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ElectricalElectromagnetics Field Theory

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For any conductor, the Maxwell's second equation is

Curl(H) = Jc

Curl(E) = Jc

Curl(E) = Jd

Curl(H) = Jd

Correct Answer

⚙️ TE • Technical Concept & PrincipleElectricalElectromagnetics Field Theory

Option A

Curl(H) = Jc

Quick Summary:

Maxwell's second equation (derived from Ampere's Circuital Law) states that the curl of the magnetic field intensity vector $\vec{H}$ is equal to the current density $\vec{J}$ . For a conductive medium, the conduction current density $\vec{J}_c$ is the dominant term.

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$\nabla \times \vec{H} = \vec{J}_c + \frac{\partial\vec{D}}{\partial t}$ — Generalized Ampere's Law

$\nabla \times \vec{H} = \vec{J}_c$ — Form for conductive media

⚙️ Working Principle

In a perfect conductor or good conductor, the displacement current density $\vec{J}_d$ (given by $\frac{\partial\vec{D}}{\partial t}$ ) is negligible compared to the conduction current density $\vec{J}_c = \sigma\vec{E}$ . Thus, the differential form of Ampere's Law simplifies from $\nabla \times \vec{H} = \vec{J}_c + \frac{\partial\vec{D}}{\partial t}$ to $\nabla \times \vec{H} = \vec{J}_c$ .

📌 Key Points

▸
Maxwell's equations are the foundation of classical electromagnetism.
▸
The second equation relates the magnetic field to the current density.
▸
For conductors, $\sigma$ is large, making $\vec{J}_c$ the dominant component.

✅ Advantages

▸
Provides a clear relationship between magnetic fields and source currents.
▸
Allows for the simplification of field problems in metallic structures.

❌ Disadvantages / Limitations

▸
Only holds true for non-time-varying displacement fields in conductors.
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Does not account for non-ohmic conduction.

🛠️ Applications / Uses

▸
Designing transmission lines and cables.
▸
Analyzing electromagnetic shielding in conductors.

📄 Additional Information

▸
In vacuum or dielectrics, $\vec{J}_c = 0$ , meaning $\nabla \times \vec{H} = \frac{\partial\vec{D}}{\partial t}$ .
▸
Option B and C are incorrect because the curl of the Electric Field intensity $\vec{E}$ describes Faraday's Law, not Ampere's Law.

📊 Diagram / Illustration

✅

A is correct — Maxwell's second equation for a conductor is expressed as $\nabla \times \vec{H} = \vec{J}_c$ .

Core Concepts Used

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Ampere's Circuital Law Conduction Current Density Maxwell's Equations

💡 EXAM TIP

Remember that Maxwell's 1st is Gauss's Law ( $abla \cdot \vec{D} = \rho$ ), 2nd is Ampere's Law ( $\nabla \times \vec{H} = \vec{J}$ ), 3rd is Gauss's Law for Magnetism ( $\nabla \cdot \vec{B} = 0$ ), and 4th is Faraday's Law ( $\nabla \times \vec{E} = -\frac{\partial\vec{B}}{\partial t}$ ).

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