Question Video: Determining the Magnetic Field Strength in a Circular Coil of Wire with Multiple Turns Physics

Video Transcript

A thin, circular coil of wire with a radius 4.2 centimeters carries a constant current of 3.9 amperes. The coil has 35 turns of wire. What is the strength of the magnetic field at the center of the coil? Give your answer in teslas expressed in scientific notation to one decimal place. Use 𝜇 naught equals four 𝜋 times 10 to the negative seven tesla meters per ampere.

In this question, we’re asked to calculate the strength of the magnetic field at the center of a coil of wire, given the number of turns in the wire, the current in the wire, and the radius of the coil.

To answer this question, we need to recall the formula for the magnetic field in the center of a coil of wire. The magnetic field strength 𝐵, at the center of a coil of wire with 𝑁 turns, is equal to 𝜇 naught 𝑁𝐼 divided by two 𝑟, where 𝐼 is the current in the wire and 𝑟 is the radius of the coil.

First, let’s check that we have all the values we need to know. We’re told to use a value of four 𝜋 times 10 to the negative seven tesla meters per amp for 𝜇 naught. There are 35 turns in the wire, so 𝑁 equals 35. And the current 𝐼 in the wire equals 3.9 amps. We also know that the radius of the coil is 4.2 centimeters. But before we can use this value, we need to convert it into the SI units of meters.

We can recall that the unit prefix centi- is equivalent to a factor of 10 to the negative two. So, let’s rewrite this value as 𝑟 equals 4.2 times 10 to the negative two meters, which is just 0.042 meters. Substituting these values into the formula, we find that the magnetic field strength at the center of the coil is equal to four 𝜋 times 10 to the negative seven tesla meters per amp times 35 times 3.9 amps divided by two times 0.042 meters.

Before we calculate this value, let’s just check that we have the correct units. In the numerator of this equation, we have units of tesla meters per amp multiplied by units of amps. Here, the amp terms cancel, leaving just units of tesla meters. In the denominator of our expression, we have units of meters. So, overall, we have units of tesla meters divided by meters. Here, the meter terms cancel out, leaving us units of teslas, exactly the unit we’d expect for a magnetic field strength. Now, all we need to do is put this expression into a calculator. Doing this gives us a value of 0.002042 and so on teslas.

For the final answer, this question asks for scientific notation to one decimal place. To write this value in scientific notation, we simply multiply by factors of 10 to the negative one until we have one nonzero digit in front of the decimal point. Doing this gives us a value of 2.042 and so on times 10 to the negative three teslas. Finally, we just need to round this to one decimal place, which leaves us with 2.0 times 10 to the negative three teslas.

So, the magnetic field strength at the center of the coil is equal to 2.0 times 10 to the negative three teslas. This is our final answer to this question.