Video Transcript
There are two square
current-carrying coils A and B. Coil A has sides of length 𝑙,
while coil B has sides of length two 𝑙. If both coils have the same number
of turns and have equal magnetic dipole moments, what is the ratio of the current in
coil A to that in coil B? (A) One to two, (B) one to one, (C)
two to one, (D) four to one.
Let’s begin by recalling that the
magnetic dipole moment of a coil or a loop of wire carrying a current in a magnetic
field is defined as the torque acting on the coil divided by the magnetic field
strength. Assuming that both coils are in a
magnetic field of the same strength, we can say that the torque acting on coil A,
which we’ll call 𝜏 sub A, is equal to the torque acting on coil B, 𝜏 sub B.
At this point, it’ll be helpful to
recall the formula for the torque on a rectangular current-carrying coil in a
magnetic field. 𝜏 equals 𝐵𝐼𝐴𝑁 sin 𝜃, where 𝐵
is the strength of the magnetic field. 𝐼 is the current in the coil. 𝐴 is the area of the coil. 𝑁 is the number of loops, or
turns, in the coil. And 𝜃 is the angle between the
magnetic field and the perpendicular to the coil.
Now, it’s safe to assume that some
of these terms have the same value for both coils. We already noted that the magnetic
field that the coils are in is the same. So the value of 𝐵 is the same for
both coils. We were also told that both coils
have the same number of loops, or turns, 𝑁. And it’s safe to assume that the
angle 𝜃 is the same for each coil.
Remember that this question is
asking us to find the ratio of the current in coil A to that in coil B. And we need to keep in mind that
the two coils have different areas, since the sides of coil B are longer than the
sides of coil A. So let’s apply the torque formula
to the coils, ignoring the three terms that we just mentioned are the same for both
coils and only considering the current and area of the coils.
So we expand upon the fact that the
torque on both coils is equal by writing that the current times area for both coils
is equal. And we can rearrange this equation
to solve for 𝐼 sub A over 𝐼 sub B, which corresponds to the ratio we want to
find. To do this, we can simply divide
both sides by 𝐴 sub A times 𝐼 sub B. Then, simplifying the expression,
we have that 𝐼 sub A over 𝐼 sub B equals 𝐴 sub B over 𝐴 sub A. This means that in order to find
the answer to the question, to find the ratio of the current in coil A to that in
coil B, all we need to do is find the ratio of the area of coil B to that of coil
A. These two ratios are
equivalent.
Since we know these coils are
square-shaped, we can find the area of each coil by multiplying its side lengths
together. Thus, coil A has an area of 𝑙
times 𝑙, which equals 𝑙 squared, and coil B has an area of two 𝑙 times two 𝑙,
which equals four 𝑙 squared. Substituting these values for area
into our equality, we have four 𝑙 squared over 𝑙 squared. And we can cancel 𝑙 squared from
the numerator and denominator. So the ratio just becomes four over
one. Adopting ratio notation for our
answer, we see that the ratio of the current in coil A to that in coil B is four to
one.
Looking at our answer options, we
can see that this answer corresponds to option (D). So option (D), four to one, is the
correct answer for the ratio of the current in coil A to that in coil B.
