Question

Describe the effect of the radius of curvature of a conductor on the concentration of charges

Answer 1

Electrons in a negatively charged conductor. The electrons all exert force on each other by Culomb's law. Since they are in a conductor they will move (f=ma) until all of the forces on each electron are balanced. When they are in the interior of the charged conductor the net force is towards the surface. However, once they get to the surface they cannot leave (unless they are heated and have a lot of KE), all they can do is move around the surface.

So, consider the force on one surface electron due to the neighboring surface electrons. If there is any curvature to the surface then the net force on the electron will not be parallel to the surface. However, since the electron is constrained to not leave the conductor, only the component of the net force which is parallel to the surface can cause motion of the electron. In an area with small radius of curvature the component of the force parallel to the surface is small, and therefore more electrons are required to exert the same force. This results in an accumulation of charges around regions with a small radius of curvature.

The force is inversely proportional to the square of the distance, but only the component of the force parallel to the surface will cause motion. As the radius of curvature decreases so does the parallel component of the force. Since that decreases either you need more charges or smaller distances, both of which result in an increase of charge density.