# 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 Vm 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-1T-2.

Now, by Ampereâ€™s law,

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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