Video Transcript
The diagram shows a rectangular
loop of current-carrying wire between the poles of a magnet. The sections of the loop 𝑎𝑏 and
𝑑𝑐 are perpendicular to the magnetic field. The sections of the loop 𝑎𝑐 and
𝑏𝑑 are aligned at an angle 𝜃 equals 33 degrees from the direction of the magnetic
field. The current in the loop is 1.75
amps, and the magnetic field strength is 0.15 teslas. Length 𝑎𝑐 equals 0.065 meters and
length 𝑎𝑏 equals 0.045 meters. Find the torque acting on the loop
to the nearest micronewton-meter.
In our diagram, we see our
current-carrying rectangular loop. It sits in between the poles of a
permanent magnet. Therefore, the loop is exposed to a
constant magnetic field pointing from the north to the south pole of the magnet. We’ll call this field 𝐵. And we were told that the magnetic
field strength equals 0.15 teslas. We’re told the current in the coil
equals 1.75 amps. We’ll call that value 𝐼.
Because this loop carries current
and exists in a magnetic field, it experiences a torque. We’ll call that torque 𝜏. We are also given the dimensions
𝑎𝑐 and 𝑎𝑏 for our rectangular coil and also the angle 𝜃. So let’s make a note of these
values. We can now clear some space to work
on our solution.
Let’s begin by recalling the
formula for the torque on a rectangular loop of current-carrying wire in a magnetic
field, like we have here. Torque, 𝜏, equals 𝐵𝐼𝐴𝑁 sin 𝜃,
where 𝐵 is the strength of the magnetic field. 𝐼 is the current in the loop. 𝐴 is the loop’s area. 𝑁 is the number of loops. And 𝜃 is the angle between the
area vector of a rectangular loop and the magnetic field. Since there’s only one loop in the
wire here, 𝑁 equals one. And we can simplify the formula to
just 𝐵𝐼𝐴 sin 𝜃.
We don’t yet know the area of the
loop, so we’ll have to calculate it. We were given the side lengths of
the loop. And since the loop is rectangular,
we can find its area simply using the formula length times width. The longer side of the wire is
0.065 meters and the shorter side is 0.045 meters. So their product gives an area of
0.002925 square meters.
Something we have to look out for
is this angle given as 𝜃 in the formula for torque. In the formula, 𝜃 represents the
angle between the area vector of the rectangular loop and the magnetic field. But that’s not the angle that’s
labeled as 𝜃 in the diagram. In the diagram, 𝜃 shows the angle
between the plane of the loop and the magnetic field. So to avoid confusion, we can
rename the angle in the formula using the symbol 𝜙.
Now to find this angle 𝜙, let’s
view the rectangular loop of wire from the side, as it’s oriented with the external
magnetic field. The area vector of the rectangular
loop is a vector that is perpendicular to the area of that rectangle. Based on the direction of current
in the loop, the area vector of this rectangular loop would point in this
direction. And so the angle 𝜙 would be this
angle labeled here.
Now to calculate a value for the
angle 𝜙, all we need to do is subtract 𝜃 from 90 degrees. Thus, 𝜙 equals 90 degrees minus 33
degrees, which is 57 degrees.
We now have values for all the
terms in the formula for torque. And since they’re all expressed in
their appropriate SI or SI-derived units, we know that we’ll end up with a torque
value in proper units of newton-meters. Substituting the values into the
formula and grabbing a calculator, we get a result of about 6.44 times 10 to the
power negative four newton-meters.
We’ve been told to give our final
answer to the nearest micronewton-meter. So we should recall that the prefix
micro- means 10 to the negative sixth. And so our answer becomes 644
micronewton-meters. Thus, our final answer is that the
torque acting on the loop is 644 micronewton-meters.
