Electromagnetic Concepts Matching

This question involves matching key concepts and laws from electromagnetic theory presented in List-I with their corresponding mathematical representations or definitions found in List-II. We need to establish the correct pairing for each item.

Displacement Current ($ \vec J_d $) Matching

Displacement current ($ \vec J_d $) is a fundamental concept in electromagnetism, introduced by James Clerk Maxwell. It represents the contribution to electric current from a time-varying electric field. The mathematical expression for displacement current density is given by:

$ \vec J_d = \epsilon_0 \frac{\partial \vec E}{\partial t} $

Here, $ \epsilon_0 $ is the permittivity of free space, and $ \frac{\partial \vec E}{\partial t} $ is the time rate of change of the electric field ($ \vec E $). This definition directly corresponds to option (IV) in List-II.

Poynting Vector ($\vec S$) Matching

The Poynting vector ($\vec S$) is a vector quantity used to describe the magnitude and direction of the flow of electromagnetic energy per unit area. It is defined as the cross product of the electric field ($ \vec E $) and the magnetic field ($ \vec B $) per unit volume, scaled by the permeability of free space ($ \mu_0 $):

$ \vec S = \frac{1}{\mu_0} (\vec E \times \vec B) $

This expression accurately represents the Poynting vector and matches with option (III) in List-II.

Energy Stored in Electric Field ($ \vec E $) Match

The energy stored in the electric field ($ \vec E $) is related to the square of the electric field strength. The energy density, which is the energy stored per unit volume, is typically expressed as:

$ u_E = \frac{1}{2} \epsilon_0 E^2 $

Option (I) provided is $ \frac{1}{2} \epsilon_0 \int E^2 dt $. While the standard representation for energy density involves $ E^2 $, the integral form $ \int E^2 dt $ might relate to accumulated energy or energy flow over time. In the context of this matching question, this expression is associated with the energy stored in the electric field, aligning with option (I) in List-II.

Gauss's Law Matching

Gauss's Law is one of Maxwell's fundamental equations that relates the electric field to the electric charges that create it. The differential form of Gauss's Law states that the divergence of the electric field ($ \vec E $) is equal to the charge density ($ \rho $) divided by the permittivity of free space ($ \epsilon_0 $):

$ \vec \nabla \cdot \vec E = \frac{\rho}{\epsilon_0} $

This mathematical statement correctly represents Gauss's Law and matches option (II) in List-II.

Summary of Electromagnetic Matches

Let's consolidate the correct pairings derived from the analysis:

Based on these matches, the correct combination is (A) - (IV), (B) - (III), (C) - (I), and (D) - (II).