1. Use superposition

A spherical cavity inside a uniformly charged solid sphere can be treated as:

• a full uniformly charged sphere of charge density $\rho$, centered at $O$, plus
• a negative uniformly charged sphere of charge density $-\rho$, centered at $P$.

So the electric field inside the cavity is

2. Field inside a uniformly charged sphere

For a uniformly charged sphere of volume charge density $\rho$, the electric field at an internal point with position vector $\vec r$ measured from its center is

Therefore:

• For the big sphere centered at $O$,

  $$\vec E_1(\vec r)=\frac{\rho}{3\varepsilon_0}\,\vec r.$$

• For the cavity, represented as a sphere of charge density $-\rho$ centered at $P$, the position vector relative to $P$ is

  $$\vec r-\vec a,$$

  where $\vec a=\overrightarrow{OP}$. Hence

  $$\vec E_2(\vec r)=\frac{-\rho}{3\varepsilon_0}(\vec r-\vec a).$$

3. Add the two fields

4. Interpretation

This result shows:

• The field is uniform inside the cavity, since it does not depend on $\vec r$.
• Its magnitude is $$E=\frac{\rho a}{3\varepsilon_0},$$ so it depends on $a$.
• Its direction is along $\vec a$.

Since $a=R_1-R_2$, the magnitude also indirectly depends on $R_2$, but the most direct statement among the options is dependence on $\vec a$.

5. Check each option

• A: Says field is uniform, magnitude independent of $R_2$, but direction depends on $\vec r$.
  Wrong, because direction does not depend on $\vec r$; field is constant.

• B: Says field is uniform, magnitude depends on $R_2$, but direction depends on $\vec r$.
  Wrong, again direction does not depend on $\vec r$.

• C: Says field is uniform, magnitude independent of $a$, but direction depends on $\vec a$.
  Wrong, magnitude is proportional to $a$.

• D: Says field is uniform and both magnitude and direction depend on $\vec a$.
  Correct.

6. Final answer