What does someone mean by the direction of magnetic flux?

Question

As far as I know Magnetic Flux is a scalar quantity being the dot product of two vectors. $$\displaystyle{\phi_B = \int{\vec{B}\cdot d\vec{A}}}$$ Now my question might not be so clear to you. So let me explain, I am high schooler learning about Electromagnetic Induction. So in one of the classes of Lenz's Law, my teacher asked me a question regarding the direction of Induced current. The question was :

If a square loop of wire in the x-y plane is given some velocity $\vec{V}$ in the positive $x$ direction and there exists a magnetic field $\perp$ to the plane in the negative $z$ direction, but only in the region bound between $x=a$ and $x=b$

And while explaining that question, he said that

As the magnetic flux would be increasing when the loop would be entering the field, so there would be induced current flowing in the loop in the direction, opposite to that of the increasing magnetic flux.

I couldn't understand what he meant when he had said the "direction of flux" ? Like flux is a scalar or is it a tensor ?

Answer 1

Flux is a scalar, but it can be either positive or negative, as given by the dot product definition that you quote. But whether it is positive or negative depends on which of the two normals to $dA$ you have chosen as the direction of $d\vec A$. [There's nothing abstruse about this: if you wanted to call the flow of water through a pipe positive or negative, you'd have to choose which direction through the pipe to call 'positive'!]

Another sign convention also applies in your second extract from your teacher. It is that the sense of a circulation around an area is counted as positive if it would cause a right handed screw to translate in the direction of the flux through the loop.

With this convention we'd say that the flux through a cross-section of an isolated coil due to a current $I$ through the coil is $$\Phi=KI$$ in which $K$ is a positive constant, whereas the emf $\mathscr E$ induced in the coil per turn when the flux through it changes is $$\mathscr E=-\frac{d \Phi}{dt}.$$

Answer 2

Let's give the square loop of current an orientation, so you can pass through the loop going "up" or going "down", and for this problem, supposed the magnetic field is pointing "up" through the loop, and that is defined to be "positive" flux, because $\Phi$ is a scalar.

If you do something to increase $\Phi$ (increase $||B||$, or increase the area, or do a rotation to decrease the angle between $\vec B$ and $d\vec A$), one can say, colloquially, you are increasing the flux in the up direction.

But it's not really a vector quantity that can point in any direction, it's a choice between up or down versus the orientation of the loop, and Lenz's Law means that the current driven by $d\vec E/dt$ needs to push (scalar) flux "down".

In summary: The "up" or "down" directions are shorthand for "positive" or "negative" increase in flux.

That was a good spot, but I will chalk it up to the imprecision of language used to describe the problem.