# Motional EMF and the flux rule contradiction

I have a metallic rod which is being rotated in a constant magnetic field. The EMF is produced in it as per motional EMF and can explained using the Lorentz force. But how can we explain the production of EMF in it using Faraday's flux rule. In this case the rod is in constant magnetic field and even though rod is rotating, the flux is not changing. So as per Faraday's law, there shouldn't be any EMF.

• I don't have time to write a good answer at the moment but peruse the following Wikipedia article, Faraday paradox, and note that the rate of change of magnetic flux through a surface involves two terms when the contour of integration is changing with time. Oct 31, 2017 at 15:31
• The have an EMF you need to have a closed loop. The two ends of the rod are at different points, and if you connect them to a voltmeter, the reading on the meter will depend on where the leads connecting the ends are, and how they are moving. Jul 12, 2021 at 15:51
• @mikestone That's an answer, a very good one actually Jan 3, 2023 at 21:27

The Lorentz force, $$q(\mathbf E +\mathbf v \times \mathbf B)$$ on a charge $$q$$, leading to an emf $$d\mathscr E=(\mathbf E +\mathbf v \times \mathbf B).d\mathbf s$$ in a directed element $$d\mathbf s$$ of a circuit, is, arguably, more fundamental than Faraday's Law and should be used when it is not clear how to apply Faraday's law, if indeed it can be applied. For a conductor moving in a magnetic field the (so-called motional) emf is a result of the $$q\mathbf v \times \mathbf B$$ force on the free charges in the conductor. For a stationary circuit and a magnetic field changing with time, the emf is a result of the $$q\mathbf E$$ force (since $$\mathbf{\nabla} \times \mathbf E=-\frac {\partial {\mathbf B}}{\partial t}$$ so $$\mathscr E=\int_c \mathbf E.d\mathbf s=-\int_S\frac {\partial {\mathbf B}}{\partial t}.d\mathbf S$$, in which $$S$$ is an area spanning the circuit path, C.)

Faraday's law, $$\mathscr E=-\frac {d\Phi}{dt}$$, rather beautifully, works for both types of emf, but $$\frac {d\Phi}{dt}$$ can be interpreted as rate of change of flux linkage only when there is a circuit in the form of a clearcut path. This is the case for most set-ups of interest to electrical engineers. In 'difficult' cases such as a rotating metal rod or a spinning metal disc with metal brushes contacting the axle and periphery we can, as an ad hoc fix, 'bend the rules' and interpret $$\frac {d\Phi}{dt}$$ as rate of cutting of flux. [The fix brings us right back to the emf derived from the magnetic Lorentz force for a directed element $$d\mathbf s$$ of conductor moving with velocity $$\mathbf v$$, since $$d\mathscr E=(\mathbf v \times \mathbf B).d\mathbf s=-(\mathbf v \times d\mathbf s).\mathbf B =-(\text{directed area swept out by}\ d\mathbf s\ \text{per second}) .\mathbf B\ ]$$

... how can we explain the production of EMF in it using Faraday's flux rule? In this case the rod is in constant magnetic field and even though rod is rotating, the flux is not changing.

Faraday’s law is derived from the observation of changing electric or magnetic fields like in transformers.

The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit. (Wikipedia

More general there are three cases of the involved components:

1. Lorentz force / electric device: An electric current in a magnetic field give a deflection of the wire (of the electrons in this wire).
2. Electric generator: The movement of a wire in a magnetic field (of course not parallel to the magnetic field) induces a current in the wire.
3. Electromagnet: A current in a coil induces a magnetic field.

Ultimately, motional EMF and Faraday's Law are two different ways to explain the same effect of an induced EMF. In certain cases, both can be used to explain the induced EMF, while in other cases only one can be used. There is no "contradiction" per se. It just depends on the context of the set-up.

Motional EMF is due to the force that acts on moving charges in a magnetic field. The direction of this force is given by Fleming's Left Hand Rule.

An example of this would be an isolated conducting rod moving through a magnetic field. Since the rod consists of free electrons, there will be a force that acts on these electrons and hence resulting in an induced EMF.

If we modify the above example such that the metal rod is now moving on a metal frame through a magnetic field, we can see how motional EMF can similarly be applied to deduce the induced EMF on the rod. Additionally, Faraday's Law can also be used to deduce the EMF induced on the rod because there is now a well-defined area to calculate the magnetic flux linkage.

However, there are cases which motional EMF cannot be used to deduce the EMF. For example, a stationary loop coil that is placed in a varying magnetic field. In this case, you can only use Faraday's law to determine the EMF because there is no motion of charges through the B-field.

In summary, both theories are ways to deduce the induced EMF given a certain set-up. In certain cases, both can be used to determine the EMF. However, in cases like the one you mentioned where there is no well-defined area, using Faraday's Law will not be that intuitive. As such, there is no real contradiction that you should worry about. Rather, the physical cause for the EMF may be different in various set-ups. (E.g., EMF induced because of moving charge in a B-Field VS EMF induced because of varying magnetic flux linkage.)

A Note on Special Relativity [OPTIONAL]:

Given this two different theories to describe an induced EMF in different situations, it then begs the question - is there an "underlying principle" that is consistent and can be used to describe every case? As a physicist, it feels weird having 2 theories to explain the same eventual result.

The answer lies in special relativity. A changing B-field (due to motion or a changing magnitude) induces an Electric field which exerts a force on the free moving charges.

In the case of a moving conductor through a B-field, if we view it through the lab frame, the B-field is stationary while the conductor is moving. If you perform certain transformations on the "EM field tensor", you will see that in the frame of reference of the conductor (in which the conductor is stationary but the lab is moving), it sees an electric field.

Faraday's law is a consequence of one of the Maxwell's equations, which are consistent with special relativity.

Faraday's law deals with a B-field changing in time; motional EMF deals with B-field changing in space (in the frame of reference of the conductor, the B-field is moving in space). Special Relativity treats space and time on equal footing. Both of this changes in B-field whether through time or space, will induce an E-field which results in a force on the charges and hence an eventual induced EMF.

Good question and observation about the two seemingly "contradicting" theories! Hope this helps.