Video Transcript
A circular loop of wire of radius
50 millimeters carries a constant current 𝐼 amperes and produces a magnetic field
of strength 𝐵 one teslas at its center. Another circular loop of wire has a
radius of 150 millimeters. Given that this wire also carries a
constant current of 𝐼 amperes, which of the following correctly shows the relation
between 𝐵 two, the strength of the magnetic field produced by the larger loop at
its center, and 𝐵 one? Is it (A) 𝐵 two equals one-third
𝐵 one? (B) 𝐵 two equals three 𝐵 one. (C) 𝐵 two equals one-ninth 𝐵
one. (D) 𝐵 two equals nine 𝐵 one. Or (E) 𝐵 two equals 𝐵 one.
In this question, we have two
different loops of wire, we’ll call them loops one and two, that are carrying the
same current but have different radii. And we want to compare the strength
of the magnetic field produced at the centers of the loops. We were told that a magnetic field
of strength 𝐵 one is produced at the center of the loop with a radius of 50
millimeters. So let’s go ahead and label this
radius value as 𝑟 one. Likewise, we know that a magnetic
field of strength 𝐵 two is produced at the center of the loop with a radius of 150
millimeters. So let’s call this radius 𝑟
two.
To answer this question, we should
recall the formula for calculating the strength of the magnetic field 𝐵 produced at
the center of a single wire loop of radius 𝑟 that’s carrying a current 𝐼, that is,
𝐵 equals 𝜇 naught 𝐼 divided by two 𝑟. Note that the term 𝜇 naught is a
known constant called the permeability of free space. Since we want to determine the
relationship between magnetic field strength and the radius of a loop of wire, a
proportionality relation will be really useful to us here.
Recall that a statement of
proportionality tells how variables change with respect to each other. So, to create a proportionality
from this formula, we simply ignore all the constant, unchanging values by setting
them equal to one. And we replace the equals sign with
this symbol, which tells us that we’re no longer strictly equating the left and
right sides of this expression. Notice too that we treat the
current 𝐼 as a constant here, since we know its value is the same for both wire
loops. Thus, our relationship reads that
𝐵 is proportional to one over 𝑟.
Another way to say this is that the
strength of the produced magnetic field is inversely proportional to the radius of
the loop, since as one quantity increases, the other must decrease. This makes sense because the
magnetic field strength decreases with distance from a current-carrying wire. Just knowing this, we can eliminate
a few answer options.
Loop two has a greater radius than
loop one. So we know that the strength of the
field at the center of loop two must be less than that of loop one. Therefore, we can eliminate answer
options (B) and (D), because they suggest that 𝐵 two is greater than 𝐵 one. We can also eliminate option (E),
since it suggests that the strengths of the magnetic fields are the same. This leaves options (A) and
(C). And here’s where that statement of
proportionality really comes in handy. 𝐵 is inversely proportional to
𝑟. And so an increase in 𝑟
corresponds to a decrease in 𝐵 of the same factor.
Compared to loop one, loop two has
a radius that’s three times greater. Thus, due to this inversely
proportional relationship, the field strength at the center of loop two will be
three times smaller. Answer choice (A) shows this
relationship. So we know this is the correct
answer. 𝐵 two equals one-third times 𝐵
one.
