The electric field outside a sphere with total charge Q distributed with spherical symmetry can be found immediately from Gauss's law. In MKS using a spherical surface,
where is the permittivity of free space, so
where is a unit radial vector. This is the same result that would be obtained from a full-blown integration of Coulomb's law, and is analogous to the result that the gravitational force from a sphere is equal to the force from a point mass equal to the total mass of the sphere and located at its center.
Coulomb's Law, Gauss's Law, Sphere Gravitational Force
References
Griffiths, D. J. Introduction to Electrodynamics, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 61-62, 1998.
