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This video is about alternating current or AC generators.

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Let's begin with a couple of key points about alternating current generators.

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As their name suggests, in these generators, current direction changes.

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Direct current generators also exist, but in the IBDP physics course, we only discuss

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alternating current generators, because these are commonly used in energy production.

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The basis of alternating current generators is a coil that rotates in a magnetic field.

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Let's examine a coil that is rotating clockwise in a magnetic field.

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To do this, we recall the following equation from subtopic 11.1.

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Here, the Greek letter phi represents magnetic flux, b stands for magnetic flux density,

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which is numerically equivalent to the magnetic field strength,

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a is the area of the coil, and theta is the angle between the direction of the magnetic field,

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and the normal to the coil's area. Let's assume that we start observing the coil

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when it is in a vertical position, like this. The magnetic field is pointing horizontally

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to the right. If we assume that the direction of the normal to the coil's area is also pointing

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to the right, then we can say that theta is equal to zero degrees. As we said, the coil is rotating

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clockwise in the magnetic field, so after a quarter of a turn, the coil will be in this position.

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Since now the normal to the area is pointing downwards, theta is 90 degrees. After another

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quarter of a turn, the coil is positioned like this. Theta is 180 degrees. Another quarter

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of a turn later, the coil looks like this. Theta is 270 degrees. Finally, after another quarter

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of a turn, the coil returns to its original position, and we can say that theta is 360 or

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simply zero degrees. Let's draw a graph to show how magnetic flux through the coil varies with time,

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as the coil completes one full rotation in the magnetic field. I am not using specific values

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for magnetic flux and time on the axes, because for now, we are just interested in the general

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shape of the graph. Let's go back to the equation, phi is equal to b times a times cosine theta.

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As the coil rotates in the field, b and a remain constant, while theta varies with time. Therefore,

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the magnetic flux versus time graph is a cosine graph. When theta is zero, cosine theta is one,

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so at t equals zero, magnetic flux is at its maximum positive value. When theta is 90 degrees,

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cosine theta is zero, so magnetic flux is zero. When theta is 180 degrees, cosine theta is negative

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one, therefore magnetic flux is at its maximum negative value, cosine 270 degrees is again zero,

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so here flux is once again zero, and when theta is 360 degrees, or in other words once again zero

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degrees, magnetic flux is again at its maximum positive value. Connecting these points, we get

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a cosine graph. Next, let's see how the induced emf in the coil varies with time. To do so,

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we'll use this equation also from subtopic 11.1. In this equation, epsilon represents the induced

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emf, n is the number of turns on the coil, delta phi is change in magnetic flux, and delta t is

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change in time. Let's assume that in our example here, the number of coil turns so n is equal to one,

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so we can rewrite the equation as negative delta phi over delta t. There are two different ways to

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think about the induced emf versus time graph. The first is in terms of the gradient of the magnetic

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flux versus time graph. You have learned in math that gradient is change in y over change in x.

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On the magnetic flux versus time graph, the y or vertical axis value is phi,

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and the x or horizontal axis value is t. So the gradient of this graph is change in flux

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over change in time, or in other words, delta phi over delta t. Since induced emf is equal to negative

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delta phi over delta t, the negative of the gradient of the magnetic flux versus time graph

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at any given point in time gives us the induced emf value at that moment.

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Let's draw a tangent to the magnetic flux versus time graph at different points,

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and examine the gradient of these tangents. The tangent at the starting point of the graph

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is horizontal, so the gradient which I will denote with the letter m is zero. This means

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that at the corresponding moment in time, so at t equals zero, the induced emf is zero.

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Let's plot this point on the emf versus time graph.

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At the first instance, when magnetic flux is zero, so here, the tangent looks like this.

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If you move along the graph, you will see that this is the tangent with the steepest possible

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negative gradient, so I will write that the gradient is at its negative maximum value.

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Let's find the corresponding point on the induced emf versus time graph.

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Since the induced emf is the negative of this gradient, it will be at its maximum

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positive value at this moment in time, so we can plot the corresponding point here.

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Let's draw the next tangent at this point, where the gradient again is zero,

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so the corresponding point on the emf versus time graph is here.

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At this point, the gradient of the tangent is at its steepest positive value,

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so I will write that m is equal to positive max. The corresponding point gives us the maximum

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negative emf value. Finally, at this point, the gradient of the tangent is once again zero,

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so the induced emf is also zero. So the emf versus time graph looks like this.

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Another way to arrive to this graph is through differentiation.

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Delta phi over delta t in calculus can be written as d phi over dt. This means that to find the

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induced emf, we can just calculate negative d phi dt, where d phi dt is the derivative of the

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magnetic flux with respect to time. Since the magnetic flux versus time graph is a cosine graph,

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its derivative is a negative sine graph. Adding in the negative sign that is present in the induced

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emf formula, the two negatives cancel, and this is why the emf versus time graph here is a positive

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sine graph. So to generalize what we have just discussed, we can write that the induced emf

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is the negative derivative of the magnetic flux with respect to time. Next, let's see what happens

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when we increase the coil's rotational speed. Let's assume that this is how the emf versus

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time graph looks like before we increase the rotational speed. Two things happen when the

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rotational speed is increased. First, the coil completes. More rotations in a given amount of

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time, so frequency increases. Second, the rate of change of flux increases, therefore the peak

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emf value increases. To better understand this increase in the rate of change of flux,

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consider the expression negative delta phi over delta t. This is the expression for the rate of

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change of flux. When rotational speed increases, delta t decreases, and since delta phi remains

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constant, delta phi over delta t increases. So the emf versus time graph after increasing the

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rotational speed looks like this. The particular graph that I included here shows how the graph

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changes when the rotational speed is doubled. As you can see, frequency is doubled,

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and the peak emf values are also doubled. This completes our introduction to alternating

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current generators. In the next video, we'll examine in more detail the current, the potential

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difference, and power dissipation for such generators.