Magnetic potential

In physics, the magnetic potential is a method of representing the magnetic field by using a potential value instead of the actual mathbf{B} vector field. There are two methods of relating the magnetic field to a potential field and they give rise to two possible types of magnetic potential.

Magnetic vector potential

This is the most popular method of defining a magnetic potential and used in most physics text books. The magnetic vector potential mathbf{A} is a three-dimensional vector field whose curl is the magnetic field in the theory of electromagnetism:

Related Topics:
Vector field - Curl - Magnetic field - Electromagnetism

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:mathbf{B} = abla imes mathbf{A}

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Starting with the above definition, calculating the divergence of both sides of the equation gives:

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: abla cdot mathbf{B} = abla cdot abla imes mathbf{A} = 0

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Note that the divergence of a curl will always give zero. Conveniently, this solves the second of Maxwell's Equations automatically, which is to say that a continuous magnetic vector potential field is guaranteed not to result in magnetic monopoles.

Related Topics:
Maxwell's Equations - Magnetic monopoles

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It should be noted that the above definition does not define the magnetic vector potential uniquely because the divergence might be anything and still have no effect on the magnetic field. Thus, there is a degree of freedom available when choosing a definition. This condition is known as gauge invariance.

Related Topics:
Magnetic field - Degree of freedom

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Coulomb gauge

In order to uniquely define the magnetic vector potential, the following equation constrains the divergence:

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: abla cdot mathbf{A} = 0

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This was named after Charles Augustin de Coulomb.

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Magnetostatic integral formulation

For magnetostatics this vector integral defines magnetic vector potential in terms of current density:

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:mathbf{A} = rac 1 {4 pi epsilon_0 c^2} int rac { mathbf{J} dV } { |r| }

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Lorentz gauge

The Lorentz gauge can also be used to uniquely constrain the magnetic vector potential and for magnetostatics gives the same result as the Coulomb gauge. The Lorentz gauge is:

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: abla cdot mathbf{A} = - rac { 1 } { c } rac { partial phi } { partial t }

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This was named after Hendrik Lorentz.

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