# What is the *cause* of the Lorentz Force

Is it possible to explain what actually causes the force on a current carrying conductor in a magnetic field. I have read that this is due to the magnetic fields 'interacting' in some way.

This diagram encapsulates the idea

Apparently the force results from the field lines wanting to 'snap back'. Is this correct? Is there a way to explain the physics behind why the concentration of field lines results in a force experienced.

• A number crunching approach is to compute the translational force and the toque on a magnetic dipole subject to a uniform external field, then repeat the computation for a non-uniform imposed field. Alas I don't think that this is what you are looking for. Related: physics.stackexchange.com/q/170130 – dmckee Jul 20 '15 at 2:12
• It would make sense that there isn't a 'simplistic' explanation to it. – Pravin Jul 20 '15 at 2:16
• I believe it relates to the 'catapult field experiment' – Pravin Jul 20 '15 at 2:44
• Are you talking about a coil or are this go straight wires? – HolgerFiedler Jul 21 '15 at 6:59
• Straight wires mate – Pravin Jul 22 '15 at 7:03

No, one cannot explain the "cause" any deeper than the explanation that Lorentz force and Maxwell equations are postulated as a description and experimentally are found to foretell correct results. One can give certain motivations why these might be the correct equations: for example, the Lorentz force law is one of the simplest Lorenz covariant laws one can write to calculate a force (it is given by the tensor equation $f_\mu = q\,F_{\mu\,\nu} \,v^{\nu}$ which is Lorentz covariant and $F$ is a tensor field describing electromagnetism, $v$ the four-velocity of a particle of charge $q$ and $f$ the 4-force).

However, there is an approach to the calculation which is vaguely like what you seek for currents and this is the method of virtual work; look this up in connexion with electrostatitcs and magnetostatics: it only applies in the low frequency limit and also only to steady state charge flows, i.e. steady state currents. In this method, you work out the total energies contained in the magnetostatic field with a conductor in two positions, infinitessimally displaced from one another. The force component / torque in a given direction is then equal to the derivative of this energy with respect to a linear displacement parameter for translation / rotation defined by this direction. In this method, to change a configuration that "squeezes" field lines together is equivalent to raising the magnetic energy density of the field, and thus requires the input of work.

As user7027 says, a current also feels a force in a uniform field, illustrating that it is very tricky to use virtual work for driven currents. In this case an outside current source is needed to keep the current constant against the back EMF that is generated by the currents motion across the field lines, so you need to include the outside source in a putative virtual work calculation. Virtual work methods are most useful for passive systems comprising intrinsic material magnetization and/ or undriven ring currents.

• Thanks. Would you say then that the force arises as a reaction force to the required to increase the magnetic field density? – Pravin Jul 20 '15 at 3:43
• @Pravin That's essentially the virtual work idea. Read up on it applied to magnetostatics and I think you'll see that. – WetSavannaAnimal Jul 20 '15 at 3:49
• Only if the wire moves into different magnetic field does the magnetic energy change. But wire in an uniform magnetic field feels a Lorentz force. – stuffu Jul 20 '15 at 6:23
• @JohnDuffield I guess it depends on what you mean by "explain". Ultimately, we're using inductive thinking grounded on experiment: we can describe our experimental results: sometimes our descriptions seem elegant enough that we feel like we're explaining them. We can make logical deductions grounded on postulates, which is perhaps what you're thinking: this for example is what I mean when I say that one can postulate Lorentz invariance and see what shakes. – WetSavannaAnimal Jul 20 '15 at 21:22
• @user7027 Which is why it is very tricky to apply virtual work to driven current: see changes to my answer. – WetSavannaAnimal Jul 20 '15 at 21:55

Apart from philosophical debates: What is the cause of any force? A gradient in the energy. I'm not in the mood to do any actual calculation, but the energy density of a magnetic field is $\sim \vec{H} \cdot \vec B \sim B^2$ (here at least). Now, we are looking at a field that is created by superposition $\vec B = \vec B_1 + \vec B_2$ with $\vec{B}_i$ being the field around wire $i$. For sufficiently large distance of the two contributions will be virtually zero, but if they are close enough than you have to compute the vectorial sum the value of which depends on two things: a) the distance (creating along the connection vector) and b) the signs of the two currents (determining the sign of the force, $(a+b)^2\neq (a-b)^2$).

What is the cause of the Lorentz Force?

The "screw" nature of electromagnetism. Minkowski referred to it in Space and Time, as did Maxwell in On Physical Lines of Force: "a motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw". This is why the right hand rule applies to both electromagnetism and to screw threads. IMHO to really "get" this, you have to take note of Jackson's Classical Electrodynamics: "one should properly speak of the electromagnetic field Fμν rather than E or B separately". Then you need to depict Fμν for an electron. One simplistic way to do it is to combine the radial electric field lines with concentric magnetic field lines like this:

It's simplistic, but now you start to appreciate the electron's "spinor" nature. And if you've taken note of Maxwell's page title, you may appreciate that counter-rotating vortices attract and co-rotating vortices repel. Whilst an electron doesn't involve some fluid motion, there is the Poynting vector and a "circulating energy flow", so the analogy works. As a result if you set down an electron near a positron they will move towards one another in a straight line. But if you throw the electron past the positron they will also move around each other, something like this:

This is what we see in positronium, and now the Lorentz force $\mathbf{F} = q\left[\mathbf{E} + (\mathbf{v} \times \mathbf{B})\right]$ looks obvious. It's just a combination of the linear and rotational force that results from "spinor" electromagnetic field interactions. And it fits with QED in that the electron and positron are "exchanging field". Positronium is like hydrogen but lighter and short-lived, and as you know hydrogen doesn't have much in the way of a field$^*$.

Is it possible to explain what actually causes the force on a current-carrying conductor in a magnetic field?

Yes. You can understand the linear and rotational forces between charged particles easily enough. The next step is to understand the rotational force on a charged particle near a current-carrying conductor. That's essentially a stationary column of metal ions and a moving column of electrons. Ever read Einstein talking about a field as a state of space? OK, see the gravitomagnetic field, which is described as "twisted space" by NASA author Tony Phillips? You can think of the electromagnetic field as something similar but a tad more intense. Only if you had motion relative to it, you might think of it as "turning space" and start talking about curl aka rot which is short for rotor. IMHO this is the key to really understanding how magnets work. The electrons all have a negative electromagnetic field, and the metal ions all have a positive electromagnetic field with the opposite chirality. If the electrons weren't moving, the opposite "twist fields" would just about cancel each other out$^*$. However the electrons are moving up the wire:

So it's like you're moving through one set of twist fields but not the other. And when you have motion relative to a twist field, you think of it as a turn field, and that's what a magnetic field is. So what you "see" is a residual turn field, a magnetic field, around the wire. An electron thrown past the wire moves in a circular fashion not because of some magical action-at-a-distance force, but because it's "a dynamical spinor in frame-dragged space".

The last step is to understand why two wires move together. For this you can think of your electron being confined in the adjacent wire. It moves upwards, and it moves in a circular fashion. This rotation means the residual turn field looks like a twist field, and since the rotations are counter-rotations to the left and right, you're again in a situation where counter-rotating vortices attract. There's a net linear attraction between the two wires. For the catapult, bend one of the wires into a loop to make a primitive solenoid, then into multiple loops for a better solenoid, which is akin to a bar magnet. Then bend it into a horseshoe shape and put the other wire between the poles like this:

Image courtesy of SPM physics

Again, it moves.

$*$ There is a residual field, but we don't call it an electric field or a magnetic field. Or an electromagnetic field. Or a gravitomagnetic field.

• Your comment raises a lot of questions. But the most important is: your model makes predictions that differ from those of mainstream physics. Do you have any experimental evidence to support your claims? – Bosoneando Jul 24 '15 at 18:40
• When did Maxwell & co say that the EM field of the proton, positron and deuteron are similar? The $F_{ij}$ components of the EM tensor of the positron are three orders of magnitude larger than those of the proton or deuteron, and that has been measured thousands of times. And I haven't said a word of QED, so you can't complain about virtual particles – Bosoneando Jul 25 '15 at 6:54
• Yes, I want you to predict the magnetic dipolar moment of the positron, proton and deuteron – Bosoneando Jul 26 '15 at 19:53
• You predicted the EM field away those particles. To describe the EM field, you have to know the lower multipole moments, i.e. electric charge and magnetic moment. With electric charge alone you only can describe, in the rest frame, the $F_{0i}$ components of the EM tensor (aka electric field E), and as you know, "one should properly speak of the electromagnetic field Fμν rather than E or B separately". – Bosoneando Jul 27 '15 at 9:12
• And for the electron stability, we know that 1) electric charge is conserved 2) energy is conserved 3) the electron is the lightest charged particle and 4) if the electron could decay, the process had to violate at least one of the previous statements, and that hasn't been observed (yet) in any experiment. – Bosoneando Jul 27 '15 at 9:25

Yes, it is possible to explain it. The reason are the magnetic dipole moments of electrons and its intrinsic spin. More see my explanations in this paper.

Electrons have magnetic dipole moment and intrinsic spin. (This spin is really a rotation due to the Einstein-de Haas experiment.) When moving electrons came into a non-parallel to the electrons movement magnetic field the electron's magnetic dipole moment get aligned. This aligne the spin too.

Due to electron's gyroscopic effect the wire get a moment and get deflected sideways. Perhaps the full process is more complicated due to photon emission (electromagnetic radiation of the wire) and spin disalignement and due to permanent disruption of the electron's movement from the atomic structur of the wire.