Calculations of Electric Potential I

According to Gauss's law, the electric field inside a conductor is zero. Since no work is done moving a test charge within the conductor, the electric potential remains constant throughout the interior and equals the potential at the surface. This uniform potential is a key property of conductors in electrostatic equilibrium.

Outside a charged sphere, the electric field follows the inverse square law. The electric potential is calculated as the integral of the electric field and displacement, with potential defined as zero at infinite distance. This results in potential that varies inversely with radial distance from the sphere's center.

Electric potential is continuous across all regions, including at conductor boundaries. However, the electric field is discontinuous at these boundaries. This discontinuity arises because the electric field depends on the spatial rate of change of potential, which changes abruptly at conductor surfaces.

For a charged ring, the ring is divided into infinitesimal arc elements. Since all elements are equidistant from a point on the ring's axis, the total potential equals the total charge divided by the distance from that point. Integration over all arc elements yields a result equivalent to placing all charge at a common distance.

The cylindrical coordinate system is used to calculate electric potential at a point on the axis of a charged ring. This system naturally describes the geometry of the ring and simplifies the integration process. Each arc element is characterized by its angular position and arc length.

To find potential at specific distances, substitute the known quantities—sphere radius, permittivity of free space, and total charge—into the potential equation. For distances inside the sphere, potential equals the surface value. For distances outside, use the inverse square relationship to determine potential at each radial distance.

Point M on the ring's axis is equidistant from all infinitesimal ring elements. Since potential is a scalar quantity, contributions from all elements add directly without vector complications. The total potential therefore simplifies to the total charge divided by the common distance, as if all charge were concentrated at one point.