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In magnetostatics, the force of attraction or repulsion between two current-carrying wires (see first figure below) is often called Ampère's force law. The physical origin of this force is that each wire generates a magnetic field , following the Biot–Savart law , and the other wire experiences a magnetic force as a consequence, following ...
In three dimensions, the derivative has a special structure allowing the introduction of a cross product: = + = + from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation.
The I symbol was used by André-Marie Ampère, after whom the unit of electric current is named, in formulating Ampère's force law (1820). [8] The notation travelled from France to Great Britain, where it became standard, although at least one journal did not change from using C to I until 1896. [9]
The third of Maxwell's equations is called the Ampère–Maxwell law. It states that a magnetic field can be generated by an electric current. [13] The direction of the magnetic field is given by Ampère's right-hand grip rule. If the wire is straight, then the magnetic field is curled around it like the gripped fingers in the right-hand rule.
Ampère's force law describes the experimentally-derived fact that, for two thin, straight, stationary, parallel wires, a distance r apart, in each of which a current I flows, the force per unit length, F m /L, that one wire exerts upon the other in the vacuum of free space would be given by .
Maxwell added displacement current to the electric current term in Ampère's circuital law. In his 1865 paper A Dynamical Theory of the Electromagnetic Field Maxwell used this amended version of Ampère's circuital law to derive the electromagnetic wave equation. This derivation is now generally accepted as a historical landmark in physics by ...
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems.
By the Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ to an integral of the "circulation of the fields" (i.e. their curls) over a surface it bounds, i.e. = (), Hence the Ampère–Maxwell law, the modified version of Ampère's circuital law, in integral form can be rewritten as ((+)) =