Video Transcript
A wire is given the form of a
solenoid 𝑆 one that has 400 turns and a length 𝑙. The current in 𝑆 one is 𝐼, and
the strength of the magnetic field produced by 𝑆 one at its center is 𝐵 one. A second wire is used to form a
solenoid 𝑆 two that has 150 turns. 𝑆 one and 𝑆 two are connected end
to end to form a solenoid 𝑆 three. The spacing of the turns of 𝑆
three is adjusted until the length of 𝑆 three is 𝑙 and the turns of 𝑆 three are
equidistant from each other. The turns of 𝑆 three are equal in
radius to the turns of 𝑆 one. The current in 𝑆 three is 𝐼, and
the strength of the magnetic field produced by 𝑆 three at its center is 𝐵 two. Which of the following describes
the relationship between 𝐵 two and 𝐵 one? (A) 𝐵 two is equal to 𝐵 one. (B) 𝐵 two is equal to eight over
11 times 𝐵 one. (C) 𝐵 two is equal to 11 over
eight times 𝐵 one. (D) 𝐵 two is equal to eight over
three times 𝐵 one. (E) 𝐵 two is equal to five over
eight times 𝐵 one.
Okay, so this question talks about
three different solenoids. We can recall that a solenoid is a
wire formed into a shape like this one, consisting of a series of equally spaced
loops or turns. Let’s suppose that this solenoid
we’ve sketched here represents 𝑆 one. We’re told that the length of 𝑆
one, that’s the length between these two ends of the solenoid, is 𝑙 and that it
consists of 400 turns. So, that’s 400 of these individual
loops of wire.
It’s worth noting that in this
sketch we’ve drawn, we’ve clearly not drawn out all of these 400 turns. Then, in addition to 𝑆 one, we’ve
also got another solenoid 𝑆 two. We’re not told the length of the
solenoid 𝑆 two, but we are told that it consists of 150 turns of wire. The question then goes on to say
that these solenoids, 𝑆 one and 𝑆 two, are connected end to end so that they form
a third solenoid 𝑆 three.
We then know that the total number
of turns of wire in this solenoid 𝑆 three must be equal to the 400 turns from 𝑆
one plus the 150 turns from 𝑆 two. Adding together 400 and 150, we
then find that the solenoid 𝑆 three has 550 turns. We’re then told that the spacing of
the turns of 𝑆 three is adjusted so that all of the turns of wire are equidistant
from each other and the overall length of 𝑆 three is 𝑙. That is, 𝑆 three is now a solenoid
with the same length 𝑙 as the solenoid 𝑆 one. But 𝑆 three has 550 turns of wire
over the same length that 𝑆 one had just 400.
Now, for the rest of this question,
we’re just considering the solenoids 𝑆 one and 𝑆 three. So, let’s clear off the solenoid 𝑆
two to give ourselves some more space. We’re told that both of these two
solenoids, 𝑆 one and 𝑆 three, carry the same current of 𝐼. As a result of the current in the
wire, we get a magnetic field inside of each of the two solenoids. In the case of 𝑆 one, we’re told
that the strength of this magnetic field at the center of the solenoid is 𝐵 one,
while for 𝑆 three, this strength is 𝐵 two.
To match this notation, let’s label
the number of turns of the solenoid 𝑆 one as 𝑁 one, so we have 𝑁 one is equal to
400. And we’ll label the number of turns
of 𝑆 three as 𝑁 two, so we have 𝑁 two is equal to 550. Let’s now recall that there’s an
equation which tells us how to calculate the strength of the magnetic field inside
of a solenoid in terms of the current in the wire, the solenoid’s length, and the
number of turns of wire used to form the solenoid.
Specifically, the field strength 𝐵
is equal to a constant 𝜇 naught, known as the permeability of free space,
multiplied by the number of turns 𝑁 multiplied by the current 𝐼 divided by the
solenoid’s length 𝑙. In this question, we’re asked which
of these five equations describes the relationship between the two magnetic field
strengths 𝐵 two and 𝐵 one. And in order to work this out, we
can apply this equation to each of the two solenoids 𝑆 one and 𝑆 three.
For the solenoid 𝑆 one, we have
that 𝐵 one is equal to 𝜇 naught multiplied by 𝑁 one multiplied by 𝐼 divided by
𝑙. Meanwhile, for 𝑆 three, we have
that 𝐵 two is equal to 𝜇 naught multiplied by 𝑁 two multiplied by 𝐼 divided by
𝑙. Looking at these equations for the
two solenoids 𝑆 one and 𝑆 three, we can notice that in each case the current 𝐼 is
the same as is the solenoid length 𝑙. And obviously the constant 𝜇
naught will have the same value in each case too.
That means that the only quantity
that differs on the right-hand side of each of these two equations is the number of
turns of wire. In the first case we have 𝑁 one
turns of wire, while in the second case we have 𝑁 two. We can then group together the
three terms 𝜇 naught, 𝐼, and 𝑙 that are the same in each of the two
equations. Next, in place of 𝑁 one and 𝑁
two, we can substitute their actual values of 400 and 550, respectively.
We have then that 𝐵 one is equal
to 400 multiplied by 𝜇 naught 𝐼 over 𝑙, while 𝐵 two is equal to 550 multiplied
by the same 𝜇 naught 𝐼 over 𝑙. If we then take this equation for
𝐵 one and divide both sides by 400 so that on the right the 400 in the numerator
and the 400 in the denominator cancel out and then with this equation for 𝐵 two we
divide both sides by 550 so that on the right the 550s cancel out, we find that 𝐵
one over 400 is equal to 𝜇 naught 𝐼 over 𝑙, while 𝐵 two over 550 is also equal
to 𝜇 naught 𝐼 over 𝑙.
Since 𝐵 one divided by 400 and 𝐵
two divided by 550 are both equal to the same quantity 𝜇 naught 𝐼 over 𝑙, then it
must be true that 𝐵 one divided by 400 is equal to 𝐵 two divided by 550. This equation we found here
describes the relationship between the quantities 𝐵 two and 𝐵 one, which is what
the question was asking us for. All we need to do now is to
simplify this equation we found to see which of these five equations that we’re
given our equation matches.
To do this, we’ll multiply both
sides of our equation by 550. On the right-hand side, the 550
cancels from the numerator and the denominator. On the left-hand side, 550 divided
by 400 simplifies to 11 divided by eight. And so we have that 11 over eight
times 𝐵 one is equal to 𝐵 two. Finally, swapping over the left-
and right-hand sides of the equation, we have that 𝐵 two is equal to 11 over eight
times 𝐵 one.
We can notice that this equation
we’ve derived matches the one that’s given to us in answer option (C). We have found that the equation
which describes the relationship between 𝐵 two and 𝐵 one is the equation from
option (C). 𝐵 two is equal to 11 over eight
times 𝐵 one.
