The scalar and vector magnetic potential:
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Scalar magnetic potential is analogous to scalar potential in electric fields (i.e. voltage). The magnetic field vector is the negative gradient of scalar magnetic potential, just as the electric field vector is the negative gradient of electrostatic potential.
The physical meaning of the magnetic vector potential is actually very similar: it's the potential energy per unit element of current. ... This formula justifies the idea that the electric scalar potential is energy per unit charge.
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