A method to determine the electric field scattered by a dielectric sphere with a permittivity and conductivity varying only in the radial direction is presented. In order to take advantage of the spherically symmetrical geometry, the interior electric field is expanded in terms of vector spherical harmonics which are orthogonal over a spherical surface. Using the orthogonality of those functions with the vector wave functions in the expanded form of the free space dyadic Green's function, one can reduce the three-dimensional (3-D) electric field integral equation into a system of one-dimensional (1-D) integral equations. The scalar coefficients of the series expansion for the interior electric field are then obtained by solving this system of equations. For the medium outside the sphere, the scattered field is evaluated through the same procedure. Comparisons with alternative methods for specific cases demonstrate that the method is reliable in determining the interior field and the scattered field, for spherical objects with layered or continuously varying radial profile.