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For any metals, the Maxwell's equation hold true is

Curl(H) = J

Curl(J) = dD/dt

Curl(H) = D

Curl(J) = dB/dt

Correct Answer

⚙️ TE • Technical Concept & PrincipleElectricalElectromagnetics Field Theory

Option A

Curl(H) = J

Quick Summary:

In good conductors (metals) at low to moderate frequencies, the conduction current density ( $J$ ) is significantly larger than the displacement current density ( $dD/dt$ ). Thus, the Ampere-Maxwell law simplifies to the Magnetomotive Force form: $\nabla \times H = J$ .

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$\nabla \times H = J + \frac{\partial D}{\partial t}$ — General Ampere-Maxwell Law

$\nabla \times H \approx J$ — Approximation for good conductors

⚙️ Working Principle

Maxwell's Fourth Equation (Ampere-Maxwell law) is $\nabla \times H = J + \frac{\partial D}{\partial t}$ . For metals, the conductivity ( $\sigma$ ) is very high, making the conduction current $J = \sigma E$ dominant. The displacement current term $\frac{\partial D}{\partial t} = \epsilon \frac{\partial E}{\partial t}$ becomes negligible compared to the conduction current, resulting in the steady-state form $\nabla \times H = J$ .

📌 Key Points

▸
The term $\frac{\partial D}{\partial t}$ represents displacement current.
▸
For metals, $\sigma$ is very high, so conduction current $J$ dominates.
▸
This simplification is valid for the 'quasi-static' approximation in conductive media.

✅ Advantages

▸
Simplifies analysis of electromagnetic waves in conductors.
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Allows for the use of the skin effect theory in power engineering.

❌ Disadvantages / Limitations

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Invalid at extremely high frequencies (e.g., optical frequencies) where displacement current becomes comparable.
▸
Neglects phase shift between conduction and displacement currents.

🛠️ Applications / Uses

▸
Design of power cables and shielding materials.
▸
Induction heating and eddy current analysis.

📄 Additional Information

▸
The displacement current density is given by $J_d = \frac{\partial D}{\partial t}$ .
▸
Option B and D involve erroneous curl relations for current and flux density.

📊 Diagram / Illustration

✅

A is correct — In metals, the displacement current term $\frac{\partial D}{\partial t}$ is negligible compared to the conduction current $J$ , reducing Maxwell's equation to $\nabla \times H = J$ .

Core Concepts Used

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Maxwell's Equations Conduction Current vs Displacement Current Electromagnetic Field Theory

💡 EXAM TIP

Always check the conductivity of the medium: if $\sigma / (\omega \epsilon) \gg 1$ , the material acts as a good conductor, and displacement current can be ignored.

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