# Magnetic Scalar Potential

The magnetic scalar potential is a useful tool in describing the magnetic field around a current source. It is defined only in regions of space in the absence of currents.
We know from Ampereâ€™s law that for a steady current. If the current density is zero in some region of space then we have,

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and so we can write the magnetic field as the gradient of a scalar quantity as,

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where

*V*_{m}is called*the magnetic scalar potential*. It is expressed in amperes.

# Magnetic Vector Potential

We know that the divergence of magnetic flux density is always zero everywhere . Hence, can be expressed as the curl of some other vector function. We designate this vector as which is known as*the magnetic vector potential*.

Magnetic vector potential is expressed in webers per metre (Wb/m) or in newtons per ampere (

*N/A*) or in volt-second per metre (

*V*-s/m); with its dimension as MLI

^{-1}T

^{-2}.

If we let, , which is called

*Coulombâ€™s gauge condition*then, we obtain,This is similar to Poissonâ€™s equation of electrostatics, , whose solution is . By comparison, we get the magnetic vector potential as,

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The concept of magnetic vector potential is extremely useful for studying radiation in transmission lines, waveguides, antennas, etc.

# Derivation of Magnetic Flux in terms of Magnetic Vector Potential

We know that the magnetic flux coming out of a surface is given as,

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where is the magnetic flux density. Writing this in terms of magnetic vector potential as and applying Stokesâ€™ theorem, we obtain,

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