## Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

## Magnetic scalar po

The scalar potential is another useful quantity in describing the magnetic field, especially for permanent magnets.

In a simply connected domain where there is no free current,

$abla imesmathbf{H}=0,$

hence we can define magnetic scalar potential ψ as[6]

$mathbf{H}=- ablapsi.$

Using the definition of H:

$ablacdotmathbf{B}=mu_{0} ablacdot(mathbf{H+M})=0,$

it follows that

$abla^{2}psi=- ablacdotmathbf{H}= ablacdotmathbf{M}.$

Here ∇ • M acts as the source for magnetic field, much like ∇ • P as the source for electric field. So analogously to bound electric charge, the quantity

$ho_{m}=- ablacdotmathbf{M}$

is called the bound magnetic charge.

If there is free current, one may subtract the contribution of free current per Biot-Savart law from total magnetic field and solve the remainder with the scalar potential method.

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