### Video Transcript

A solenoid formed from a length of
wire has 80 turns. The solenoid carries a constant
current of 13 amperes, and the strength of the magnetic field produced is measured
to be 7.3 times 10 to the negative three tesla at its center. Calculate the length of the
solenoid, giving your answer to the nearest centimeter. Use a value of four 𝜋 times 10 to
the negative seven tesla meters per ampere for 𝜇 naught.

This question is asking us to think
about a solenoid, which is a wire formed into a shape like this one, consisting of a
series of equally spaced loops or turns. In this case, we’re told that that
number of turns, so that’s the number of these individual loops of the wire forming
the solenoid, is equal to 80. And let’s label this total number
of turns as capital 𝑁. We’re also told that the solenoid
carries a constant current, which we’ll label as 𝐼. This current 𝐼 is equal to 13
amperes. As a result of this current in the
wire, a magnetic field is produced inside of the solenoid. We’re told that the strength of
this magnetic field, which we’ve labeled as 𝐵, is measured at the center of the
solenoid to be equal to 7.3 times 10 to the negative three tesla.

Given this information about the
solenoid, we’re asked to calculate its length. We’ll label this length as 𝐿. And it’s worth pointing out that
this is the length between the two ends of the solenoid, not the total length of the
wire that’s used to form it. Now it turns out that there’s an
equation which relates the length 𝐿 of a solenoid, the number of turns of wire,
capital 𝑁, used to form it, the current 𝐼 in that wire, and the strength 𝐵 of the
magnetic field produced inside of the solenoid. Specifically, that equation says
that 𝐵 is equal to 𝜇 naught multiplied by 𝑁 multiplied by 𝐼 divided by 𝐿, where
𝜇 naught is a constant known as the permeability of free space.

Since in this case we’re trying to
find the value of the length 𝐿, then that means we need to take this equation and
rearrange it to make 𝐿 the subject. To do this, we’ll begin by
multiplying both sides of the equation by the length 𝐿 so that on the right-hand
side, the 𝐿 in the numerator then cancels with the 𝐿 in the denominator. Then we’ll divide both sides of the
equation by 𝐵, the strength of the magnetic field. On the left-hand side, the 𝐵 in
the numerator and the 𝐵 in the denominator cancel out. We end up with an equation that
says 𝐿 is equal to 𝜇 naught multiplied by 𝑁 multiplied by 𝐼 divided by 𝐵.

On the right-hand side of the
equation, we’ve been given a value for the constant 𝜇 naught. And we also know the values of the
other three quantities 𝑁, 𝐼, and 𝐵 for this particular solenoid. That means we can go straight ahead
and substitute in these values in order to calculate our length 𝐿. When we do this, we find that 𝐿 is
equal to four 𝜋 times 10 to the negative seven tesla meters per ampere, that’s the
constant 𝜇 naught, multiplied by 80, the number of turns of wire 𝑁, multiplied by
the current 𝐼 of 13 amperes divided by 7.3 times 10 to the negative three tesla,
the strength of the magnetic field 𝐵 inside of the solenoid.

Looking now at the units on the
right-hand side of the equation, we can see that in the numerator the amperes and
the per ampere cancel out, and then the teslas cancel from the numerator and the
denominator. This just leaves us with units of
meters. And so with these values, we’ll
calculate a length 𝐿 of the solenoid in units of meters. When we then evaluate this
expression, we get a result of 0.1790 et cetera meters. However, we should notice that
we’re asked to give our answer to the nearest centimeter. That means that, first, we’ll need
to convert our answer from units of meters into units of centimeters. And then we’ll need to round it to
the nearest whole number of centimeters.

Let’s recall that one meter is
equal to 100 centimeters. That means that to convert a value
from units of meters into units of centimeters, we need to take the value in meters
and multiply it by a factor of 100. Multiplying the value in meters for
the length 𝐿 by 100. And evaluating the result, we find
that in units of centimeters, the length 𝐿 is equal to 17.90 et cetera
centimeters. We now just need to round this to
the nearest centimeter. When we do this, the result rounds
up to 18 centimeters. And so our answer is that, to the
nearest centimeter, the length of this solenoid is 18 centimeters.