Ampère's circuital law

On Physical Lines of Force

On Physical Lines of Force

Ampère's circuital law, named after the French physicist and mathematician André-Marie Ampère who formulated it in 1826, is a fundamental principle in electromagnetism that relates the integrated magnetic field around a closed loop to the electric current passing through the loop. This law is a cornerstone in the study of electromagnetic fields and has profound implications in both theoretical and applied physics, particularly in the design of electrical and electronic devices.

Formulation

Ampère's circuital law can be expressed in its integral form as:

\[ \oint_{\text{loop}} \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} \]

where:

• \(\oint_{\text{loop}}\) denotes the line integral around a closed loop,
• \(\mathbf{B}\) is the magnetic field vector,
• \(d\mathbf{l}\) is an infinitesimal vector element of the loop,
• \(\mu_0\) is the magnetic constant or permeability of free space, and
• \(I_{\text{enc}}\) is the total current passing through any surface bounded by the loop.

This law indicates that the total magnetic field around a closed loop is directly proportional to the current flowing through the loop.

Maxwell's Extension

James Clerk Maxwell later extended Ampère's circuital law to include time-varying electric fields as sources of magnetic fields, leading to what is known as Maxwell's equations. The modified version, often called Ampère-Maxwell law, incorporates the displacement current term, allowing the law to hold under all circumstances, including during the presence of changing electric fields. The extended form is given by:

\[ \oint_{\text{loop}} \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right) \]

where:

• \(\varepsilon_0\) is the permittivity of free space,
• \(\frac{d\Phi_E}{dt}\) is the rate of change of electric flux through the loop.

Applications

Ampère's circuital law is instrumental in the design and analysis of electrical circuits, electromagnetic induction devices, transformers, and motors. It is also essential in the study of magnetic fields around conductors and in the development of theories related to magnetic materials.

See Also

• Electromagnetic field
• Magnetic field
• Electric current
• Maxwell's equations
• Electromagnetic induction

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