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: Understanding the Concept:
This question requires matching fundamental concepts and equations in electromagnetism. We need to identify the correct mathematical expression for each term in List-I from the options in List-II.
Step 2: Detailed Explanation:

(A) Displacement current (\(J_d\)): Maxwell's addition to Ampere's law, the displacement current density \( \vec{J}_d \) is defined as \( \vec{J}_d = \epsilon_0 \frac{\partial \vec{E}}{\partial t} \). This term accounts for the production of a magnetic field by a time-varying electric field. This matches with (IV).

(B) Poynting vector (\(\vec{S}\)): The Poynting vector represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. It is defined as \( \vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B}) \). This matches with (III).

(C) Energy stored in electric field (\(\vec{E}\)): The energy density (energy per unit volume) stored in an electric field is \( u_E = \frac{1}{2}\epsilon_0 E^2 \). To find the total energy stored in a volume \( \tau \), we integrate this density over the volume: \( U_E = \int u_E d\tau = \int \frac{1}{2}\epsilon_0 E^2 d\tau = \frac{\epsilon_0}{2} \int E^2 d\tau \). This matches with (I).

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