球形和圆柱形电容器

A spherical capacitor consists of two concentric conducting spherical shells with radii R1 (inner) and R2 (outer), separated by an insulator. The shells carry equal and opposite charges of +Q and −Q. The electric field between the shells is directed radially outward and can be calculated using Gauss's law applied to a spherical Gaussian surface.

Capacitance is found by integrating the electric field along a radial path between the shells to determine potential difference, then taking the ratio of charge to potential difference. For an isolated spherical capacitor with infinite outer radius, the capacitance depends only on the inner shell radius R1, making it simpler to calculate than finite geometries.

Potential difference is derived from the electric field by integration along a path between the capacitor plates or shells. The electric field is directly proportional to the enclosed charge and inversely proportional to the radius squared. This relationship allows you to calculate potential difference from the field magnitude and use it to determine capacitance.

A cylindrical capacitor consists of two concentric conducting cylinders of length l and radii R1 (inner) and R2 (outer), with equal and opposite charges. Like the spherical case, Gauss's law determines the radially outward electric field. The cylindrical geometry produces a different capacitance formula that depends on length and the logarithm of the radius ratio.

Gauss's law relates the electric field to the enclosed charge through a Gaussian surface. For spherical capacitors, a concentric spherical surface is used; for cylindrical capacitors, a coaxial cylindrical surface is used. The symmetry of each geometry ensures the electric field is perpendicular to the surface, simplifying the calculation and yielding the field magnitude directly.

To calculate cylindrical capacitor capacitance, you need the length l of the cylinders and the inner radius R1 and outer radius R2. For example, a capacitor with length 5 cm and radii 2 mm and 4 mm yields a capacitance of 4.02 pF. These parameters are substituted into the cylindrical capacitance formula to obtain the result.

Capacitance measures a capacitor's ability to store charge at a given potential difference. The ratio of charge Q to potential difference V defines this storage capacity. For both spherical and cylindrical geometries, this ratio depends only on the physical dimensions and the insulating medium, making it a fundamental geometric property independent of the applied voltage.