Video: Eq17S1-Physics-Q31

Which of the following graphs represents the current generated by a dynamo consisting of a number of coils having equal small angles between their planes?


Video Transcript

Which of the following graphs represents the current generated by a dynamo consisting of a number of coils having equal small angles between their planes?

We see there are four different options for this representation of current: option a), b), c), and d). In each case, there’s a graph with current on the vertical axis and time on the horizontal axis. The difference between the graphs of course is the shape of the curve in each one. We want to figure out which one represents the current generated by a dynamo consisting of a large number of coils separated by equal small angles.

Let’s start out on this by looking at a dynamo that just has a single coil to it. Here, we have the basic elements of a dynamo, also called an AC generator. There is a coil of wire arranged in a flat plane. And that coil is between the poles of a permanent magnet. And each end of the coil of wire is attached to what are called slip rings. These are rings that maintain constant electrical contact with the end of the wire even as the coil rotates. The two slip rings are then attached to another circuit which is powered by the current generated through the dynamo.

Unlike a motor which converts electrical energy into mechanical energy, a generator goes in the opposite direction. It takes mechanical energy and converts it to electrical energy. That’s what happens here with our dynamo, though we haven’t drawn where the mechanical energy comes in. Nonetheless, it’s that mechanical energy input from the outside that leads to the rotation of our coil of wire. Having a coil of conducting material rotating doesn’t by itself generate current. But in the presence of a magnetic field like we have here between our bar magnet, such movement does generate current.

The reason behind this is that as our coil rotates, the magnetic flux through that plane of the coil is changing in time. And we know from Faraday’s law that changing magnetic flux over time leads to an EMF, which itself leads to the creation of electrical current. We can even draw a graph of the current generated in this coil of wire as it rotates over time. As we create this graph of current generated in our loop versus time, we can align that current generated with the particular orientation of our coil of wire relative to the magnetic field that it’s contained in. We see we have the north and south pole of a magnet marked out here. And the first orientation of our coil is perpendicular to the field lines created by that magnet.

Looking back over at our sketch of the dynamo, this would correspond to an orientation of the coil like this up and down. At this moment in time, the change in magnetic flux through our loop is at its lowest point; it’s at zero. Zero change in magnetic flux over time, it means zero EMF induced which means zero current induced. But then looking ahead at the next orientation of our coil, we see that it’s rotated 90 degrees. This coil orientation is like the one we started with. That’s flat and is parallel with the magnetic field lines. The magnetic flux is changing rapidly through the coil as it’s at this point. And so, the EMF is at a high point and so is the current generated.

The coil continues to rotate another 90 degrees. Once again, with the coil oriented vertically, we have zero EMF induced and therefore zero current induced. The coil then continues to rotate another 90 degrees. This time, the magnetic flux through the loop is in the opposite direction that it was before. There is induced EMF, but it’s of a reversed polarity than it was originally. That is whereas before our EMF led to what we’ll call a positive current, now the EMF induced leads to what we’ll call a negative current. This negative induced current has the same magnitude as our positive induced current, just the opposite sign. That opposite sign comes from the fact that our rotating coil has now moved through more than 180 degrees. This means that the magnetic flux lines from the magnet are moving through the coil in the opposite direction as they were before. It’s not the field lines that have changed, but rather the orientation of our coil.

Moving on, our coil continues to rotate until it’s completed one full revolution. Once more, in its vertical orientation, zero EMF is induced and therefore zero current. We’ve therefore moved through one complete cycle of rotation of our coil. And we see that the current induced in the coil fits a sine curve. That is, it moves in a sinusoidal pattern as the coil rotates.

If we were to take a glance at our answer options, we’d see that this graph we’ve generated is a good match for graph b). Indeed, that would be our answer choice if we wanted to find a graph which represents the current generated by a dynamo consisting of a single coil. After all, that’s the dynamo we have here and the Dynamo we’ve created a graph of current versus time for. But of course, we wanna find the graph that corresponds to a dynamo that has a number of coils separated by equal small angles between their planes. So we know then that actually answer b) is not the one we want to pick.

Now before we go further, we can talk about a modification that’s often made to this setup of a dynamo. The modification has to do with the slip rings. Notice that the way that they’re connected up, we get current which is positive at some points and negative at others. That is, we get AC power supply. But what if we want a DC generator? In that case, we would use what is called a commutator in place of our slip rings. A commutator is a split ring, not a slip ring which helps to ensure that no matter how our coil is rotating in the magnetic field, the current it generates always comes out in the same direction.

In other words, if we changed nothing about our dynamo setup except replacing our slip rings with a commutator, then here’s the effect that would have on our current versus time graph. The positive current that we generated before wouldn’t change. But the negative current would be flipped or rectified about the horizontal axis. We can see that using a commutator, our current is always in the same direction. It’s always positive in this case.

Looking once more at our answer options, we see that our graph now corresponds to the graph in option a). But remembering that this graph shows current generated using a single coil, we realize that this also can’t be our answer. So then, we’re down to answer options c) and d). At this point, let’s go back to our dynamo and imagine making some more changes.

Let’s say that instead of just having one coil, we now add a second coil at right angles to the first one. Now then, our dynamo would look something like this, with two coils rather than one. Going over to our current versus time graph, here’s how we can say this would change. As the coils in our dynamo rotate, we would still get the current generated by the single coil that we had before. But now, our second coil would also generate current, which we’ve shown here in the green curve. By adding the second coil, what we’re seeing is that the overall current output by our dynamo is becoming more stable. It’s not going up and down quite so much as it did when there is just one coil. That’s because we can say that the total current output follows this orange curve which traces out the maximum of the pink and the green curves.

This raises a question though: what if we we’re to add even more coils to our dynamo? If we were to do that and the coils were separated by equal angles, notice what starts to happen if we plot the current generated by each one of our coils. Well, one thing that starts to happen is our graph begins getting cluttered. But if we trace out the maximum current generated by any of our rotating coils, we see that’s moving along at a fairly high level.

This shows us that as we add more and more coils separated by equal small angles to our dynamo, then our generator becomes more and more like a DC generator. And more than that, we see among our choices of the graph a), b), c), or d) that best matches this output that graph d) is a close fit. It’s this one that represents the current generated by a dynamo with a number of coils that are separated by equal small angles.

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