# Lorentz force separation from induction in rotating loop within AC magnetic field

I was wondering , is it possible to create a loop of any kind of geometry that could rotate in a non homogeneous AC magnetic field and have no net induction within the loop but only have current generated from the effect of Lorentz force on the electrons in the moving conductor ? It is achieved easily in a static magnetic field. In changing magnetic field this in theory can be achieved either by avoiding forming loops around areas of flux or by forming loops with canceling flux between them. I asked this to chatGPT and got an answer that in theory it could be possible but then i tried to draw out loops on paper and soon realized it might not be possible after all.

Another interesting aspect is that when i tried to see the direction of current in the loop as it rotates within a AC magnetic field that is bipolar for example like those found in universal motor stators , it seems to me the direction of the Lorentz force generated current in the loop is the same as that generated by induction, is this always the case in loops rotating in AC fields?

• > have no net induction within the loop but only have current generated from the effect of Lorentz force on the electrons in the moving conductor ? What do you mean by no net induction, that circulation of electric field is zero? What do you mean by Lorentz force, electric plus magnetic force or just the magnetic part? Jan 1 at 16:19
• @Ján Lalinský I mean just the magnetic part, because all other parts and additions form induction within a loop, my curiosity is about whether one can have a loop without induction effects of any kind where any current generated is solely from the deflection force on electrons created by movement through a magnetic field, whether static or time varying Jan 1 at 17:41

EMF in a loop of wire is, in general, not exactly the integral of the electric field $$\mathbf E$$ and $$\mathbf v\times\mathbf B$$, as sometimes tacitly assumed:

$$\mathscr{E} \approx \oint_\gamma \mathbf E \cdot d\mathbf s + \mathbf v\times \mathbf B\cdot d\mathbf s.$$

For example, consider a rigid planar loop of wire translating in electrostatic field of a stationary point charge, the lines of force of electric field lying in the plane of the loop. Thus the charge is at rest in the lab reference frame, and the loop is translating. Since in the loop frame there is, in general, a changing magnetic flux through the loop, there is non-zero net EMF in the loop, and appropriate non-zero current. But in the lab frame, the above integral vanishes; so the integral does not give value of EMF accurately in this case.

Thus we see that even when the electric part of the above integral vanishes, this does not mean that the EMF is due to motional EMF due to magnetic field. In the example, EMF is still due to electric field, just the circulation integral does not quantify it correctly.

A better formula for the net EMF is based on electric field $$\mathbf E'$$ in the frame of the wire loop(assuming it moves uniformly):

$$\mathscr{E} = \oint_\gamma \mathbf E' \cdot d\mathbf s.$$

Since approximately for small velocities we have $$\mathbf E' \approx \mathbf E + \mathbf v \times \mathbf B$$, we get the approximate integral above, but we see why it is only approximate.

This shows that all EMF can be thought as due to electric field only, in the frame of the wire element; there the element has zero velocity, so there is no motional EMF due to motion in magnetic field.

We can try to transform this into the lab frame, but the exact formula will be more complicated that the integral above, taking into account length contraction and field values at different times at different positions, and distinguishing the EMF contribution due to electric field and due to motion in magnetic field becomes, in general, quite difficult.

• I don't see why the integral vanishes in the lab frame, can you elaborate? Charge is induced on the loop while it passes by the stationary charge. Currents result since the loop is in motion, yielding a time-varying magnetic flux through the loop. On an unrelated note, I find it a little easier to visualize the fields and the currents by replacing the point charge with a uniformly charged line segment that runs along the direction of motion. Its motion in the loop frame emulates current flow in an ordinary wire, when it's close to the loop.
– Puk
Jan 2 at 2:24
• @Ján Lalinský I am not sure I understood your answer. I was analyzing this from a B field perspective. on a piece of paper making drawings I concluded that once you form a loop , there is no possible geometry that doesn't get varying flux through it as it rotates in a non uniform , non homogeneous magnetic field therefore if the field is static the loop will get induced current that has a frequency that matches to rotational frequency of the loop and if the B field is AC then the current will have a frequency that is the sum of RPM+ AC field frequency. Jan 2 at 7:53
• @Puk the integral vanishes when $\mathbf E$ is field of the static charge, because this field is conservative. In the frame of the loop, field of the charge $\mathbf E'$ is not conservative, and the integral does not vanish. This latter integral doesn't depend on the induced surface charge on the loop surface; this charge can be made arbitrarily small by assuming the loop is made of an arbitrarily thin wire; thus it has arbitrarily small contribution to electric field, and cannot explain the EMF. The EMF is due to external electric field. Jan 2 at 13:00
• @Girts The point of my answer is that it is not always correct to classify the EMF as "due to motion in magnetic field", as opposed "due to induced electric field". Thus the question seems based on a misconception that we can easily define, for a moving loop, which part is due to induced electric field, and which part is motional. In general, this is not easy. Where it is easy is in the frame of the translating loop - non-zero EMF is purely due to an induced electric field. Jan 2 at 13:03
• @Girts In the case of a rotating loop, the loop is a non-inertial frame, and EMF in its frame is due to both effective electric field there, and (a very minor effect in practice) due to centrifugal force. Jan 2 at 13:10