This video is about alternating current or AC generators.
Let's begin with a couple of key points about alternating current generators.
As their name suggests, in these generators, current direction changes.
Direct current generators also exist, but in the IBDP physics course, we only discuss
alternating current generators, because these are commonly used in energy production.
The basis of alternating current generators is a coil that rotates in a magnetic field.
Let's examine a coil that is rotating clockwise in a magnetic field.
To do this, we recall the following equation from subtopic 11.1.
Here, the Greek letter phi represents magnetic flux, b stands for magnetic flux density,
which is numerically equivalent to the magnetic field strength,
a is the area of the coil, and theta is the angle between the direction of the magnetic field,
and the normal to the coil's area. Let's assume that we start observing the coil
when it is in a vertical position, like this. The magnetic field is pointing horizontally
to the right. If we assume that the direction of the normal to the coil's area is also pointing
to the right, then we can say that theta is equal to zero degrees. As we said, the coil is rotating
clockwise in the magnetic field, so after a quarter of a turn, the coil will be in this position.
Since now the normal to the area is pointing downwards, theta is 90 degrees. After another
quarter of a turn, the coil is positioned like this. Theta is 180 degrees. Another quarter
of a turn later, the coil looks like this. Theta is 270 degrees. Finally, after another quarter
of a turn, the coil returns to its original position, and we can say that theta is 360 or
simply zero degrees. Let's draw a graph to show how magnetic flux through the coil varies with time,
as the coil completes one full rotation in the magnetic field. I am not using specific values
for magnetic flux and time on the axes, because for now, we are just interested in the general
shape of the graph. Let's go back to the equation, phi is equal to b times a times cosine theta.
As the coil rotates in the field, b and a remain constant, while theta varies with time. Therefore,
the magnetic flux versus time graph is a cosine graph. When theta is zero, cosine theta is one,
so at t equals zero, magnetic flux is at its maximum positive value. When theta is 90 degrees,
cosine theta is zero, so magnetic flux is zero. When theta is 180 degrees, cosine theta is negative
one, therefore magnetic flux is at its maximum negative value, cosine 270 degrees is again zero,
so here flux is once again zero, and when theta is 360 degrees, or in other words once again zero
degrees, magnetic flux is again at its maximum positive value. Connecting these points, we get
a cosine graph. Next, let's see how the induced emf in the coil varies with time. To do so,
we'll use this equation also from subtopic 11.1. In this equation, epsilon represents the induced
emf, n is the number of turns on the coil, delta phi is change in magnetic flux, and delta t is
change in time. Let's assume that in our example here, the number of coil turns so n is equal to one,
so we can rewrite the equation as negative delta phi over delta t. There are two different ways to
think about the induced emf versus time graph. The first is in terms of the gradient of the magnetic
flux versus time graph. You have learned in math that gradient is change in y over change in x.
On the magnetic flux versus time graph, the y or vertical axis value is phi,
and the x or horizontal axis value is t. So the gradient of this graph is change in flux
over change in time, or in other words, delta phi over delta t. Since induced emf is equal to negative
delta phi over delta t, the negative of the gradient of the magnetic flux versus time graph
at any given point in time gives us the induced emf value at that moment.
Let's draw a tangent to the magnetic flux versus time graph at different points,
and examine the gradient of these tangents. The tangent at the starting point of the graph
is horizontal, so the gradient which I will denote with the letter m is zero. This means
that at the corresponding moment in time, so at t equals zero, the induced emf is zero.
Let's plot this point on the emf versus time graph.
At the first instance, when magnetic flux is zero, so here, the tangent looks like this.
If you move along the graph, you will see that this is the tangent with the steepest possible
negative gradient, so I will write that the gradient is at its negative maximum value.
Let's find the corresponding point on the induced emf versus time graph.
Since the induced emf is the negative of this gradient, it will be at its maximum
positive value at this moment in time, so we can plot the corresponding point here.
Let's draw the next tangent at this point, where the gradient again is zero,
so the corresponding point on the emf versus time graph is here.
At this point, the gradient of the tangent is at its steepest positive value,
so I will write that m is equal to positive max. The corresponding point gives us the maximum
negative emf value. Finally, at this point, the gradient of the tangent is once again zero,
so the induced emf is also zero. So the emf versus time graph looks like this.
Another way to arrive to this graph is through differentiation.
Delta phi over delta t in calculus can be written as d phi over dt. This means that to find the
induced emf, we can just calculate negative d phi dt, where d phi dt is the derivative of the
magnetic flux with respect to time. Since the magnetic flux versus time graph is a cosine graph,
its derivative is a negative sine graph. Adding in the negative sign that is present in the induced
emf formula, the two negatives cancel, and this is why the emf versus time graph here is a positive
sine graph. So to generalize what we have just discussed, we can write that the induced emf
is the negative derivative of the magnetic flux with respect to time. Next, let's see what happens
when we increase the coil's rotational speed. Let's assume that this is how the emf versus
time graph looks like before we increase the rotational speed. Two things happen when the
rotational speed is increased. First, the coil completes. More rotations in a given amount of
time, so frequency increases. Second, the rate of change of flux increases, therefore the peak
emf value increases. To better understand this increase in the rate of change of flux,
consider the expression negative delta phi over delta t. This is the expression for the rate of
change of flux. When rotational speed increases, delta t decreases, and since delta phi remains
constant, delta phi over delta t increases. So the emf versus time graph after increasing the
rotational speed looks like this. The particular graph that I included here shows how the graph
changes when the rotational speed is doubled. As you can see, frequency is doubled,
and the peak emf values are also doubled. This completes our introduction to alternating
current generators. In the next video, we'll examine in more detail the current, the potential
difference, and power dissipation for such generators.