### 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 π naught equals four π times
10 to the negative seven tesla meters per ampere.

In this question, weβre 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 π΅, at
the center of a coil of wire with π turns, is equal to π naught ππΌ divided by
two π, where πΌ is the current in the wire and π is the radius of the coil.

First, letβs check that we have all
the values we need to know. Weβre told to use a value of four
π times 10 to the negative seven tesla meters per amp for π naught. There are 35 turns in the wire, so
π equals 35. And the current πΌ 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, letβs rewrite this value as π
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,
letβs 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 weβd 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.