Video Transcript
A conducting rod moves at a
velocity 𝐯 between the poles of a magnet in a time 𝑡 equals 0.15 seconds,
generating an emf of 775 microvolts across its length. The magnetic field between the
poles has a magnitude of 18 milliteslas. The cross section of the magnet is
square. Find the distance 𝑑 that the rod
moves. Give your answer in centimeters to
one decimal place.
In this question, we have a
conducting rod moving between the poles of a magnet. And we want to calculate the
distance 𝑑 that the rod moves.
Let’s begin by visualizing the
problem in a 2D perspective. We have our conducting rod, and it
moves with a velocity 𝐯 to the right. The magnetic field is directed from
north to south. So from this perspective, we can
represent the magnetic field with 𝑥, which means into the screen. We are also told that the cross
section of the magnet is square with side length 𝑑.
Now, let’s recall that when a
straight conducting rod moves through a uniform magnetic field, an electric
electromotive force, or emf, is induced across the rod with a magnitude of 𝐿𝑣𝐵
sin 𝜃, where 𝐿 is the length of the rod. 𝑣 is the speed of the rod. 𝐵 is the strength of the magnetic
field. And 𝜃 is the angle between the
rod’s velocity and the magnetic field.
In this question, the rod is moving
to the right and the magnetic field points into the screen. So the rod’s velocity is
perpendicular to the magnetic field, and 𝜃 equals 90 degrees. We know that the sin of 90 degrees
simply equals one, which means that an emf of magnitude 𝐿𝑣𝐵 will be induced
across the rod as it moves.
We have not been given the velocity
in the question. But we know that this is equal to
distance over time. We can also see that the length of
the rod 𝐿 is equal to 𝑑. So we now have the equation 𝜀
equals 𝑑 squared 𝐵 over 𝑡. We can now rearrange this equation
to make 𝑑 the subject. We do this by multiplying both
sides of the equation by 𝑡 over 𝐵 and then taking the square root of both
sides.
This will leave us with 𝑑 equals
the square root of 𝜀𝑡 over 𝐵. The emf is given as 775 microvolts,
which is equal to 775 times 10 to the power of negative six volts. We also have the time 𝑡 equals
0.15 seconds. And the magnetic field is equal to
18 milliteslas, which is equal to 18 times 10 to the power of negative three
teslas.
We can now substitute these values
into our equation for the distance 𝑑. Completing the calculation, we find
that this is equal to 0.080 meters, which is equal to 8.0 centimeters. And this is our answer. The distance 𝑑 that the rod moves
is 8.0 centimeters.
