# Equivalence between bound electric currents & bound magnetic charge in magnetization

I have been trying to remember if in classical EM it is equivalent to describe magnetization through bound electric currents
A. $\vec{j_b} = \nabla \times \vec M$
$\vec{k_b} = \vec M \times \vec{\hat{n}}$
OR bound magnetic charges
B. $\rho_b = -\nabla \cdot \vec M$
$\sigma_b = \vec M \cdot \vec{\hat{n}}$

The topic originated https://www.physicsforums.com/showthread.php?t=447805", there someone already suggested this is not valid inside matter but I am still not seeing it. From what I recall once bound sources are introduced (in place of matter) one can replace magnetization and polarization in Maxwell equations and boundary conditions and solve, right? What am I missing?

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DrDu

I did not see this explicitly, but I think the important point is the assumption that the magnetization is due to bound currents only. The important point is what precisely you understand under bound charges.
I think this is equivalent to assuming that the relation between B and M is local.
The important point is that your two systems of equations state that the system can be equivalently be described knowing either only the longitudinal part or the transversal part of the magnetization. Hence you need some equation which links the two.

Meir Achuz
Homework Helper
Gold Member

They are alternate ways of doing the same thing.
The (fictitious) magnetic charge is easier, which is why it is intorduced.

So it works for both internal and external field, right?

anybody?

what is the field due to bound magnetic charges of an arbitrarily long uniformly magnetized bar magnet?
is it not arbitrarily small everywhere (except at the very ends)?​

what is the field of the same bar magnet due to bound electric currents?
is it not uniform inside the bar magnet and arbitrarily small everywhere else (except at the very ends)?​

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Meir Achuz