Two non-conducting spheres of radii R_1 and R_2 and carrying uniform volume charge densities + \rho and - \rho , respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region

The electrostatic field is constant in magnitude, The electrostatic field has same direction

The electrostatic field is zero

The electrostatic potential is constant

The electrostatic field is constant in magnitude

The electrostatic field has same direction

Electrostatics PYQ — JEE Advanced 2013 · Shift 2 · Q41 with Solution

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Correct answer: D, C

For a sphere with uniform volume charge density $\rho$ , the electric field at an interior point whose position vector from the center is $\vec r$ is

$\vec E=\frac{\rho}{3\varepsilon_0}\,\vec r$

For a sphere with charge density $-\rho$ , the field becomes

$\vec E=-\frac{\rho}{3\varepsilon_0}\,\vec r$

This result is valid for all interior points.

Let the centers of the two spheres be $O_1$ and $O_2$ .

Take any point $P$ in the overlapping region.

Let

Since $P$ lies inside both spheres, the field at $P$ due to each sphere is given by the interior formula.

So,

due to the positively charged sphere: $\vec E_1=\frac{\rho}{3\varepsilon_0}\vec r_1$
due to the negatively charged sphere: $\vec E_2=-\frac{\rho}{3\varepsilon_0}\vec r_2$

Hence total field is

=\frac{\rho}{3\varepsilon_0}(\vec r_1-\vec r_2)$$ But $$\vec r_1-\vec r_2=\overrightarrow{O_1P}-\overrightarrow{O_2P}= \overrightarrow{O_1O_2}$$ which is a constant vector, independent of the point $P$. Therefore, $$\boxed{\vec E=\frac{\rho}{3\varepsilon_0}\overrightarrow{O_1O_2}}$$ So throughout the overlapping region, the electric field is **constant as a vector**. That means: - its **magnitude is constant** - its **direction is the same everywhere** Thus **C** and **D** are correct. --- 3. **Check whether the field is zero** Since $$\vec E=\frac{\rho}{3\varepsilon_0}\overrightarrow{O_1O_2}$$ this is zero only if the centers coincide, which is not the case for partial overlap. So **A is false**. --- 4. **Check whether potential is constant** Using $\vec E=-\nabla V$, if $V$ were constant in the overlap region, then $\vec E$ would have to be zero. But the field is nonzero constant. Hence the potential is **not constant**; instead it varies linearly with position. So **B is false**. --- 5. **Option-wise conclusion** - **A:** False - **B:** False - **C:** True - **D:** True Therefore the correct options are $$\boxed{C, D}$$