Video Transcript
A 3.3-centimeter conducting rod moves through a 55-millitesla uniform magnetic field,
as shown in the diagram. The rod travels at 8.5 centimeters per second, and the potential difference across
the rod is 110 microvolts. Find the angle 𝜃.
In this question, we’re asked about the angle 𝜃 between the direction that a moving
rod is traveling in and the direction of a uniform magnetic field, 𝐵, that it
passes through. Note that the angle is not measured between the direction of the field and the
direction the rod points in, but rather between the direction of the field and the
direction in which the rod moves.
We’re told that the magnitude of the rod’s velocity is 8.5 centimeters per
second. And the magnetic field has a magnitude of 𝐵 equals 55 milliteslas. We also know the potential difference across the rod, which is 110 microvolts. Actually, this is the induced electromotive force, or emf, 𝜖. Finally, there’s the length of the conductor, which we’ll call 𝑙, of 3.3
centimeters.
Let’s remember the equation that relates the electromotive force, 𝜖, with velocity,
𝑣, the uniform magnetic field, 𝐵, and the length of the conductor, 𝑙. 𝜖 is equal to 𝑙 times 𝐵 times 𝑣 times the sin of 𝜃, where 𝜃 is the angle
between the direction of movement and the direction of the magnetic field, 𝐵. This will give us the answer to our question.
Before substituting in our numbers, we must be consistent with the units of
measurement and convert them to SI base units. Our speed, 8.5 centimeters per second, can be converted to meters per second by
dividing by 100. Our magnetic field can be converted to units of teslas by dividing by 1000, giving us
0.055 teslas. The emf can be converted to units of volts by dividing by 10 to the six, or one
million, giving us 1.1 times 10 to the negative four volts. And lastly, the rod’s length can be converted to units of meters by dividing by
100.
We’re now set to use these values in our equation. Clearing space on screen, 1.1 times 10 to the negative four volts equals 0.033 meters
times 0.055 teslas times 0.085 meters per second times the sin of 𝜃. We can rearrange the equation to find that the sin of 𝜃 equals 0.07130. And from here, we can find the inverse sine of this number. To the nearest degree, it is equal to 45 degrees.
So 𝜃, the angle between the rod’s motion and the external magnetic field, is 45
degrees.
