Maxwell’s first equation in free space is

Question

Examoogle · Accepted Answer

Option B: $
abla 	imes H = \frac{\partial D}{\partial t}$

Explanation & Concepts:
Maxwell’s first equation, derived from the Ampere-Maxwell Law, relates the magnetic field intensity to the displacement current density in the absence of conduction currents. In free space, where the conductivity $\sigma = 0$, the conduction current density $J = 0$, leading to the relation $
abla 	imes H = \frac{\partial D}{\partial t}$.

Working Principle: The principle stems from the generalization of Ampere's Law to include time-varying electric fields. Faraday's discovery of electromagnetic induction necessitated the addition of the displacement current term, $\frac{\partial D}{\partial t}$, to maintain the consistency of the continuity equation. In a vacuum (free space), charge density $ho = 0$ and current density $J = 0$, making the time-variation of the electric displacement field the sole source of the circulating magnetic field.

Key Formulas: $
abla 	imes H = J + \frac{\partial D}{\partial t}$ — General Ampere-Maxwell Law; $
abla 	imes H = \frac{\partial D}{\partial t}$ — Free space condition

Key Points: Maxwell's equations are the foundation of classical electromagnetism.; The term $\frac{\partial D}{\partial t}$ represents the displacement current density.; In free space, conductivity is zero, hence $J=0$.; This equation implies that a time-varying electric field generates a magnetic field.

Advantages: Predicts electromagnetic wave propagation.; Provides a unified theory for electric and magnetic fields.

Disadvantages: Does not account for quantum electrodynamic effects.; Requires knowledge of vector calculus for implementation.

Applications: Radio wave transmission.; Design of antennas and waveguides.

Additional Info: The constant $\epsilon_0$ is the permittivity of free space, approximately $8.854 	imes 10^{-12} 	ext{ F/m}$.. Option A is the general form including conduction current $J$. Option C represents a static field without time variation. Option D is Ampere's law for static currents (magnetostatics).

Answer: B is correct — In free space, the conduction current density $J$ is zero, simplifying the Maxwell-Ampere equation to the curl of the magnetic field intensity equal to the time derivative of the electric displacement field $
abla 	imes H = \frac{\partial D}{\partial t}$.

Exam Tip: Always verify the medium conditions (e.g., free space, conductor, or dielectric) before applying Maxwell's equations, as $J$ and $ho$ variations change the equation form significantly.

Examoogle · Answer

Option A: $
abla 	imes H = J + \frac{\partial D}{\partial t}$

Examoogle · Answer

Option C: $
abla 	imes H = 0$

Examoogle · Answer

Option D: $
abla 	imes H = J$

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