Video Transcript
A solenoid has a length of 3.2
centimeters and consists of 90 turns of wire. The wire carries a current of 1.2
amperes. Calculate the strength of the
magnetic field at the center of the solenoid. Give your answer in teslas
expressed in scientific notation to one decimal place. Use a value of four 𝜋 times 10 to
the negative seventh tesla meters per ampere from 𝜇 naught.
Okay, so in this exercise, we’re
working with a solenoid, which is a series of turns or loops of wire. Now, we’re told that our solenoid,
has 90 turns, and we can see there definitely aren’t 90 here, but we can just
pretend there are. So this solenoid has 90 turns, and
we’ll say that’s represented by the letter capital 𝑁. Along with this, we know that the
length of our solenoid, we’ll call it capital 𝐿, is 3.2 centimeters and also that a
current, we’ll call it 𝐼, of 1.2 amperes is carried by this wire.
Because of this current, the turns
in the solenoid will create a magnetic field around themselves. We want to calculate the strength
of that cumulative or overall field at the center of the solenoid. So, that would be about at this
point along its length. And if we were looking at the
solenoid end on through its circular cross section, we would find that point right
here. So it’s there that we want to solve
for the strength of the magnetic field, and we’ll call that field strength capital
𝐵.
At this point, we can recall a
mathematical relationship describing this magnetic field. The strength of the field at the
center of the solenoid is equal to 𝜇 naught, a constant called the permeability of
free space, multiplied by the total number of turns the solenoid has times the
current that exists in this wire all divided by the overall length of the solenoid
from end to end. In our problem statement, we’re
told what 𝑁, 𝐼, and 𝐿 are, and we’re also given the value to use for 𝜇
naught. So, we can substitute those given
values into our equation for 𝐵. So then, the value we use for 𝜇
naught is four 𝜋 times 10 to the negative seventh tesla meters per ampere. Capital 𝑁 is 90. The current 𝐼 is 1.2 amperes. And the length of the solenoid 𝐿
is 3.2 centimeters.
Before we calculate this magnetic
field strength, notice that the units of amperes, A, here cancel out in our
numerator and that also, in the numerator, we have these units of meters while in
the denominator we have a distance unit in centimeters. We’re going to give our answer in
teslas. We see that’s the unit right here
represented by capital T. And to do this, we’ll need our distance unit to cancel
out. To help that process along, let’s
convert the length of our solenoid from centimeters into meters. 100 centimeters is equal to one
meter. And therefore, 3.2 centimeters is
0.032 meters. And now, the distance units in our
numerator and denominator, written in meters, will indeed cancel out and we’re left
with units of teslas.
And when we calculate 𝐵, we get a
result in scientific notation to one decimal place of 4.2 times 10 to the negative
third teslas. This is the strength of the
magnetic field at the center of solenoid.
