# How does an induced magnetic field generate a current, that causes a force?

I was reading about induction motors and stumbled across topics I used to tutor in college (namely, electricity and magnetism). I recalled all of the right-hand-rule-esque mechanisms from tutoring the course, and I was trying to wrap my head around how the concepts I learned in college coincided with the practical function of the induction motor. Here are the general steps

1. Battery/power source produces DC current
2. Inverter changes it to AC current
3. AC current causes a 3 phase current to flow through coils in the stator
4. The current through each wire produces a magnetic field (magnetic field A) around the wire; which varies in strength because of AC
5. The varying magnetic field from each wire effectively creates a rotating magnetic field around the rotar
6. Because of Lenz Law, a magnetic field is induced (magnetic field B) to counter the increasing/decreasing strength and direction of the magnetic field A. This magnetic field B is associated with/caused by/etc... a current around the rotar bars (electric field B)
7. The Lorentz Force law (F = qv X B) causes forces on these rotar bars and this causes the rotar to rotate, thereby producing power

My question is related to the final step 7... My understanding is that the specific electric field (q) and magnetic field (B) in the equation are electric field B and magnetic field A... but I am conceptually confused as to why this is the case. Magnetic field A effectively causes electric field B, so how can something caused by magnetic field A interact with magnetic field A to produce a force? Doesn't that violate some concept of conservation of energy? If X can produce Y, and X can interact with Y to produce force (effectively, energy) couldn't that create an infinite supply? I'm obviously missing something; what is it?

One of Maxwell's equations, Faraday's Law of Induction, states that the negative rate of change of magnetic flux $$\Phi_B$$ through a conductor produces a potential difference across its ends. If the conductor forms a loop, a current $$i$$ will flow. $$\epsilon = -\frac{\partial\Phi_B}{\partial t}$$

Therefore, a time-varying magnetic field (also read: flux) will produce a time-varying current through a conducting loop.

Another one of Maxwell's Equations is the Ampere-Maxwell Law of Induction. This is essentially the opposite of Faraday's Law as it describes how time-varying electric fields (currents) induce magnetic fields. $$\vec{\nabla} \times \vec{B} \propto \frac{\partial \vec{E}}{\partial t} \propto \vec{J}$$ Where $$\vec{J}$$ is the current density.

Now, because the time-varying current is the negative of the initial time-varying magnetic field, the induced magnetic field produced by this current is in the opposite direction to the initial magnetic field. This results in it opposing the initial field changing.

As you say, this is known as Lenz' Law and the negative sign is necessary, in fact, to maintain conservation of energy.

What seems to be the confusion for you is over the Lorentz force equation which describes the force $$\vec{F}$$ that acts on a charged particle of electric charge $$q$$ moving through a magnetic field $$\vec{B}$$ at velocity $$\vec{v}$$. The force is evaluated for each point in space: $$\vec{F} = q \, \vec{v} \times \vec{B}$$

This form of the equation is perhaps not very instructive when analysing the forces acting on a rotor in a motor. The better form comes when you consider the fact that currents are simply moving charges.

Therefore, we can relate the charge $$q$$ moving through a length $$\vec{L}$$ of a conducting wire in time $$t$$ as $$q = it$$

and the velocity $$\vec{v}$$ of the electrons in the conductor can be given by $$\vec{v} = \frac{\vec{L}}{t}$$

Combining the last two equations and substituting them into the Lorentz Force law, you get the familiar equations for the force acting on a current-carrying wire in the rotor: $$\vec{F} = i \, \vec{L} \times \vec{B}$$

So hopefully full knowledge of these terms will clear up any confusion for you.

• I understand the point you are making but I feel as though you aren't clarifying my conclusion. I realize its the charged particles flowing through the loop that are being acted on, but to me it doesn't seem to make much sense that the magnetic field B is causing the the flow/movement of those particles (via Lenz Law) AND acting on them to create a force (Lorentz Law). How can X BOTH create AND act on Y to produce a force? – Alek Piasecki Oct 24 '19 at 17:40
• Okay, I'll try to go through each step in order in the context of a transformer. An AC source produces an oscillating current $i$ in a primary coil. This oscillating current produces an oscillating magnetic field, $\vec{B_1}$ in the primary coil AND a secondary coil in its vicinity. This oscillating B-Field induces an oscillating current in the secondary coil which also induces a secondary magnetic field $\vec{B_2}$ in the secondary coil which passes through the primary coil. This new field acts in the opposite direction to oppose the change in the first field (Lenz' Law). – Jamie Smith Oct 25 '19 at 8:07
• In the context of an induction motor, an AC current source produces an oscillating current in the stator (eg. 3-phase). This current induces an oscillating magnetic field in and around the stator, passing through the rotor. An oscillating current is induced in the rotor which, by Lenz's law, induces a secondary magnetic field which acts to oppose the change in magnetic flux in the stator. This is achieved by rotating to minimise the 'motion' between the two fields.The energy to do work on the rotor is transferred from the initial magnetic field (which itself came from the initial AC current). – Jamie Smith Oct 25 '19 at 8:38