Calculation of charges screening an external coaxial electric field on the surface of a conducting axial symmetric body

A one-dimensional integral equation for finding charges on the surface of a conducting axially symmetric body is given. For the case of an ellipsoid of rotation in the electric field with polynomial values on the axis of symmetry an exact solution is obtained. The axis of symmetry of the body and the axis of the external field coincide with each other. A numerical algorithm, based on a combination of the projective method and the method of iterative regularization for solving the first kind Fredholm equations, is proposed. The projectors were chosen as B-splines. The charges calculated for the ellipsoid of rotation are close to analytical ones.