Video Transcript
A conducting coil has an area of 8.68 times 10 to the negative three square meters. The coil moves perpendicularly to a magnetic field that increases in strength from 12 milliteslas to 16 milliteslas in 0.14 seconds, during which an electromotive force of magnitude 18.6 millivolts is induced in the coil. How many turns does the coil have?
Let’s say that this is our conducting coil with some number of turns. The coil moves in a perpendicular magnetic field — we’ll call it 𝐵 — that gets stronger over time. Because of the change in magnetic flux through the coil, an electromotive force is induced in it. Faraday’s law is an equation that tells us how induced emf relates to change in magnetic flux. Here, the electromotive force is represented using the Greek letter 𝜀. It’s equal to negative the number of turns in a coil multiplied by ΔΦ sub 𝐵, the change in magnetic flux through the coil, divided by Δ𝑡, the time taken for that magnetic flux to change.
In our scenario, it’s not the electromotive force that we want to solve for, but rather the number of turns in our coil 𝑁. We can start to do this by multiplying both sides of our equation by Δ𝑡 over ΔΦ sub 𝐵. This means that on the right-hand side, ΔΦ sub 𝐵 cancels from numerator and denominator, as does Δ𝑡. If we then multiply both sides of this equation by negative one, that will give us a negative sign on the left-hand side and an overall positive on the right. If, finally, we swap the sides of this equation, we have an equation where the number of turns 𝑁 is the subject.
At this point, let’s recall that magnetic flux Φ sub 𝐵 is equal to magnetic field strength 𝐵 multiplied by the area 𝐴 exposed to that field. Therefore, we can replace Φ sub 𝐵 in our equation with 𝐵 times 𝐴. Here, 𝐵 represents our magnetic field strength and 𝐴 represents the cross-sectional area of our coil exposed to that field. Considering our coil, we know that the area exposed to the field 𝐴 doesn’t change over time. The magnetic field though does, increasing in strength from 12 milliteslas to 16 milliteslas. We can write then that 𝑁 is equal to negative 𝜀 times Δ𝑡 divided by Δ𝐵 multiplied by 𝐴.
As we think about what Δ𝐵 is, that is, how the magnetic field strength changes, we know that its final value is 16 milliteslas and its initial value is 12 milliteslas. The overall change in 𝐵 then, Δ𝐵, is positive four milliteslas. All of this takes place in an amount of time we’ve called Δ𝑡 of 0.14 seconds. Along with all this, we know the area of our coil exposed to the magnetic field.
With all of these values substituted into our equation, the only substitution left is for the electromotive force 𝜀. Notice we’re given the magnitude of the emf, 18.6 millivolts. Since we know that the number 𝑁 can’t be a negative number, we can say that the true value of 𝜀 is negative 18.6 millivolts. This is consistent with what we’re told in the problem statement, since there it only reports the magnitude of emf, meaning it could be negative. And note that now that we’re multiplying a negative by a negative, the overall result is a positive.
So then, we’re just about ready to calculate capital 𝑁. Before we do though, we’ll want to change our units of millivolts into volts and our units of milliteslas into teslas. Remembering that the prefix milli- indicates 10 to the negative three or one one thousandth of a quantity, we can write then that 18.6 millivolts is equal to 18.6 times 10 to the negative three volts. Likewise, four milliteslas is equal to four times 10 to the negative three teslas.
Considering now the units in this expression, both in numerator and denominator, we can recall that one tesla is defined as a volt times a second divided by a meter squared. If we make this substitution of units, we find that the units of volts cancel from numerator and denominator, as do the units of seconds. And the units of meters squared cancel out as well. That is, all of the units in this expression cancel out. We’re solving then for a unitless or a pure number. Entering this expression on our calculator, we get a result of exactly 75. This is how many turns there are in our conducting coil.
