Force Magnetic Dipole: Electric Current vs Magnetic Pole Model

[da_willem]

There has been some dispute in the past about the validity of the electric current model of a magnetic dipole producing a force $$\nabla (\vec{m}\cdot\vec{B})$$ versus the magnetic pole model producing $$(\vec{m}\cdot \nabla)\vec{B}$$ (see e.g. Boyer `87). I think for elementary particles this dispute is now settled in favour of the electric current loop model.

The difference between these two force terms is using some vector relation $$\vec{m} \times (\nabla \times \vec{B})$$. But this only vanishes if the magnetic dipole moment is parallel to the curl of B or B itself is rotationless.

But for rotationless magnetic fields, magnetic fields are already solenoidal, what is left? Aren't the only solenoidal rotationless fields constant fields?

So aren't we always making errors when using the second force expression, e.g. in calculating the force on magnetized objects? Or is this error usually very small? Any thoughts?

 

[da_willem]

I guess for example the field of an electromagnet is both solenoidal and irrotational, I had the Helmholtz decomposition wrongly in mind. But still, is the term $$\vec{m} \times (\nabla \times \vec{B})$$ often/usually negligible or does one generally have to use the first term $$\nabla (\vec{m}\cdot\vec{B})$$ for a magnetized material?

 

[pam]

For a permanent magnet or an electromagnet, curl B is only non zero on the surface of the magnet, so either force expression should work. There is a delta function term for the energy of two dipoles, which is different for magnetic or electric dipoles. The magnetic form for the delta function is the correct one for elementary particles.

 

[da_willem]

Thanks for your response, so as long as there are no free currents at the position of the dipole the expressions are the same. The first of the expressions stated includes an interaction associated with an inhomogeneous magnetization whereas the second does not. Wouldn't it imply that there can be only inhomogeneous magnetization when there are currents within the material?

 

[pam]

Each of those expressions refers to the force on a point dipole, or the force on an extended body using the dipole approximation.
The question of spatial dependence of the magnetization M never enters.
If you want the force due to a field B(r) acting on a finite body with magnetization M(r),
you need different equations.

 

[da_willem]

U're probably right. I'm just looking for an easy way from the expression we have for the force on a single dipole towards the Kelvin force on a magnetized material consisting of many dipoles.

 

[pam]

The easiest way to get the force on a body having permanent magnetization M(r) in an external magnetic field is to introduce (fictitious) magnetic charge.
This is the same as the bound charge in electric polarization P.
The magnetic charge is given by $$\rho_m=-\nabla\cdot{\vec M}$$
and surface charge $$\sigma_m=M_n$$, where $$M_n$$ is the normal component of M at the surface.
Then $${\vec F}=\int\rho{\vec B}d^3r$$.

 

[granpa]

The easiest way to get the force on a body having permanent magnetization M(r) in an external magnetic field is to introduce (fictitious) magnetic charge.
This is the same as the bound charge in electric polarization P.
The magnetic charge is given by $$\rho_m=-\nabla\cdot{\vec M}$$
and surface charge $$\sigma_m=M_n$$, where $$M_n$$ is the normal component of M at the surface.
Then $${\vec F}=\int\rho{\vec B}d^3r$$.

what happened to the surface charge in the formula for force?

edith. I guess its just considered to be bound charge which happens to be at the surface.