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Two non-conducting spheres of radii R1 and R2 carry uniform volume charge densities +ρ and -ρ respectively. They are placed so that they partially overlap. In the overlapping region, which statement is true?

1. the electrostatic field is zero
2. the electrostatic potential is constant
3. the electrostatic field is constant in magnitude
4. the electrostatic field has the same direction everywhere

Correct answer: the electrostatic field has the same direction everywhere

Solution

Inside the +ρ sphere E1 = ρ r1/(3ε0); inside the -ρ sphere E2 = -ρ r2/(3ε0). Superposing, the net field in the overlap = ρ(r1 - r2)/(3ε0) = ρ d/(3ε0), where d is the constant vector joining the two centres. This is uniform: same magnitude and same direction everywhere in the overlap. The strongest single given statement is that the field has the same direction everywhere (it is in fact a uniform field).

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