Question

A dielectric sphere of radius \( R \) has constant polarization \( \mathbf{P} = P_0 \hat{z} \) so that the field inside the sphere is \( \mathbf{E} = - \frac{P_0}{3 \epsilon_0} \hat{z} \). Then, which of the following is (are) correct?

• A. The bound surface charge density is \( P_0 \cos\theta \).
• B. The electric field at a distance \( r \) on the z-axis varies as \( \frac{1}{r^2} \) for \( r \gg R \).
• C. The electric potential at a distance \( 2R \) on the z-axis is \( \frac{P_0 R}{12 \epsilon_0} \).
• D. The electric field outside is equivalent to that of a dipole at the origin.

Hint:

For a polarized dielectric sphere, the electric field outside behaves like a dipole field, decaying as \( \frac{1}{r^2} \) at large distances.

Answer 1

The Correct Option is B

Step 1: Understanding the polarization of the dielectric sphere.
For a dielectric sphere with constant polarization, the field outside the sphere behaves like the field of a dipole. However, at distances large compared to the radius, the electric field falls off as \( \frac{1}{r^2} \), just like the field of a dipole. The surface charge density and potential outside depend on the polarization.
Step 2: Conclusion.
Thus, the correct answer is option (B).