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Force on a current-carrying wire in a uniform magnetic field

Suppose there is a wire carrying a current through a uniform magnetic field, as shown in the figure above. The force felt by the wire is represented by the downwards orange arrow.

Is this force the result of the interaction between the wire's magnetic field (concentric rings; not shown) and the uniform magnetic field? In other words, in order for there to be a magnetic force, must there always be two magnetic fields interacting?

Another way to imagine this: is the Lorentz force (magnetic) experienced by a point charge moving through a uniform magnetic field always the result of the interaction between the magnetic field generated by the moving charge and the uniform field through which it is traveling?

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  • $\begingroup$ very similar question here (physics.stackexchange.com/questions/217194/…) but the answer doesn't make sense to me. How can the downward force not be a result of interacting magnetic fields? $\endgroup$
    – M. A.
    Commented Feb 8, 2019 at 19:49

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Older textbooks often attributed the force on a current-carrying conductor in an external magnetic field to an interaction between the external field and the conductor's field.

The idea goes back to Faraday, who conceived of magnetic field lines as 'lines of force', under tension and pushing out sideways. Think of the pattern of field lines between two bar magnets placed in line, with the North Pole of one facing the South Pole of the other with a gap between them. Can't you just feel those lines of force pulling the magnets together? Even more interesting is the resultant field pattern that you get for the set-up you've shown in your diagram. Many of the resultant field lines (co-inciding at each point with the direction of the resultant field of wire and magnet) run in curved paths from the N to the S of the magnet. But the lines are closer together (as the fields re-inforce) above the wire than they are underneath it (where the fields tend to cancel). So the lines above, under tension and pushing out sideways, act like catapult strings, forcing the wire downwards, in accordance with your Fleming's left hand rule!

Maxwell developed the theory of Faraday's lines of force mathematically. The Maxwell stress tensor, defined for each point in a magnetic &/or electric field, was the outcome. I believe that the motor effect force on the wire can be correctly predicted using this concept.

For simple set-ups such as yours, it is, of course, far easier to consider the force to be due to a direct interaction between the magnetic field and moving charges in the wire – the good old Lorentz force law.

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  • $\begingroup$ When you say that the "motor effect force on the wire can be correctly predicted using [Maxwell's stress tensor] concept", do you mean that we wind up with the same result as predicted by the Lorentz force law? If so, the force on the wire resulting from magnetic field interactions between conductor and magnet fields simplifies down to Lorentz force law (the two are not different). Am I understanding correctly? Thank you! $\endgroup$
    – M. A.
    Commented Feb 8, 2019 at 22:32
  • $\begingroup$ I believe that is the case, but I have to confess that I've never worked with the Maxwell stress tensor, so I can't be as definite as I'd like to be. Sorry to let you down. $\endgroup$
    – Philip Wood
    Commented Feb 8, 2019 at 22:42
  • $\begingroup$ Thank you, your answer is very helpful and gives me a new direction in my search for a definitive answer. $\endgroup$
    – M. A.
    Commented Feb 8, 2019 at 22:49
  • $\begingroup$ Thank you. Do tell us what you find out. I have, incidentally just looked at your parallel wires question. Note that the fields of the individual wires tend to cancel between the wires but reinforce in the region beyond them. So, in the Faraday-Maxwell picture, the wires are forced together by the closer-packed resultant field lines in the outer region. Incidentally one A-level Physics syllabus in the UK requires students to learn this approach. I believe that electrical engineers are also taught it. $\endgroup$
    – Philip Wood
    Commented Feb 8, 2019 at 23:37
  • $\begingroup$ Please let me know what you think of the discussion here: physics.stackexchange.com/questions/459674/… $\endgroup$
    – M. A.
    Commented Feb 9, 2019 at 6:25
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The Lorentz force experienced by a moving charged particle or a wire in which a current is flowing is not the result of the interaction between both magnetic fields, but only between the external magnetic field and the intensity of the current. You can see this explicitly in the expression for the Lorentz force.

For a charged particle:

$$\vec{F}=q\vec{v}\times \vec{B}$$

For a wire:

$$\vec{F}=I\int\vec{dl}\times \vec{B}$$

In both cases, only the external field is the one interacting directly with the current of the system. While it is true that a charged particle or a current by themselves produce their own magnetic field, but this is usually neglected in calculations.

If the current is strong enough, the magnetic field of the charge or wire will actually interact with the magnetic moment of whatever is causing the external field (i.e. a magnet) and produce additional interactions.

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  • $\begingroup$ Thank you. Do you have a source for this? I have the opposite elsewhere: " Lorentz force, develops due to the interaction of the applied magnetic field and the magnetic field generated by the particle in motion. " nationalmaglab.org/education/magnet-academy/watch-play/… $\endgroup$
    – M. A.
    Commented Feb 8, 2019 at 20:00
  • $\begingroup$ Is there proof of an experiment where the current is set to be extremely high and the additional interactions are visible? $\endgroup$
    – M. A.
    Commented Feb 8, 2019 at 20:01
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    $\begingroup$ I saw the link you posted. It is a bit weird, as all the literature I've consulted usually stated only the interaction between the magnetic field and the current (see for example, Resnick Vol. 2, Wangness, Jackson, or here web.mit.edu/sahughes/www/8.022/lec10.pdf). Regarding the experiment you're interested in, I'm currently not aware of one you can read online. I'll let you know if I find it. $\endgroup$
    – Charlie
    Commented Feb 8, 2019 at 20:06
  • $\begingroup$ Your link does not explicitly suggest that the Lorentz force is separate from any force due to magnetic field interaction. In fact, the MIT document has figures of the concentric magnetic field formed around a wire carrying current. Take the example of the attractive force between two parallel wires: is the attractive force purely due to the Lorentz force or is there any additional effect due to the interaction between the magnetic fields of the wires? $\endgroup$
    – M. A.
    Commented Feb 8, 2019 at 20:17
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    $\begingroup$ Right, each wire feels a magnetic force that is proportional to its own current and the magnitude of the other field. So my question boils down to: is there any other magnetic force at play due to wire field interaction, or does it all reduce to Lorentz forces? Does your original answer need to be modified now? Thanks for the extended discussion. $\endgroup$
    – M. A.
    Commented Feb 8, 2019 at 20:44

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