Video Transcript
A conducting coil has a radius 𝑟 equals 18 centimeters and 25 turns. The coil is moved so that half of its area is within a uniform magnetic field of strength 0.12 teslas, directed out of the plane of the diagram shown along the axis of the coil. An electromotive force of 0.33 volts is induced while the coil moves. What is the average speed at which the coil moves?
In this situation, we have a circular conducting coil that used to be entirely outside of this uniform magnetic field. But then, over some amount of time, the coil moved so that now half of its area is within that field. This means that the magnetic flux through the coil changed over time. A law of physics known as Faraday’s law tells us that when there’s a change in magnetic flux through the area of a conductor that induces an emf, represented by the Greek letter 𝜀.
Because there is a change in magnetic flux through this conducting coil as it moves halfway into the magnetic field, it too will experience an induced emf. Faraday’s law tells us that the emf induced in the coil has to do with how fast the magnetic flux through that coil changes. The greater △𝜙 sub 𝐵, the change in magnetic flux, over △𝑡, the change in time, the more emf, 𝜀, will be induced in the coil. This means that the faster this coil moves into the field, the more emf will be induced in it.
We’re told that electromotive force, 0.33 volts. And we want to use this information to solve for the average speed at which this coil moves into the field. To get started doing this, let’s record some of the information we’re given. The radius of our conducting coil 𝑟 is 18 centimeters. The coil has 25 turns in it. We’ll call that number 𝑁. The strength of the uniform magnetic field is 0.12 teslas. That’s 𝐵. And lastly, the emf induced in the coil 𝜀 is 0.33 volts.
Knowing all this, let’s clear some space and begin to think about just how it is we want to use Faraday’s law to solve for the average speed of our conducting coil. Let’s recall that over some time interval we’ve called △𝑡, our conducting coil moves a linear distance of 𝑟 equal to the radius of the coil. If we recall that, in general, the average speed of an object equals the distance that object travels divided by the time taken to travel that distance, then we can say that the average speed 𝑣 of our coil is equal to the radius of the coil divided by △𝑡. This average speed of our coil corresponds to an induced emf of 0.33 volts in the coil. If the average speed were faster, we would get more of an emf induced, and if it were less, the emf induced would be smaller.
We now have an expression for the average speed 𝑣 of our coil. But we don’t see that expression anywhere in Faraday’s law. However, let’s recall that the magnetic flux 𝜙 sub 𝐵 through some area equals the magnetic field strength 𝐵 through that area multiplied by the area 𝐴. This means we can replace 𝜙 sub 𝐵 with 𝐵 times 𝐴 in this expression. In our particular scenario, we know that the magnetic field strength 𝐵 is constant. It doesn’t change over time. But the area of our conducting coil exposed to that constant magnetic field does change in time. This means we can rewrite this value △𝐵 times 𝐴 as the magnetic field strength 𝐵 times △𝐴, the change in the area exposed to that field. And what is that change in area △𝐴?
We see that, initially, the area of our coil exposed to the field is zero. None of the coil is in the field. However, after the coil has moved, half of that cross-sectional area of the loop is exposed to the uniform magnetic field. Since the area of an entire circle is 𝜋 times the radius of the circle squared, we can replace △𝐴 with one-half the area of our circle, one-half 𝜋𝑟 squared, minus zero, because this is the initial area of our coil exposed to the magnetic field. Subtracting zero from our numerator doesn’t change it. So we can leave off that part.
And now if we write one-half 𝜋𝑟 squared as one-half times 𝜋 times 𝑟 times 𝑟, notice that if we take one of these 𝑟’s and divided by △𝑡, then we indeed have found a place where the average speed 𝑣 of our coil appears in this equation. In this equation as it is, we’re considering that the emf induced could be positive or negative. That is, its direction could be indicated. Let’s say though that we’re only considering the magnitude of this emf, in which case the negative sign on the right side of our equation can be removed.
We can now work with this equation to solve for the average speed 𝑣 of our coil. If we multiply both sides of our equation by two, canceling that factor out on the right, and then divide both sides by 𝑁 times 𝐵 times 𝜋 times 𝑟, canceling out all of those factors on the right, we find, switching the sides of our equation, that 𝑣 equals two times the magnitude of 𝜀, that’s the emf induced in our coil, divided by the number of turns in the coil times the magnetic field strength times 𝜋 times the radius of the coil.
When we consider all the values on the right-hand side of this expression, we actually know all of them. The magnitude of the emf induced is 0.33 volts. The number of turns in the coil 𝑁 is 25. The magnetic field strength 𝐵 is 0.12 teslas. And the radius of the coil 𝑟 is 18 centimeters.
Before we calculate 𝑣, we’ll want to make one change in units, converting the units of centimeters into meters. One centimeter, we recall, is equal to one one hundredth of a meter. And so to convert 18 centimeters into meters, we’ll take the decimal point and we’ll move it two spots to the left. 18 centimeters is 0.18 meters. Entering this expression on our calculator, to two decimal places, we get a result of 0.39 meters per second. This is the average speed of our conducting coil as it moves into the magnetic field.
