Faraday's law: How to interpret flux on a time/voltage graph?

Question

Background

This is related to a homework assignment, but my question is more on the conceptual side. I will therefore only paraphrase the problem.

Problem

The question begins with the idea of dropping a magnetic rod through a coil. You know, the usual. The problem then provides a time/voltage graph of the induced voltage.

(I should specify; time on the horizontal axis, voltage on the vertical.)

We are then provided with the unit sizes of the dimensions, i.e. the area of each square on the graph, and are asked to estimate the magnetic flux.

My thoughts

Given Faraday's law, we know that $$\epsilon = -\phi'(t)$$ and integrating both sides, we realize that the magnetic flux would be the integral of the induced voltage wrt. time, and is therefore the area under the curve. I can therefore give an estimate by just roughly counting how many squares are covered by the area under the graph.

My question

What's a more or less conceptually correct way to provide such an answer? Do I sum up the absolute values of the areas under and over the curve, providing a purely positive result, or do I consider the negative flux to cancel out the positive flux, yielding roughly zero?

The is telling me that "you should only consider $A_1$ as the largest flux happens when the voltage graph crosses the time axis". That doesn't feel right, but I don't know enough to say why.

Answer 1

If I understand the problem, the rod is dropped all the way through a coil. A2 is when it comes out the other side. Is this correct?

If that is correct, the flux should go from zero to zero over the whole process, and that is only possible by using both portions with signs.

In general, the sign of the voltage is related to the sign of the flux change, where the flux has a directionality to it.

Answer 2

I do not know exactly how the magnetic flux linked with the coil, $\phi$, varies with position relative to the centre of the coil, $x$, but I suspect that it will have the shape of a normal distribution graph.
What suggests that the magnetic flux is very small / zero when $|x|$ is very large and a maximum at $x=0$?

The graph of induced emf, $\mathcal E$, against time, $t$.

Faraday says $\mathcal E = (-) \dfrac {d\phi}{dt} = \dfrac {d\phi}{dx}\cdot \dfrac {dx}{dt} = \dfrac {d\phi}{dx}\cdot v$ where $v$ is the velocity of the rod.

Except at the start when the rod is released from rest the rod is in motion and yet the induced emf becomes very small when the rod is exiting the coil and even though the speed of the rod is increasing long after it passes the centre of the coil the induced emf is falling which suggests a falling away of the magnetic flux as the rod moves further from the centre of the coil.
When the rod is central to the coil with $x=0$ and at time $t=0$, the induced emf is zero and as $v$ is finite this means that $\dfrac{d\phi}{dx}$ is zero and this is a position of maximum magnetic flux.

Thus the maximum flux occurs when $x=0\,/\, t=0$.

So, as $\phi = \int \mathcal E\,dt$, either the magnitude of area $A_1$ or area $A_2$ is equal to the maximum flux.