Spherical faraday cage for the case of equal point charges and chebyshev-type quadrature on the sphere

It is shown that configuratin of N point charges 1/N on the unist sphere S,which generate a small electrostatic field inside the unit ball, correspond to good N-tuples of nodes for Chebyshev-type quadrature on S.Quadraure results for S are obtained with the aid of spherical harmonics and an existence theorem of S.N. Bernstein for Chebyshev-type quadrature on an interval. It follows in particular that there exist so-called shperical pdesigns consisting of O(p3 )points. A more physically motivated approach to good nodes on S is briefly indicated. Finally, the above correspondence is used to give lower and upper bounds for the smallest possible electrostaic field on ballsB(0,γ) with γ < 1, due to N point charges 1/N on S