### Video Transcript

A rectangular coil with 100 turns and a cross-sectional area of 0.06 square meters rotates at a frequency of 50 Hertz in a uniform magnetic field of flux density 0.1 teslas. The plane of the coil is initially parallel to the magnetic field. Calculate the average emf induced through one-quarter of a revolution.

If we were to sketch out our rectangular coil and this magnetic field from a couple of different vantage points. If we looked at the coil front-on so that its full area, we can call it capital ๐ด, is facing us squarely. Weโre told that the uniform magnetic field that the coil was in, we can call that capital ๐ต, is parallel to the plane of the coil. That means if we were to look at the coil from the bottom up, then our perspective on it might look like this. We would see the edge of the coil and the magnetic field lines parallel to it.

Weโre told that this coil goes through a rotation of one-quarter of a revolution. In other words, itโs turned 90 degrees from its initial position. We want to calculate the average emf induced in our coil through this turn. Physically, the key thing going on here is that the magnetic flux through our coil is changing as it turns. That change in magnetic flux over time is what induces an emf in the coil. This happens according to Faradayโs law of electromagnetic induction

This law tells us that the emf induced in a conducting loop is equal to negative the number of turns in the loop multiplied by the change in magnetic flux, ฮ๐ท sub ๐, divided by the change in time, ฮ๐ก. This means that, in order to calculate the average emf we want to solve for induced in our coil, weโll need to know three things. First, weโll need to know ๐, the number of turns in our coil. Second, we want to solve for ฮ๐ท sub ๐, which is the change in magnetic flux through the coil. And finally, weโll need to know ฮ๐ก, the time over which this change occurs.

Weโve got a bit of a head start because weโre told in our problem statement that the number of turns in our coil, which is capital ๐, is 100. Knowing that, letโs move on to solve for ฮ๐ท sub ๐, the change in magnetic flux through this coil. To help us out with that, we can recall that magnetic flux, ๐ท sub ๐, is equal to the strength of the magnetic field involved multiplied by the area through which that field moves.

Now, in our case, weโve got a magnetic field ๐ต which is constant, and we have an area ๐ด. But we see the area is not constant because our coil is rotating. This tells us that, indeed, there will be a change, a ฮ, of magnetic flux through our coil. To find that out to solve for the change in magnetic flux over this rotation, we can use two values. Weโll use the final magnetic flux through our coil. Weโll call it ๐ท sub ๐ ๐. And weโll use the initial magnetic flux for our coil, ๐ท sub ๐ ๐.

If we take the difference between these two values, that will give us the change in magnetic flux. So then, what are these values? Whatโs the final and the initial magnetic flux through our coil? Letโs start by figuring out the initial magnetic flux through our coil.

We saw that, initially, the position of the plane of our coil is parallel to the uniform magnetic field itโs in. By visual inspection, we can see that no magnetic field lines cross through this coil. Because no magnetic field lines cross through the coil, that means the flux through it is zero. So, ๐ท sub ๐ ๐, the initial magnetic flux through our coil, is zero.

But then, as we look to the final magnetic flux through the coil, we see that the coil doesnโt remain in the same position. We know that, rather, it rotates until finally its plane is perpendicular to the magnetic field lines. In this case, the coil is maximally exposed to the magnetic field lines. Its magnetic flux is maximum. Or yet at this way, to one another, that flux is simply the product of the magnetic field ๐ต in the cross-sectional area of the coil ๐ด.

All this indicates then that the change magnetic flux through our coil as it goes through one-quarter of a revolution is ๐ต times ๐ด minus zero, or simply ๐ต times ๐ด. Now that we know the change in magnetic flux through our coil, letโs move on to consider the change in time over which this occurs.

In our problem statement, weโre not told anything directly about time. But we are told the frequency of rotation of the coil, that it moves at 50 hertz. That is, it rotates 50 times every second. Itโs true that the frequency of rotation is, in itself, a time value. But if we take the inverse of that, one over frequency, that is a time. Itโs called the period, symbolized capital ๐. The period of rotation is how much time it takes a rotating object to go through one complete revolution.

That fact is helpful to us because weโre told that our coil goes through one-quarter of a revolution. In other words, ฮ๐ก, the change in time over which this occurs, is the period, capital ๐, divided by four. We donโt know the period ๐, but we do know that itโs equal to one over ๐. So, we can replace ๐ in this equation with one over ๐. With that substitution made, if we then multiply both the numerator and denominator of this equation by one over four, then in the denominator multiplying four by one over four gives us one. And our answer simplifies to one over four ๐. Thatโs ฮ๐ก, the time over which this 90-degree rotation occurred.

So, now we have everything we need in order to calculate the average emf induced in this coil. That emf is equal to negative the number of loops in the coil, capital ๐, multiplied by the change in magnetic flux through the coil, ๐ต times ๐ด, all divided by the time over which the process occurred, one over four ๐. And notice that if we multiply both numerator and denominator by four ๐, that factor cancels out with one over four ๐ in the denominator, leaving us with this final symbolic expression for the emf induced, negative four ๐ times ๐ times ๐ต times ๐ด.

In our problem statement, weโre told that the frequency ๐ is 50 hertz, the number of turns in our coil is 100, the magnetic field strength is 0.1 tesla, and the cross-sectional area of our coil is 0.06 square meters. Plugging all these terms in, the frequency, the number of turns, the magnetic field strength, and the cross-sectional area, weโre ready to calculate the emf induced. Multiplying these numbers together, we find a result of negative 120 volts. Thatโs the average emf induced in this coil as it goes through one-quarter of a revolution.