Question

A point charge \( q \) is placed at the origin, inside a linear dielectric medium of infinite extent, having relative permittivity \( \epsilon_r \). Which of the following option(s) is/are correct?

• A. The magnitude of the polarization varies as \( \frac{1}{r^2} \)
• B. The magnitude of the polarization varies as \( \frac{1}{r^3} \)
• C. The magnitude of the screened charge due to the dielectric medium is less than the magnitude of the point charge \( q \) for \( \epsilon_r>1 \)
• D. The magnitude of the screened charge due to the dielectric medium is more than the magnitude of the point charge \( q \) for \( \epsilon_r = 1 \)

Hint:

In a dielectric medium, the polarization reduces the effective charge felt outside the medium. The relative permittivity plays a crucial role in determining the screening effect.

Answer 1

The Correct Option is A, C

Step 1:
The polarization \( \mathbf{P} \) in a dielectric medium is related to the electric field \( \mathbf{E} \) and the relative permittivity \( \epsilon_r \). In the case of a point charge \( q \) placed at the origin, the electric field behaves as \( \frac{1}{r^2} \), but the polarization depends on the medium and varies differently.

Step 2:
The polarization \( \mathbf{P} \) is proportional to the electric field \( \mathbf{E} \), and the magnitude of the polarization varies as \( \frac{1}{r^3} \) in this case (this matches option B). However, as the medium screens the charge, the magnitude of the polarization varies as \( \frac{1}{r^2} \), which matches option (A).

Step 3:
The dielectric medium screens the charge, reducing the effective charge felt by any external observer. For \( \epsilon_r > 1 \), the magnitude of the screened charge is less than the original point charge \( q \), which makes option (C) correct.

Step 4:
For \( \epsilon_r = 1 \), the medium behaves as vacuum, so the magnitude of the screened charge equals the original point charge, confirming that the magnitude of the screened charge is the same as \( q \) in this case, making option (D) incorrect.