Question

The law/Theory and equations are given in the table below. Match List-I with List-II

• A. A - I, B - II, C - III, D - IV
• B. A- IV, B- I, C- II, D- III
• C. A- IV, B- II, C- I, D- III
• D. A- I, B- IV, C- III, D- II

Hint:

Remember the physical meaning of the differential operators in Maxwell's equations: - \textbf{Curl (\(\nabla \times\))}: Relates a circulating field to a source (current or changing field). E.g., Curl of E from changing B (Faraday), Curl of H from J (Ampere). - \textbf{Divergence (\(\nabla \cdot\))}: Relates a field flowing out of a point to a source at that point. E.g., Divergence of D from charge \(\rho\) (Gauss), Divergence of J from changing charge (Continuity).

Answer 1

The Correct Option is A

Step 1: Match each law/theory with its corresponding Maxwell's equation in differential form.

(A) Continuity equation: This equation expresses the principle of conservation of charge. It states that the divergence of the current density (\(\vec{J}\)) is equal to the negative rate of change of the charge density (\(\rho_v\)). This matches (I) \(\nabla \cdot \vec{J} = -\frac{\partial \rho_v}{\partial t}\).
(B) Ampere's law (modified): Maxwell's modification to Ampere's circuital law states that the curl of the magnetic field intensity (\(\vec{H}\)) is equal to the sum of the conduction current density (\(\vec{J}\)) and the displacement current density (\(\frac{\partial \vec{D}}{\partial t}\)). This matches (II) \(\nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}\).
(C) Displacement current: The concept of displacement current was Maxwell's key contribution. The displacement current density itself is defined as the rate of change of the electric displacement field (\(\vec{D}\)). This matches (III) \(\vec{J}_D = \frac{\partial \vec{D}}{\partial t}\).
(D) Faraday's law: This law of induction states that a time-varying magnetic field creates an electric field. In differential form, the curl of the electric field (\(\vec{E}\)) is equal to the negative rate of change of the magnetic flux density (\(\vec{B}\)). This matches (IV) \(\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}\).

Step 2: Combine the matches. The correct matching is A-I, B-II, C-III, D-IV. This corresponds to option (A).