### Video Transcript

A conducting coil with four turns has a diameter ๐ equals 25 centimeters. The coil moves 1.5 centimeters at a velocity ๐ฃ equals 7.5 centimeters per second parallel to the axis of a stationary bar magnet, as shown in the diagram. An emf of magnitude 3.6 millivolts is induced in the coil while it moves past the magnet. Find the change in the magnetic field strength between the points where the coil started to move and where it stopped moving. Give your answer in scientific notation to one decimal place.

In our diagram, we see these two components: thereโs the conducting coil that has four turns and thereโs a stationary permanent magnet. The bar magnet, we know, generates a magnetic field. And as the conducting coil moves along with a velocity ๐ฃ, it experiences a change in the magnetic field due to the magnet. That change in magnetic field through the area of the conducting coil creates a change in magnetic flux, and that by Faradayโs law induces an emf in the coil. In this example, we want to find the change in the magnetic field strength between the points where the coil started to move and where it stopped. We can begin by writing down some of the information given to us.

Weโre told that the coil moves at a velocity of 7.5 centimeters per second. Along with this, the distance the coil moves, weโll call it ๐ , is 1.5 centimeters. And this movement through the bar magnetโs magnetic field induces an emf, weโll call ๐, of 3.6 millivolts in the coil. Along with all this, the coil diameter ๐, weโre told, is 25 centimeters.

Clearing some space on screen, we write down the law we mentioned earlier, Faradayโs law. This law connects a change in magnetic flux, ฮ๐ sub ๐ต, through a conductor with an emf, represented by ๐, induced in the conductor. In this equation, capital ๐ is equal to the number of turns in the conductor, and ฮ๐ก is the time interval over which this change in magnetic flux ฮ๐ sub ๐ต occurs. In general, magnetic flux, ๐ sub ๐ต, is equal to the strength of a magnetic field, ๐ต, multiplied by the area exposed to that field, ๐ด. This means that as we go to apply Faradayโs law, we can replace ๐ sub ๐ต here with the quantity ๐ต times ๐ด. Here, ๐ต is the magnetic field created by our bar magnet and ๐ด is the area of one of the loops of our conducting coil.

We can see that as our coil moves, the area ๐ด of those loops doesnโt change. Specifically, the area exposed to the magnetic field doesnโt change. However, because the magnetic field around the bar magnet is not uniform, as the coil moves, the magnetic field strength ๐ต does change. The fact that in this quantity ๐ต changes while ๐ด does not means that we can rewrite the equation like this. Itโs this variable ฮ๐ต representing the change in magnetic field experienced by the coil that we want to solve for. Knowing that, letโs rearrange this equation so that ฮ๐ต is the subject. Multiplying both sides of the equation by ฮ๐ก divided by negative ๐ times ๐ด causes negative ๐, ๐ด, and ฮ๐ก to cancel out on the right.

If we then switch the sides of the remaining expression, we find that ฮ๐ต equals ๐ times ฮ๐ก over negative ๐ times ๐ด. In our problem statement, weโre given values for ๐, ๐, and ๐ด. We didnโt write it down earlier, but ๐, the number of turns in our coil, is four. We donโt, however, have a value for ฮ๐ก, the time over which our conducting coil is moving. But we do know how far the coil moved, 1.5 centimeters, and we know how fast it moved, 7.5 centimeters per second.

If we recall that, in general, the average speed of an object equals the distance traveled by that object divided by the time taken to travel that distance, then multiplying both sides of this equation by ๐ก divided by ๐ฃ โ so that on the left, ๐ฃ cancels and on the right, the time ๐ก cancels โ we find that this equation for average speed also says that ๐ก equals ๐ divided by ๐ฃ. Using the variables in our scenario then, we can write that ฮ๐ก equals ๐ , the distance traveled by the coil, divided by ๐ฃ, its speed. Making this substitution in our equation, it then reads that ฮ๐ต equals ๐ times ๐ divided by negative ๐ times ๐ด times ๐ฃ.

Weโre given values for all of the variables on the right side of this expression, except for the cross-sectional area ๐ด. Note, though, that we do know the diameter of each of the individual loops of our coil. We can remember that the area of a circle, in terms of that circleโs diameter ๐, is equal to ๐ divided by four times ๐ squared. Replacing the area ๐ด in our equation with this expression, we can note that this is mathematically equivalent to multiplying numerator and denominator by four, giving us this expression.

At this point, letโs note that this negative sign in our equation, which comes from the negative sign in Faradayโs law, helps to indicate the direction of the emf induced in a conductor. In this example, though, weโre only told the magnitude of the emf induced. And therefore, when we calculate ฮ๐ต, weโll only be able to calculate the magnitude of its change. Because of this, we can remove the negative sign from this expression. Our answer will be positive. So we donโt need to consider the direction of this change. Weโre now ready to substitute in the known values for ๐, ๐ , ๐, ๐, and ๐ฃ. With these substitutions made, notice that none of our units are in SI base units. That is, weโll want to convert millivolts to volts and centimeters to meters.

Before we do this, notice that thereโs a factor of four in numerator and denominator that can therefore cancel out. When it comes to our unit conversions, we recall that one millivolt equals 10 to the negative three or one one thousandth of a volt. This means that 3.6 millivolts equals 3.6 times 10 to the negative three volts. Likewise, one centimeter equals 10 to the negative two or one one hundredth of a meter. And so, we can convert all of our centimeter values to meters by multiplying by 10 to the negative two.

The units in this expression, and feel free to confirm this, work out to volt seconds per meter squared, which is exactly equivalent to teslas. Calculating this result to one decimal place, we find that ฮ๐ต is 3.7 times 10 to the negative three teslas. This is the magnitude of the change in magnetic field experienced by our moving coil.