State one factor that decreases the emf induced in a wire that moves perpendicularly to a uniform magnetic field.
Here we have our wire which we’re told is in motion. And we’ll give it a speed we’ll call 𝑣. And we’ll also say that the wire has a length 𝑙. Our wire we know is moving perpendicularly to a uniform magnetic field. We can call this field strength 𝐵. And here we’ve drawn it coming out of the page. We could have drawn it into the page just as well.
This wire we imagine is running along a fixed track of conducting material. It’s around this loop formed by the wire as well as part of the track that emf is induced and induced current flows. To figure out just how much emf is induced, we can recall Faraday’s law, which tells us that the magnitude of that induced emf is equal to the change in magnetic flux Φ sub 𝐵 divided by the change in time Δ𝑡. And we recall further that Φ sub 𝐵 the magnetic flux is equal to the magnetic field 𝐵 multiplied by the area over which that field acts 𝐴.
Combining these two equations, we can write that the magnitude of emf induced in our wire is equal to Δ 𝐵 times 𝐴 divided by Δ𝑡. Since the magnetic field 𝐵 in our example is uniform and constant, we can rewrite the right-hand side to read 𝐵 times Δ𝐴 divided by Δ𝑡.
Let’s now consider this term Δ𝐴 and locate it on our diagram. We know we can highlight the area of our loop at the instant shown. But when we consider Δ𝐴, that’s the change in area of this loop due to the motion of the wire. We can recall that when moving at a constant speed 𝑣, that speed is equal to the distance travelled divided by the time it takes to travel that distance.
This means that if we call the horizontal section of this rectangular Δ𝐴 area 𝑑, then 𝑑 is equal to 𝑣 the speed of the wire times Δ𝑡. Recalling further that the length of the wire is 𝑙, this means we can write Δ𝐴 the change in area of our loop as 𝑙 times 𝑣 times Δ𝑡.
Taking this result and plugging it in for Δ𝐴 in our equation for emf magnitude, we see that the factors of Δ𝑡 in numerator and denominator cancel out. So the magnitude of the emf induced is equal simply to 𝐵, the magnetic field strength, times 𝑙, the length of the wire, times 𝑣, its speed.
In answer to our question then, we found not one, but three different factors, which we could reduce in order to decrease the emf induced in the wire. We could decrease the magnetic field strength or decrease the speed of the wire or decrease the length of the wire. Each of these factors would decrease the emf induced.