GATE 2025 PH – Question 57
A point charge $q$ is placed at the origin inside an infinite linear dielectric with relative permittivity $\epsilon_r$. Which option(s) is/are correct?
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Correct answer: (A) $|P|\propto1/r^2$; (C) The screened charge magnitude is smaller than $|q|$ for $\epsilon_r>1$
Explanation
A point charge $q$ at the origin is embedded in an infinite linear dielectric with relative permittivity $\epsilon_r$.
**Fields.** The displacement field is $D=\dfrac{q}{4\pi r^2}$. The electric field is
$$E=\frac{D}{\epsilon_0\epsilon_r}=\frac{q}{4\pi\epsilon_0\epsilon_rr^2},$$
and the polarisation is
$$P=\epsilon_0\chi_eE=\epsilon_0(\epsilon_r-1)E=\frac{(\epsilon_r-1)q}{4\pi\epsilon_rr^2}\;\propto\;\frac1{r^2}.$$
- **A. $|P|\propto1/r^2$.** True. ✓
- **B. $|P|\propto1/r^3$.** False.
- **C. The screened charge is smaller than $|q|$ for $\epsilon_r>1$.** True. The net charge seen from outside (free plus bound charge near the origin) is $q/\epsilon_r<q$. ✓
- **D. The screened charge is larger than $|q|$ for $\epsilon_r=1$.** False: for $\epsilon_r=1$ (vacuum) there is no screening and the charge is $q$.
Answer **A and C**.