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.

A spherical capacitor consists of two oppositely charged concentric spherical shells separated by an insulator. The inner shell radius is R₁, and the outer shell radius is R₂.

Considering a spherical Gaussian surface of radius r, the radially outward electric field can be expressed using the  Gauss Law. The electric field is directly proportional to the charge enclosed and inversely proportional to the radius square.

Recall that potential difference can be derived from the electric field. Therefore, integrating the electric field along a radial path between the shells gives the potential difference for a spherical capacitor.

Now, the ratio of charge to the potential difference gives the capacitance for a spherical capacitor.

When the concentric spherical shells are replaced with concentric conducting cylinders, a cylindrical capacitor is formed.

Applying Gauss Law, the electric field directed radially outward from the common axis of the cylinder is calculated.

The potential difference calculated from the electric field can be applied to estimate the capacitance for a cylindrical capacitor.