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When considering the force on a current carrying wire in a magnetic field, is $B$ in the Lorentz force $F=IlB$ the field inside the wire where the current is flowing, or is it the field immediately outside the wire?

To put it another way, is the Lorentz force greater on a piece of iron than a piece of copper (because iron has a high permeability), given the same current and H field? And if not, then does a wire experience a greater force when the entire circuit is in air rather than submerged in water (because air has a higher permeability than water)?

If the Lorentz force depends on $B$ immediately outside of the wire, how does this not violate localism? Why are the electrons being pushed to the side by the magnetic field outside the wire when they're inside the wire?

To be absolutely clear, I'm not talking about the production of the H field due to the motion of electrons in the wire. You may assume the H field is uniform everywhere. The B field only differs as the permeability of the substance differs.

Joseph Summerhays
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2 Answers2

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Neither. The magnetic field in the Lorentz force equation is the magnetic field due to all current sources other than the wire itself.

One could in principle look at the forces on each microscopic element of the wire. Each element of the wire would feel forces from two sources: from external fields, and from the currents (free and bound) inside the wire. Wires made of different materials will certainly have different amounts of bound currents. But as far as the net force on the wire goes, the bound currents don't matter; because when we sum up all the microscopic forces on the elements of the wire to obtain the net force, all of the forces internal to the site will cancel out (thanks, Newton!) and only the forces from the external fields will matter.

Michael Seifert
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In the formula for magnetic force on current-conducting wire element of length $L$:

$$ \mathbf F = I\mathbf L \times \mathbf B, $$ $\mathbf B$ refers to external magnetic field $\mathbf B_{ext}$, due to sources other than those in the element of the wire of length $L$; and it refers to value of this external field at the point of space where the current flows. Usually, most of the current flows inside the wire (as opposed to its surface), thus $B$ in the formula refers to external magnetic induction inside the wire. However, since $\mathbf B_{ext}$ is continuous in the region where the wire element is present, and source of the external magnetic field is usually quite distant (compared to length $L$), value of this external magnetic field at the surface is almost the same as inside the wire, so usually the difference is negligible.

If the wire is thick or the source of the external magnetic field is very close, so $\mathbf B_{ext}$ varies throughout the cross-section of the wire, then we can use the integral formula taking this into account:

$$ \mathbf F = \int \mathbf j \times \mathbf B~ d^3\mathbf x. $$ Here $\mathbf B$ is assumed to be a function of position, and integral goes over all points of space where current density $\mathbf j$ is non-zero. Thus macroscopic magnetic force on a wire can be expressed as sum of forces on all current elements in the wire (due to moving charge), and is proportional to product of current density and external magnetic field at the position of the element.

Ján Lalinský
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