Displacement current density is

Question

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Option C: $\frac{\partial D}{\partial t}$

Explanation & Concepts:
Displacement current density, denoted as $J_d$, is defined as the time rate of change of electric flux density. It was introduced by James Clerk Maxwell to maintain the continuity equation and satisfy the Ampere-Maxwell law in time-varying electromagnetic fields.

Working Principle: In regions where the electric field changes over time, a time-varying electric displacement $D$ generates a magnetic field even in the absence of conduction current. According to the Maxwell-Ampere law, the total current density is the sum of conduction current density $J$ and displacement current density $J_d = \frac{\partial D}{\partial t}$. This mechanism ensures that the divergence of the total current is zero, consistent with the charge conservation principle.

Key Formulas: $J_d = \frac{\partial D}{\partial t}$ — Definition of displacement current density; $
abla 	imes H = J + \frac{\partial D}{\partial t}$ — The Ampere-Maxwell law

Key Points: Displacement current is not a flow of charge carriers like conduction current.; It physically arises from the change in electric flux in free space or dielectrics.; Essential for the propagation of electromagnetic waves.; Displacement current density has the units of $A/m^2$.

Advantages: Allows for the existence of electromagnetic waves in vacuum.; Ensures consistency with the continuity equation $
abla \cdot J + \frac{\partial ho}{\partial t} = 0$.

Disadvantages: Often neglected in DC circuits where fields are static.; Requires knowledge of permittivity $\epsilon$ to compute $D$ in media.

Applications: Radio wave propagation; Capacitor operation in AC circuits; Design of high-frequency communication systems

Additional Info: The term 'displacement current' was coined by Maxwell, though it is technically a time-varying electric field flux rather than a moving charge.. Option B represents conduction current density ($J$), while Option D is the time derivative of conduction current, which is not a standard Maxwell equation term.

Answer: C is correct — Displacement current density is mathematically defined as the partial derivative of the electric displacement field with respect to time, expressed as $J_d = \frac{\partial D}{\partial t}$.

Exam Tip: Always remember that displacement current allows current to 'flow' through a capacitor even though no physical charges jump across the gap.

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Option A: $D$

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Option B: $J$

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Option D: $\frac{\partial J}{\partial t}$

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