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 𝐴, is facing us squarely. We’re told that the uniform magnetic field that the coil was in, we can call that capital 𝐵, 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.
We’re told that this coil goes through a rotation of one-quarter of a revolution. In other words, it’s 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 Faraday’s 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, Δ𝛷 sub 𝑚, divided by the change in time, Δ𝑡. This means that, in order to calculate the average emf we want to solve for induced in our coil, we’ll need to know three things. First, we’ll need to know 𝑁, the number of turns in our coil. Second, we want to solve for Δ𝛷 sub 𝑚, which is the change in magnetic flux through the coil. And finally, we’ll need to know Δ𝑡, the time over which this change occurs.
We’ve got a bit of a head start because we’re told in our problem statement that the number of turns in our coil, which is capital 𝑁, is 100. Knowing that, let’s move on to solve for Δ𝛷 sub 𝑚, the change in magnetic flux through this coil. To help us out with that, we can recall that magnetic flux, 𝛷 sub 𝑚, is equal to the strength of the magnetic field involved multiplied by the area through which that field moves.
Now, in our case, we’ve got a magnetic field 𝐵 which is constant, and we have an area 𝐴. But we see the area is not constant because our coil is rotating. This tells us that, indeed, there will be a change, a Δ, 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. We’ll use the final magnetic flux through our coil. We’ll call it 𝛷 sub 𝑚 𝑓. And we’ll use the initial magnetic flux for our coil, 𝛷 sub 𝑚 𝑖.
If we take the difference between these two values, that will give us the change in magnetic flux. So then, what are these values? What’s the final and the initial magnetic flux through our coil? Let’s 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 it’s 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, 𝛷 sub 𝑚 𝑖, 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 doesn’t 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 𝐵 in the cross-sectional area of the coil 𝐴.
All this indicates then that the change magnetic flux through our coil as it goes through one-quarter of a revolution is 𝐵 times 𝐴 minus zero, or simply 𝐵 times 𝐴. Now that we know the change in magnetic flux through our coil, let’s move on to consider the change in time over which this occurs.
In our problem statement, we’re 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. It’s 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. It’s called the period, symbolized capital 𝑇. 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 we’re told that our coil goes through one-quarter of a revolution. In other words, Δ𝑡, the change in time over which this occurs, is the period, capital 𝑇, divided by four. We don’t know the period 𝑇, but we do know that it’s equal to one over 𝑓. So, we can replace 𝑇 in this equation with one over 𝑓. 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 𝑓. That’s Δ𝑡, 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 𝑁, multiplied by the change in magnetic flux through the coil, 𝐵 times 𝐴, all divided by the time over which the process occurred, one over four 𝑓. And notice that if we multiply both numerator and denominator by four 𝑓, that factor cancels out with one over four 𝑓 in the denominator, leaving us with this final symbolic expression for the emf induced, negative four 𝑓 times 𝑁 times 𝐵 times 𝐴.
In our problem statement, we’re told that the frequency 𝑓 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, we’re ready to calculate the emf induced. Multiplying these numbers together, we find a result of negative 120 volts. That’s the average emf induced in this coil as it goes through one-quarter of a revolution.