Question

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \text{Displacement current } (J_d) & (I) \; \frac{\epsilon_0}{2} \int E^2 d\tau \\ (B) \; \text{Poynting vector} & (II) \; \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \\ (C) \; \text{Energy stored in electric field } (\vec{E}) & (III) \; \frac{1}{\mu_0}(\vec{E} \times \vec{B}) \\ (D) \; \text{Gauss's Law} & (IV) \; \epsilon_0 \frac{\partial \vec{E}}{\partial t} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

• (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
• (A) - (IV), (B) - (III), (C) - (I), (D) - (II)
• (A) - (I), (B) - (II), (C) - (III), (D) - (IV)
• (A) - (IV), (B) - (III), (C) - (II), (D) - (I)

Hint:

For matching questions, it's often efficient to identify the one or two pairings you are most confident about first. For example, Gauss's Law (\( \nabla \cdot \vec{E} = \rho/\epsilon_0 \)) is a very standard definition. Finding this match (D → II) can help eliminate incorrect options quickly.

Answer 1

The Correct Option is B

Step 1: Understand the Task
This question requires associating fundamental electromagnetism concepts with their corresponding mathematical equations. The objective is to correctly match each item in List-I with an option from List-II.
Step 2: Detailed Explanation

(A) Displacement current (\(J_d\)): This is Maxwell's modification to Ampere's law. The displacement current density is \( \vec{J}_d = \epsilon_0 \frac{\partial \vec{E}}{\partial t} \), representing the magnetic field produced by a time-varying electric field. This corresponds to (IV).

(B) Poynting vector (\(\vec{S}\)): This vector describes the direction and magnitude of electromagnetic energy flux (energy transfer rate per unit area). Its definition is \( \vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B}) \). This matches with (III).

(C) Energy stored in electric field (\(\vec{E}\)): The energy density in an electric field is \( u_E = \frac{1}{2}\epsilon_0 E^2 \). The total energy \( U_E \) stored in a volume \( \tau \) is found by integrating this density: \( U_E = \int \frac{1}{2}\epsilon_0 E^2 d\tau = \frac{\epsilon_0}{2} \int E^2 d\tau \). This corresponds to (I).

(D) Gauss's Law: As one of Maxwell's equations, Gauss's Law relates the divergence of the electric field to the charge density \( \rho \). In differential form, it is \( abla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \). This matches with (II).
Step 3: Final Answer
The correct pairings are as follows:
(A) corresponds to (IV)
(B) corresponds to (III)
(C) corresponds to (I)
(D) corresponds to (II)
This corresponds to option (B).