The graph shown represents the relationship between the induced EMF in a dynamo coil and the angle between the normal to the plane of the coil and the magnetic flux direction 𝜃. What is the maximum value of the induced EMF?
Okay, so lots of words in this question, so let’s break it down bit by bit. So first of all, we’ve been told we’ve got this graph which shows the relationship between the induced EMF in a dynamo coil — so that’s the EMF on the vertical axis here — and the angle between the normal to the plane of the coil and the magnetic flux direction 𝜃. So the angle 𝜃 is on the horizontal axis. We’ve been asked to find the maximum value of the induced EMF. Now because we’ve got the maximum EMF on the vertical axis, we need to find the maximum value of the graph — that’s this value here — and we need to read off or calculate whatever that value is on the vertical axis. So naturally, we can’t read it off because there are no numbers there, but we can calculate it. But before we do that, let’s look more closely at our dynamo coil.
So here’s our rotating dynamo coil and it’s rotating between the north pole and south pole of a magnet. Now at this point, we can recall that magnetic field lines originate at the north pole and terminate at the south pole. So we can actually draw the magnetic field lines off this magnet. They’re going to be going left to right as we’ve drawn it from the north pole to the south pole. And of course, we can label the line or the vector that is normal to the plane of the dynamo coil. Now, in the orientation that we’ve drawn it, this is the plane of the dynamo coil. And hence, this upper pointing arrow is pointing in the direction that is perpendicular to the plane of the coil.
Now, it’s worth noting that as dynamo coil rotates, the arrow pointing in the direction normal to the plane of the coil rotates along with it. And hence, the angle between the normal line and the direction at which the magnetic field points — we’ll call this angle 𝜃 — keeps changing as the coil rotates as well so just to clarify once again 𝜃 is the angle between the normal to the coil of the plane and the magnetic field.
Now the graph that we’ve been given in this question shows the relationship between this angle 𝜃 and the EMF induced in the dynamo coil, which is shown on the vertical axis. Now we’ve been asked to find the maximum value of the induced EMF. And there are a couple of ways to think about this. Firstly, we can recall that the EMF induced in a rotating dynamo coil is equal to the number of turns in the dynamo coil multiplied by the magnetic field strength of the magnets used in the setup multiplied by 𝐴, the surface area of the coil within the magnetic field, multiplied by 𝜔, the angular frequency of rotation of the coil, multiplied by sin 𝜃, where of course 𝜃 is this angle here.
And luckily, on this graph, we can see the value of the EMF for a specific value of 𝜃. Therefore, we can use this information and actually substitute it into our equation. But before we do that, let’s realise that in our setup the only things that are changing are the value of 𝜃 and the EMF induced because as the coil rotates, the angle between the normal to the coil’s plane and the magnetic field will change and hence so will the induced EMF. However, we are not changing the number of turns on the coil or the magnetic field strength or the area of the coil or the rate at which the coil rotates. Hence, the value of 𝑁 multiplied by 𝐵 multiplied by 𝐴 multiplied by 𝜔 is going to stay constant. So for our setup, we can say that the EMF is equal to 𝐶 multiplied by sin 𝜃, where 𝐶 is a constant; that’s the product of 𝑁, 𝐵, 𝐴, and 𝜔. So let’s not forget to write that down here.
Now, we can substitute in the values that we’ve read off the graph. Because we know that when the EMF is 10 volts — so that’s this value here — that is going to be equal to the constant 𝐶 multiplied by the sine of the value of 𝜃 at that point. And at that point, 𝜃 is 45 degrees. And of course, we should include the unit volts on the left-hand side as well. Therefore, we get 10 volts is equal to the constant 𝐶 — whatever that may be — multiplied by sin of 45 degrees. Now if we divide both sides of the equation by sin of 45 degrees, we can see that it cancels on the right-hand side. And therefore, what we’re left with is that 10 volts divided by sin of 45 degrees is equal to the value of the constant 𝐶. But then, sin of 45 degrees is one over root two. And so, what we have is 10 volts divided by one over root two is equal to 𝐶. Or if we simplify this, 10 root two volt is equal to 𝐶. Therefore, we found the value of the constant 𝑁𝐵𝐴𝜔.
Now the important thing is that we know the value of the constant. We don’t need to know the value of these individual components of the constant because our master equation EMF is equal to 𝐶 sin 𝜃 only cares about the value of 𝐶, not the value of these individual bits. And in fact, we can substitute 10 root two volts for 𝐶 in our equation. So now, what we’re left with is an equation for this graph here that the EMF is equal to 10 root two volts which is a constant multiplied by the sin of 𝜃.
So if we want to find the maximum value of the EMF, we need to plug in the value of 𝜃 for which the maximum value occurs. And from the graph, we can see that the maximum value of the EMF is here. So this will occur for a 𝜃 value of 90 degrees; that’s this value here. And hence, we can say that the EMF subscript max that’s the maximum value of the EMF, which is what we would read off on the vertical axis if there were any numbers to read, is given by the constant 𝐶 that’s 10 root two volts multiplied by the sine of the angle at which the maximum value occurs.
Now at this point, before we do anything else, we can realise that it’s a good thing that we kept the unit volts in the value of 𝐶. Because regardless of what the value of sine of 𝜃 is, it’s only ever going to be a unitless number because sin of zero degrees or 45 degrees or 90 degrees or whatever degrees is never going to have a unit. It’s only going to be a multiplication factor. And so in order for the left-hand side to be an EMF, we need the unit of volts or some equivalent unit on the right-hand side. And in fact, it is the value of this constant 10 root two volts that gives us the unit.
But anyway at this point, we can evaluate that sin of 90 degrees is one. And so, all we have on the right-hand side is 10 root two volts multiplied by one. Or in other words, we have EMF max — that’s the maximum EMF — is equal to 10 root two volts. And hence, our final answer is that the maximum value of the induced EMF is 10 root two volts.
Now before we go, there’s a slightly quicker way of answering this question. And for this, we don’t need to delve into the physics of how a dynamo coil works. Instead, we can realise that this graph just looks like a sine curve but with some multiplying factor because normally a sine curve looks something like this. If we’ve got the 𝜃 values on the horizontal axis and the values of sin 𝜃 on the vertical axis. Then, we can see that sin of 𝜃 starts out at zero then goes to its maximum value when 𝜃 is equal to 90 degrees and then dips back down to zero once again at 180 degrees.
That is exactly the same as the shape of this curve that we’ve been given in the question, except a normal sine curve peaks at the value of sin 𝜃 is equal to one, whereas in this case we’ve got some value greater than 10 volts. And of course, that value is the amplitude of the sine curve. That’s the distance between the maximum value and zero. And so essentially, we can come to the same conclusion that the EMF curve given at the question is equal to some constant that’s the maximum value. So we call that EMF subscript max. And that’s the value of the EMF when it’s maximum or in other words it’s amplitude. And we multiply this by sin of 𝜃.
And then from here, we simply sub in the known values 10 volts and 45 degrees. And we’ll get that the value of the EMF when it’s 10 volts has to be divided by sin of 45 degrees because that’s the corresponding 𝜃 value. And that is going to give us the value of EMF subscript max. And of course, this once again works out to be 10 root two volts. And it also allows us to realise that the value of EMF subscript max — that’s the maximum induced EMF value — is basically the same thing as the amplitude of the curve. Makes sense right? The distance between the maximum value and zero is the amplitude. And that’s basically the same as the maximum value itself.
So those were two different ways to find the same answer: one which relied on our understanding of physics a little bit more, while the other relied on a mathematical understanding.