Ampere’s Law

Introduction

Ampere’s Law, named after its founder André-Marie Ampère, is a fundamental law in electromagnetism that relates magnetic fields to the electric currents that generate them. It is one of Maxwell’s four equations, which together form the basis of classical electrodynamics.

Ampere’s Law in Integral Form

Ampere’s law can be stated in its integral form as follows:

where is the line integral of the magnetic field around a closed loop, is the permeability of free space, and is the electric current passing through the area enclosed by the loop.

Ampere’s Law with Maxwell’s Addition

Maxwell added a term to Ampere’s law to account for the case where the electric field changes with time, creating a displacement current. This gives the differential form of Ampere’s law:

where is the curl of the magnetic field, is the current density, are the permeability and permittivity of free space respectively, and is the rate of change of the electric field.

Applications

Ampere’s law is crucial in many areas of physics and engineering, including the design of electromagnets and electric motors, the analysis of magnetic fields in materials, and the understanding of phenomena such as electromagnetic waves and electromagnetic induction. It is also used in the derivation of the Biot-Savart law, another key law in electromagnetism that describes the magnetic field generated by a steady electric current.

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