25.2 : Spherical and Cylindrical Capacitor

A spherical capacitor consists of two concentric conducting spherical shells of radii R₁ (inner shell) and R₂ (outer shell). The shells have  equal and opposite charges of +Q and −Q, respectively. For an isolated conducting spherical capacitor, the radius of the outer shell can be considered to be infinite.

Conventionally, considering the  symmetry, the electric field between the concentric shells of a spherical capacitor is directed radially outward. The magnitude of the field, calculated by applying Gauss’s law over a spherical Gaussian surface of radius r concentric with the shells, is given by,

Substitution of the electric field into the electric field-capacitance relation gives the electric potential as,

However, since the radius of the second sphere is infinite, the potential is given by,

Since, the ratio of charge to potential difference is the capacitance, the capacitance of an isolated conducting spherical capacitor is given by,

A cylindrical capacitor consists of two concentric conducting cylinders of length l and radii R₁ (inner cylinder) and R₂ (outer cylinder). The cylinders are given equal and opposite charges of +Q and -Q, respectively. Consider the calculation of the capacitance of a cylindrical capacitor of length 5 cm and radii 2 mm and 4 mm.

The known quantities are the capacitor’s length and inner and outer radii. The unknown quantity capacitance can be calculated using the known values.

The capacitance of a cylindrical capacitor is given by,

When the known values are substituted into the above equation, the calculated capacitance value is 4.02 pF.