# Significance of relative motion in Faraday's law of electromagnetic induction

If a magnet is moved through space with some finite speed then the magnetic field strength at each point of space varies with time and thus there is an induced electric field at each point of space. Now if we place a conducting loop in the space near the magnet then we notice a current (since the charges experience force along the loop) .

If the coil is moved with the same velocity as that of the magnet , there is no observed current. But why ?

I know that from the frame of the magnet there should be no current but I am not sure why there is no current if it is observed from the frame of ground.

This time also there are induced electric fields all around and these fields should apply forces on the charges in the loop (I know the loop interacts with each closed field line for a shot interval as it is moving but it does interact) and gradually the electrons should drift along the loop and hence there should be a current but of course it's not true.

Where is my reasoning wrong ?

## Edit :-

As per Philip's answer , there are two opposing emf's and thus no net current is observed . But I couldn't get from where does the emf of the magnetic field is coming from. Consider the figure below :- The yellow rod is a big magnet and the blue loop is a small conducting loop and the other two loops represent any two of the many induced electric field lines. This time the magnetic field lines are almost uniform. Now as per the picture, I don't see any emf of magnetic fields .

So is it really due to two opposing emf's ?

• This is the paradox that special relativity was designed to resolve. Dec 6, 2021 at 11:17
• @Connor Behan so I wasn't wrong with my reasoning ?? Dec 6, 2021 at 12:29
• Relativity resolves this paradox in the frame of the magnet, but from the frame of the stationary observer there is clearly a force on every singe charge , Obviously... faradays law says nothing about forces individually , it states that the Emf about a closed loop would be zero. The forces cancel eachother out in the loop Dec 6, 2021 at 15:37
• Yes, motional emf due to motion in magnetic field is still present, because conductor is moving in presence of magnetic field, so there is force $q\mathbf v\times \mathbf B$ acting on mobile charges in the conductor, and this force is perfectly cancelled by force of induced electric field $q\mathbf E_i$. Vectors $\mathbf v$ and $\mathbf B$, despite your picture, are not completely co-linear in reality, as the magnetic field decreases with distance from the magnet and its lines of force are curved. The cross product $\mathbf v\times \mathbf B$ is thus generally non-zero. Dec 13, 2021 at 17:10
• Is that microsoft paint? O_o Mar 12, 2022 at 22:13

Using the pre-Relativity approach (valid even if unsatisfying), when you move both the magnet and the coil in the same direction you induce two emfs that cancel.

There is, as you say, a non-conservative electric field due to the magnet's motion. This provides electric Lorentz forces ($$q\mathbf E$$) on the charge carriers in the coil. The motion of the loop through the magnet's (non-uniform) field provides a magnetic force ($$q\mathbf v \times \mathbf B$$) on the charge carriers. The emfs due to these forces cancel.

The comment by Ján Lalinský is the best answer. I wonder why he wrote it as a comment rather than as an the answer.

Vectors v and B are not co-linear, the magnetic field starts diverging as soon as it is gets away from the magnet. Therefore there is a magnetic force that exactly cancels the electric force from the induced currents.

Why does it exactly cancel it ? Because the equations that determine the induced electric currents are such that the total force exerted on a cahrged particle is the same in any frame. Since in the frame where the magnet is static, there is no force on a static particle, in a different frame there will also be no force on a particle comoving with the magnet.

So to your question : Is it really due to two opposing EMF's the answer is definitely YES

Your own argument next to your drawing, namely "Here the magnetic field lines are almost parallel" contains its own contradiction. Almost but not exactly. And since the loop is very small, the electric force on it is also very small, and precisely opposite to the magnetic one. Why exactly ? Because there is no force in the comoving frame, therefore no force in any frame.

because there is no relative motion between the 2 objects if you think about Lenz's law $$\epsilon=-N\frac{dB}{dt}$$ it requires a changing field from the point of view of the wire, so it doesn't matter if they are both moving from the perspective of the ground the difference between them is what matters in any frame.

• I know that the relative motion is what matters but I want to know why exactly it is from theoretical perspective ? Dec 6, 2021 at 14:59

The magnetic field at the coil is constant in any frame, so there is no induction.

• what about the electric field loops which exist in the space ? Dec 13, 2021 at 17:40
• @Antik In the comoving frame there is no electric field, because in that frame the magnetic field is constant. Dec 15, 2021 at 20:41