Video Transcript
Fill in the blank. A rod with a length of 40
centimeters starts to slide on a track that makes an angle of 45 degrees with a
magnetic field of strength 0.3 teslas. If the induced emf at its terminals
is measured to be 2.5 volts, then the velocity of the rod is blank. Give your answer to one decimal
place. (A) 75.4 meters per second, (B)
29.5 meters per second, (C) 20.8 meters per second, or (D) 0.2 meters per
second.
To begin answering this question,
let’s draw a diagram so that we can visualize what’s happening. Now, we weren’t told a specific
direction that the magnetic field points in. We can just choose to draw it
pointing from left to right. Next, we were told that the rod is
moving at a 45-degree angle to the magnetic field, so let’s add the rod to the
sketch, along with an arrow to show its direction of motion.
At this point, we should recall the
formula for the electromotive force, or emf, induced across the rod. For a straight conductor of length
𝑙, moving at a velocity 𝑣 through a magnetic field of strength 𝐵, with the
velocity at an angle of 𝜃 relative to the field, the emf is given by 𝑙 times 𝑣
times 𝐵 times the sin of 𝜃. Here, 𝜃 equals 45 degrees.
In this question, we want to find
the velocity of the rod. So let’s rearrange this equation to
make 𝑣 the subject. To do this, we simply divide both
sides of the equation by 𝑙, 𝐵, and sin 𝜃 to get 𝑣 by itself on one side of the
equals sign. Thus, the expression can be written
as 𝑣 equals the induced emf divided by 𝑙𝐵 sin 𝜃. Now, we already have values for all
of the terms on the right-hand side of the equation, so next, we should make sure
that all of them are expressed in the appropriate SI units.
The emf is given as 2.5 volts, so
that’s good to go, and so is the strength of the magnetic field, written as 0.3
teslas. The only units that need to be
converted here are centimeters for the length of the conducting rod, which should be
in meters instead. We know that one centimeter is one
hundredth of a meter, so let’s rewrite 40 centimeters as 0.40 meters.
With all the values on the
right-hand side of this equation expressed in SI-derived units, we can be sure that
this formula will give a final velocity value in SI-derived units of meters per
second. So let’s go ahead and substitute
all the values into the equation. And grabbing a calculator, we get a
result of 29.4628 and so on meters per second. To one decimal place, our final
answer becomes 29.5 meters per second, which corresponds to answer option (B). We have found that the rod moves at
a velocity of 29.5 meters per second at an angle of 45 degrees to the magnetic
field.
