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Electricity and Magnetism
Charge Density and Curvature of Surface
A sphere of radius R, having a charge of magnitude Q, has a charge density (charge per unit surface area) D, is given by
The potential, V, at the surface of the sphere is therefore given by
If two charged metal spheres are placed in contact, as shown in the diagram below, they must have the same potential.
Therefore, we can write
This means that the smaller sphere has the greater charge density.
This helps to explain why the charge on a conductor is concentrated at places where the radius of curvature is smaller (for example, the point of a pin).
An equipotential in a field is a line (or surface) joining all points which have the same potential.
An equipotential is therefore a line (or surface) along which a charge can be moved without work being done against (or by) the electric field. This means that equipotentials must always be at 90° to electric field lines so equipotentials near a single point charge are spherical.
For a charge of about 1×10-12C, the 1V equipotential is a sphere of radius 1cm, the 0·5V is a sphere of radius 2cm etc.