Question

Two non-conducting spheres of radii R_1​ and R₂​ and 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,

• A. The electrostatic field is zero
• B. The electrostatic potential is constant
• C. The electrostatic field is constant in magnitude
• D. The electrostatic field has the same direction

Answer 1

The correct option is C

The electrostatic field is constant in magnitude

Electrostatic field

Electrostatic fields arise from a potential difference or voltage gradient, and can exist when charge carriers, such as electrons, are stationary.

1. When two spheres overlap, overall electrical field is vector sum of individual fields.
2. The electrical fields formed are symmetrical if both positive and negative charges are equal. Hence gives constant electrostatic field in the overlapping region.

Explanation

Correct option is C.

With reference to the given figure, let d be the vector joining centres of two spheres. With respect to the centre of spheres, vector r₁ and r₂ are position vectors of point P on overlapping region.

1. In triangle PC₁C_{2 ,} by triangle law of vector addition, $\overrightarrow{r_1}-\overrightarrow{r_2}=\overrightarrow{d}$.
2. Electric field at P due to the positively charged sphere (E₁) is a function of he total charge enclosed in the sphere (q) of radius r_1​.$E_1=\frac{1}{4\pi {\epsilon }_0}\frac{q}{{r_1}^3}\overrightarrow{r_1}=\frac{\rho }{3{\epsilon }_0}\overrightarrow{r_1}\left(asq=\rho x\frac{4}{3}\pi {r_1}^3\right)$
3. Similarly, Electric field due to negatively charged sphere (E₂​) = $-\frac{\rho }{3{\epsilon }_0}\overrightarrow{r_2}$
4. Net electric field at P is, $E=E_1+E_2=\frac{\rho }{3{\epsilon }_0}\left({\overrightarrow{r}}_1-{\overrightarrow{r}}_2\right)=\frac{\rho }{3{\epsilon }_0}\overrightarrow{d}$

That explains that direction and magnitude of the electric field are constant in the overlap region. Hence option C is correct.

Incorrect options

1. Option A: For unlike charges, the electric field is zero outside of the smaller magnitude charge.
2. Option B: The electrostatic potential is constant inside the conductor because of constant potential at all points.
3. Option D: The electrostatic field has the same direction. At any point the direction of an electrical field is the same as the direction of the electrical force acting on a positive test charge at that point.