### Video Transcript

A thin, circular coil of wire with
a radius of 22 millimeters that has π turns carries a constant current of 0.45
amps. The strength of the magnetic field
produced by the current is 2.3 times 10 to the power of negative four tesla at the
center of the coil. Calculate π to the nearest whole
number of turns. Use a value of four π times 10 to
the power of negative seven tesla meters per amp for π naught.

In this question, we need to work
out the number of turns of a circular coil of wire, given the radius of the coil,
the current in the wire, and the strength of the magnetic field at its center. To answer this question, we need to
recall the formula for the magnetic field at 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. π naught is a constant known as
the vacuum permeability.

In this question, weβve been given
the magnetic field strength π΅, the current πΌ, the radius π of the coil, and a
value for the vacuum permeability. We need to use this information to
find the number of turns in the coil. To do this, weβll first need to
rearrange this equation to make π the subject. To do this, we start by multiplying
both sides of the equation by two π. This leaves us with two ππ΅ on the
left-hand side of the equation. On the right-hand side, the two π
terms in the numerator and the denominator cancel, leaving us with π naught
ππΌ.

To get π on its own, we now just
divide both sides of the equation by π naught times πΌ. This leaves us with the formula π
equals two ππ΅ divided by π naught πΌ. Now all we need to do is substitute
in the values weβve been given for these quantities and then calculate the
answer.

Before we do this though, we need
all the quantities to be in their SI base units. The SI base unit for distance is
meters, not millimeters. So, we need to rewrite this as π
is equal to 22 times 10 to the power of negative three meters. If we sub all of these quantities
into our equation, we find that π is equal to two times 22 times 10 to the power of
negative three meters multiplied by 2.3 times 10 to the power of negative four tesla
divided by four π times 10 to the power of negative seven tesla meters per amp
times 0.45 amps.

Before we calculate this value,
letβs check what units our answer is going to be in. We can see that in the numerator,
we have units of teslas multiplied by meters. In the denominator, we have units
of tesla meters per amp multiplied by amps. Here, the ampere terms cancel out,
again giving us units of tesla meters. So in our expression for π, both
the numerator and denominator have the same units: tesla meters. Overall, these units cancel,
leaving us with no units at all. This is exactly what weβd expect,
seeing as π is just a number.

Now, if we plug all these values
into a calculator, we find that π is equal to 17.896. Since weβre asked to give π to the
nearest whole number, our final answer to this question is 18. There are 18 turns of wire in this
coil.