Video Transcript
A conducting rod that is within a uniform magnetic field moves at a constant speed along a circular path, where the direction of the circular motion is perpendicular to the length of the rod throughout the motion. When the rod is at the positions A and C shown in the diagram, the direction of the circular motion is along the line of the direction of the magnetic field. When the rod is at the positions B and D shown in the diagram, the direction of the circular motion is perpendicular to the line of the direction of the magnetic field. The graph shows lines of four different colors. Each line could represent the change in the potential difference across the length of the rod as it moves from A to B to C to D and back to A. Which color correctly represents how the potential difference varies with time? (A) Blue, (B) orange, (C) green, (D) red, or (E) none of these lines.
These answer options refer to this graph of potential difference plotted against time. Each one of these different colors could represent the potential difference across the end of this rod as it moves through the magnetic field. The rod, we’re told, starts out at position A, here, and then moves to position B, then C, and then D and then back to A, completing one circuit.
Because our rod is made of conducting material as it moves through this uniform magnetic field, an emf or a potential difference is actually induced across its ends. That emf, we can represent it using the Greek letter 𝜀, is equal to the magnitude of the velocity of the rod multiplied by the rod’s length times the strength of the magnetic field 𝐵 the rod is moving through multiplied by the sin of this angle 𝜃. 𝜃 is the angle between the direction of the rod’s motion and the external magnetic field.
Studying this factor of sin 𝜃 as our rod moves through this circular path can help us understand what potential difference might be generated at these four points. For example, let’s consider the rod when it’s at position A, moving parallel with the external magnetic field. At this point, the velocity of the rod and the magnetic field both point in the same direction. Since these two vectors are parallel, 𝜃, the angle between them, is zero degrees.
Knowing this, if we insert zero degrees into the argument of a sine function, the sin of zero degrees is zero. Therefore, when the rod is at position A, no emf, no potential difference, is induced across its ends. The rod’s starting position at position A corresponds to a time on our time axis of zero. Since the potential difference induced across the rod at this moment is zero, we know that the green line can’t represent the potential difference induced across the rod. The other three colors blue, red, and orange all have values of zero at a time of zero. We can eliminate option (C) from our answer choices then.
And now let’s consider the rod when it’s at position B, moving perpendicularly to the external magnetic field. Now the angle 𝜃 between the rod’s velocity and the external magnetic field is 90 degrees. The sin of 90 degrees is equal to one. This is the maximum value that the sine function attains. Therefore, for our rod moving at a constant speed in a circle with a constant length in a constant magnetic field, the emf or potential difference induced across the rod when it’s at position B is a maximum value. This tells us that the orange line on our graph, because its value never changes from zero, can’t be the correct answer. The correct line must start at zero but then reach some higher value later on.
We’re left then with the red and the blue curves as well as the option that none of these lines applies. Returning to our diagram, let’s continue on, now with the rod at position C so that it’s moving antiparallel to the external field. This means that the angle between the rod’s velocity and that external field 𝜃 is 180 degrees. The sin of 180 degrees, like the sin of zero degrees, is zero. So, when the rod is at position C, once again the emf induced across it is zero. Note that this doesn’t help us differentiate between our red and blue curves because both of these lines return to zero after reaching a maximum value.
Let’s continue on to position D of our rod, where it’s moving, now from our perspective, into the screen. Because the angle between our rod’s velocity and the external field was 90 degrees when the rod was at position B, now when the rod is at position D, that angle will be 270 degrees. This is 180 degrees different from the 𝜃 value at position B. The sin of 270 degrees is equal to negative one. And this tells us that at position D the emf induced in the rod will have its maximum negative value. Once again, both the red and the blue curves show us this trend. And then when a rod completes the circuit and returns to position A, it once again experiences an induced potential difference of zero volts, and both our blue and red lines indicate this.
To see which one of these lines, if either one, is correct, let’s consider the form of our equation for emf. If we were to plot this equation for all values of 𝜃, say between zero and two 𝜋, then our function would have the shape of a sine function. That is, it would vary according to the sin of 𝜃. All by itself, with 𝜃 ranging from zero to two 𝜋, the sin of 𝜃 looks like this. Because our conducting rod is moving at a constant speed, that means that this angle 𝜃 varies linearly with the time 𝑡. Therefore, in our plot of potential difference induced across the rod, we expect a similar shape, that is, a sinusoidal shape.
We can clearly see that the blue line does not have a sinusoidal shape, but the red line does. Because the shape of the red line matches up with the form of our equation for the induced emf across our rod and because the red line meets our expectations as far as potential difference induced at certain key points in the rod’s rotation, we choose option (D) the red line as our answer. This is the line that indicates the change in potential difference across the rod as it moves from point A to B to C to D and back to A.
