Two non-conducting spheres of radii R1 and R2 carrying uniform volume charge densities +ρ and -ρ, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region:

Let the vector joining the centers of the two spheres be →d and consider a point P in the overlap region, whose position vector be →r1 and →r2 with respect to the centers of each sphere respectively. Thus, by triangle law of vector addition, →r1 − →r2 = →d
Electric field at P due to the positively charged sphere is E1 = 1/4πε0 q/r1³ →r1
Here, q is the total charge enclosed in the sphere of radius r1
Thus, q = ρ × 4/3 πr1³
Hence, E1 = ρ/3ε0 →r1
Similarly, Electric field due to negatively charged sphere is E2 = −ρ/3ε0 →r2
Thus, net electric field at P is E = E1 + E2 = ρ/3ε0 (→r1 − →r2) = ρ/3ε0 →d
Both, direction and magnitude of the electric field are constant in the overlap region.