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= 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:

<math>
\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
</math>

In integral form, the same law is written as:

<math>
\oint_{\partial S} \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} + \mu_0 \varepsilon_0 \frac{d}{dt} \int_{S} \mathbf{E} \cdot d\mathbf{A}
</math>

== Explanation of Terms ==

* <math>\mathbf{B}</math>: The magnetic field vector
* <math>\nabla \times \mathbf{B}</math>: The curl of the magnetic field
* <math>\mu_0</math>: The permeability of free space (vacuum), approximately <math>4\pi \times 10^{-7} \, \text{N/A}^2</math>
* <math>\varepsilon_0</math>: The permittivity of free space, approximately <math>8.854 \times 10^{-12} \, \text{F/m}</math>
* <math>\mathbf{J}</math>: The current density vector
* <math>\frac{\partial \mathbf{E}}{\partial t}</math>: The time rate of change of the electric field
* <math>I_{\text{enc}}</math>: The total current enclosed by the loop
* <math>\int_{S} \mathbf{E} \cdot d\mathbf{A}</math>: The electric flux through surface <math>S</math>

== Physical Significance ==

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

<math>
\oint_{\partial S} \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}
</math>

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:

<math>
\mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
</math>

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]]