# Faraday's law or Lorenz force

A student in my physics class posted, in a group, a wrong answer to a question.

The situation was: A plane has a wire extended between the tips of its wings and flies through a magnetic field, perpendicularly, while accelerating.

The question was: What will be the induced current?

His answer was 0, and he explained that as the plane was flying through a uniform field, there would be no change in Magnetic flux density, so no change in flux linkage and ultimately no induced current.

I stated, after clarifying with a question on here earlier today, that instead of consulting Faraday's laws, he should think about Fleming's left hand rule.

My teacher told me I was wrong, and said as the plane is accelerating, it cuts the field lines at a greater rate, so according to Faraday's law, a voltage will be induced, but he also says this voltage will be increasing.... Even if it was Faraday's laws, as this acceleration was constant, wouldn't the voltage be constant?

My next response is a lengthy one:

Faraday's law dictates proportionality between an induced EMF and rate of change of flux linkage. The rule of a wire cutting through fields lines is contrarily a result of Fleming's left hand rule, where a magnetic force is exerted on the electrons inside the wire that is cutting the field, in the direction of the wire; this leads to a potential difference between both sides of a wire, or induced current for a closed circuit. Also, flux linkage, and thus Faraday's law, refer to solenoids, don't they? Ultimately, the induced voltage will indeed increase due to the increase in the rate at which the wire cuts the field, but this isn't to due with Faraday's law which relates the change in flux linkage on a 2-dimensional conductor. the students understandable reasoning for his answer of an induced EMF of 0 came from him observing that the flux linkage is of constant magnitude, so will have 0 rate of change, no matter the acceleration. Using the left hand rule, and the fact that magnetic force, F, is the sum of the BQv, it can be seen that the electrons will be moved by a greater force, making for a greater potential difference, EMF or current, as the wire accelerates.

I also realised that with constant acceleration the voltage would be constant, if this was to do with Faraday's law, and will add that to what I have say.

I'm asking as I want to help this person, while not feeding them false information, wrongfully undermining my teacher and ultimately embarrassing myself.

## 1 Answer

There are two questions here: is there an EMF in the wire and is there a current in the wire? You calculated the force on the charges in the wire as $$F = qvB.$$ The EMF across the wire can be calculated from this: $$\varepsilon = \int_0^L E\,dx = \frac{1}{q}\int_0^L F\,dx = vBL$$ where $L$ is the length of the wire/wing. So, there is a non-zero EMF. But, because the wire is not a closed circuit, there is nowhere for the charges to go once they hit the end of the wire. If the plane is traveling at a constant speed, then the electrons in the wire will shift a bit until the total EMF, both from the magnetic field and the charge distribution, is zero. Then, the current will stop.

As the plane accelerates, the charges will continue to shift to one side, creating a current, but the current will not be constant. Judging by some initial scribbles of mine, the expression for the current along the wire will be a complicated function of position along the wire and time.