Video Transcript
The motion of an
alternating-current generator at successive instants 𝑡 one, 𝑡 two, and 𝑡 three is
shown in three images. The output of the current is
rectified using a commutator. Which color line on the graph
correctly shows the output of the generator between 𝑡 one and 𝑡 three? Green arrows represent induced
current.
In order to answer this question,
let’s recall how induced current works in a coil like this. The current induced in the coil by
a magnetic field is maximum when the plane of the coil is parallel to the magnetic
field lines. From looking at this diagram, we
can see that the figure at time 𝑡 two corresponds to this position, because the
magnetic field lines go from north to south. This means that the maximum
possible induced current occurs at time 𝑡 two, and since 𝑡 two occurs between 𝑡
one and 𝑡 three, the order in which these three times occur on the horizontal axis
of our graph will look like this.
Since the induced current must also
be at its maximum at time 𝑡 two, that specific instant must correspond to the very
center of the graph. The time 𝑡 two cannot appear at
either of the ends of the time axis because it is not the smallest time — 𝑡 one is
smaller — and it’s also not the largest; 𝑡 three is larger. If we look now at the coil
positions at the two instants 𝑡 one and 𝑡 three, we see that the coil is angled
off of a horizontal line by the same amount in both cases. That is, the coil is the same
angular distance away from being horizontal. Therefore, the same magnitude of
current will be induced in the coil at these two instants. We should expect then that the
graph will show the same current value at these two times.
With this in mind, we can identify
likely positions of 𝑡 one and 𝑡 three on the graph. The current at these times must
have the same magnitude and must also be indicated by just one of the curves on the
graph. Since 𝑡 one is less than 𝑡 two
and 𝑡 three is greater, 𝑡 one must be somewhere to the left of 𝑡 two, while 𝑡
three is somewhere to its right. We can now see that the only way
for current at 𝑡 two to be a maximum, while the induced current at 𝑡 one and 𝑡
three is equal and also less than that maximum value, is for these times to indicate
the curve colored in blue.
The correct option then must be
this line. The color of the line on the graph
that correctly shows the output of the generator between times 𝑡 one and 𝑡 three
is blue.
