Video Transcript
A conducting coil with a radius of 4.5 centimeters has 120 turns. The coil moves perpendicularly to a magnetic field that decreases in strength from 15 milliteslas to 11 milliteslas. A 12.5-millivolt emf is induced while the coil moves. Find the time for which the coil moves. Give your answer to two decimal places.
Let’s say that this ring is our conducting coil. And we’re told that it moves perpendicularly, say like this, through a magnetic field. While this happens, the magnetic field decreases in strength from 15 to 11 milliteslas. That means there’s a change in magnetic flux through our conducting coil. That change in flux induces an emf, an electromotive force, in the coil. A change in magnetic flux through a conductor is related to the emf induced in that conductor by a law called Faraday’s law. This law tells us that the emf induced, represented by the Greek letter 𝜀, is equal to negative the number of turns in the conductor 𝑁 multiplied by the change in magnetic flux through the conductor ΔΦ sub 𝐵 all divided by the time over which this change occurs Δ𝑡.
In our situation, it’s the time for which the coil moves that we want to solve for, and that’s represented in our equation by Δ𝑡. To make Δ𝑡 the subject of our equation, let’s multiply both sides by Δ𝑡 divided by 𝜀. That way, the induced emf 𝜀 cancels on the left and the time Δ𝑡 cancels on the right. Here then is a rearranged version of Faraday’s law.
As we think about solving for the time during which our coil moved, let’s remember that magnetic flux Φ sub 𝐵 is defined as magnetic field strength 𝐵 multiplied by the area of a conducting loop 𝐴. If we make that substitution into this equation, we can note that in our scenario, it’s the magnetic field strength 𝐵 that changes, while the area of the loop 𝐴 stays constant. Therefore, we can write our equation for Δ𝑡 as negative 𝑁 times Δ𝐵, the change in the magnetic field, multiplied by 𝐴 all divided by 𝜀. Values we can use for these variables are given to us in our problem statement. The number of turns 𝑁 in our coil is 120. The final value of the magnetic field strength is 11 milliteslas, while the initial value is 15 milliteslas. That indicates a change of negative four milliteslas.
And then though we’re not directly told the area 𝐴 of our coil, we do know that it’s a circular loop with a radius of 4.5 centimeters. We recall that the area of a circle is equal to 𝜋 times the radius of that circle squared. And that means that 𝐴 is 𝜋 times 4.5 centimeters squared. All of this is divided by an emf of 12.5 millivolts.
Before we can calculate Δ𝑡, we’ll want to change some units in this expression. We’ll change milliteslas to teslas, centimeters to meters, and millivolts to volts. Let’s recall that the prefix milli- corresponds to 10 to the negative three, or one one thousandth of some unit. That means, for example, that negative four milliteslas equals negative four times 10 to the negative three teslas. And likewise, in our denominator, 12.5 millivolts equals 12.5 times 10 to the negative three volts. Lastly, we recall that one centimeter equals one one hundredth of a meter. This tells us that 4.5 centimeters is equal to 0.045 meters.
Before we calculate Δ𝑡, notice that the negative sign here and the negative sign here in our numerator will give us a positive overall result. Notice also that the 10 to the negative three in our numerator and the 10 to the negative three in our denominator will effectively cancel one another out, giving us numerically a simplified result to calculate. That cancelation though only affects the numerical value of our result and not the units.
In any case, when we enter this expression on our calculator and round it to two decimal places, we get a result of 0.24 seconds. This is how long the coil would need to be moving when the magnetic field strength changes in the given way for an emf of 12.5 millivolts to be induced in the coil.
