The law/Theory and equations are given in the table below. Match List-I with List-II
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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).
Step 1: Match each law/theory to its corresponding Maxwell's equation in differential form.
(A) Continuity equation: Expresses charge conservation, stating that the divergence of current density (\(\vec{J}\)) equals the negative rate of change of charge density (\(\rho_v\)). Matches (I) \(abla \cdot \vec{J} = -\frac{\partial \rho_v}{\partial t}\). (B) Ampere's law (modified): Maxwell's modification states that the curl of magnetic field intensity (\(\vec{H}\)) equals the sum of conduction current density (\(\vec{J}\)) and displacement current density (\(\frac{\partial \vec{D}}{\partial t}\)). Matches (II) \(abla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}\). (C) Displacement current: Maxwell's key contribution. Displacement current density is defined as the rate of change of the electric displacement field (\(\vec{D}\)). Matches (III) \(\vec{J}_D = \frac{\partial \vec{D}}{\partial t}\). (D) Faraday's law: Induction law stating that a time-varying magnetic field creates an electric field. The curl of the electric field (\(\vec{E}\)) equals the negative rate of change of magnetic flux density (\(\vec{B}\)). Matches (IV) \(abla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}\).
Step 2: Combine the matches.The correct match is A-I, B-II, C-III, D-IV, corresponding to option (A).
