ISRO EC 2025 Q.54: solution and answer

Official answer: ASource: ISRO's final answer key of 15 May 2026

1. Write Ampere's law for a medium of conductivity $\sigma$ and permittivity ${\epsilon }^{\prime }$: $\nabla \times \vec{H}=(\sigma +j\omega {\epsilon }^{\prime })\vec{E}$. With the complex permittivity $\epsilon ={\epsilon }^{\prime }-j{\epsilon }^{\prime \prime }$, the curl equation is $\nabla \times \vec{H}=j\omega ({\epsilon }^{\prime }-j{\epsilon }^{\prime \prime })\vec{E}=(\omega {\epsilon }^{\prime \prime }+j\omega {\epsilon }^{\prime })\vec{E}$. Comparing the two forms gives ${\epsilon }^{\prime \prime }=\sigma /\omega$.

2. Hence $$\tan \delta =\frac{{\epsilon }^{\prime \prime }}{{\epsilon }^{\prime }}=\frac{\sigma }{\omega {\epsilon }^{\prime }}=\frac{|{\vec{J}}_{\text{cond}}|}{|{\vec{J}}_{\text{disp}}|}.$$

3. $\tan \delta \to \infty$ means $\sigma \gg \omega {\epsilon }^{\prime }$: conduction current completely dominates displacement current. This is the good-conductor limit, and in the ideal limit it is a perfect conductor. Option A is therefore right. Waves in such a medium are strongly attenuated, with $\alpha \approx \beta \approx \sqrt{\pi f\mu \sigma }$ and a very small skin depth.

4. Option B is wrong: a lossless dielectric has ${\epsilon }^{\prime \prime }=0$, so $\tan \delta =0$.

5. Option C is wrong: a perfect insulator has $\sigma =0$ and no loss, so ${\epsilon }^{\prime \prime }=0$ and $\tan \delta =0$.

6. Option D is wrong: it describes the opposite limit. Displacement current dominating means $\sigma \ll \omega {\epsilon }^{\prime }$, that is $\tan \delta \ll 1$.