity (??)

Vacuum permittivity is a fundamental constant that quantifies the ability of free space to permit electric field lines. It plays a critical role in determining the capacitance of a capacitor in a vacuum, linking the physical geometry of the capacitor to its electrical characteristics.

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