Ampère-Maxwell Law:

Ampère-Maxwell Law

The Ampère-Maxwell Law is one of the four equations in the set of Maxwell's Equations, which form the foundation of classical electrodynamics. It is a generalization of Ampère's Law, accounting for the contribution of the changing electric field to the magnetic field.

Statement of the Law

In differential form, the Ampère-Maxwell Law is expressed as:

$\mathrm{\nabla }\times \mathbf{𝐁}={\mu }_0\mathbf{𝐉}+{\mu }_0{\epsilon }_0\frac{\partial \mathbf{𝐄}}{\partial t}$

In integral form, the same law is written as:

${\oint }_{\partial S}\mathbf{𝐁}\cdot d\mathbf{𝐥}={\mu }_0{I}_{\text{enc}}+{\mu }_0{\epsilon }_0\frac{d}{dt}\underset{S}{\int }\mathbf{𝐄}\cdot d\mathbf{𝐀}$

Explanation of Terms

• $\mathbf{𝐁}$: The magnetic field vector
• $\mathrm{\nabla }\times \mathbf{𝐁}$: The curl of the magnetic field
• ${\mu }_0$: The permeability of free space (vacuum), approximately $4\pi \times 10^{-7}{\text{N/A}}^2$
• ${\epsilon }_0$: The permittivity of free space, approximately $8.854\times 10^{-12}\text{F/m}$
• $\mathbf{𝐉}$: The current density vector
• $\frac{\partial \mathbf{𝐄}}{\partial t}$: The time rate of change of the electric field
• $I_{\text{enc}}$: The total current enclosed by the loop
• $\underset{S}{\int }\mathbf{𝐄}\cdot d\mathbf{𝐀}$: The electric flux through surface $S$

Physical Significance

Originally, Ampère's Law related the magnetic field in a loop to the electric current passing through the loop:

${\oint }_{\partial S}\mathbf{𝐁}\cdot d\mathbf{𝐥}={\mu }_0{I}_{\text{enc}}$

However, James Clerk Maxwell noticed that this form was inconsistent with the continuity equation for electric charge. To correct this, he introduced the concept of displacement current, represented by the term:

${\mu }_0{\epsilon }_0\frac{\partial \mathbf{𝐄}}{\partial t}$

This term accounts for changing electric fields in regions where there is no conduction current, such as between the plates of a charging capacitor.

Applications

• Describes how a changing electric field can produce a magnetic field, even in the absence of conduction current.
• Explains the propagation of electromagnetic waves in free space.
• Crucial in the operation of capacitors in AC circuits.
• Forms the theoretical basis for technologies like wireless communication and electromagnetic waveguides.

Related Concepts

• Maxwell's Equations
• Displacement Current
• Electromagnetic Waves
• Faraday's Law of Induction