Video Transcript
A small aircraft flies at 150
meters per second through a region where the strength of Earth’s magnetic field
perpendicular to its wings is 35 microteslas. The wingspan of the aircraft is 12
meters. What potential difference is
induced across the aircraft’s wingtips?
If we picture this plane from an
aerial perspective, we can imagine it flying to the right with a speed we’ll call
𝑣. The wingspan of the aircraft, what
we’ll call 𝑙, is 12 meters. And we know that as this plane
flies along, it moves through Earth’s magnetic field. Specifically, that field is
perpendicular to the wings of the plane. That could mean that the magnetic
field points out of the screen at us or into the screen away from us. We actually don’t need to decide
which of these two it is. In either case, whichever way the
magnetic field points, it’s perpendicular to the plane’s velocity 𝑣. Since the magnetic field, though,
does only point in one of these two directions, we can pick one at random and then
move on to answering our question of what the potential difference induced across
the aircraft’s wingtips is.
The idea here is that the wings of
the aircraft are made of some metal. Therefore, they are conductors with
mobile electric charge. Given a magnetic field we’ll call
𝐵 that points into the screen, we would expect positive electrical charge to
accumulate at this end of the aircraft’s wingspan and negative charge down here. However, it’s not the polarity of
charge across the wingspan that we’re interested in, but rather the total potential
difference across that wingspan. For a straight conductor of length
𝑙 moving at a speed 𝑣 through a magnetic field of strength 𝐵, the potential
difference, also called the emf, induced across that conductor is 𝑙 times 𝑣 times
𝐵 times the sin of this angle called 𝜃. 𝜃, it turns out, is the angle
between the velocity vector and the magnetic field vector involved in the
scenario.
We can see from the fact that this
equation uses the sin of 𝜃 that when 𝜃 is 90 degrees, the emf, or potential
difference induced, will be a maximum value. This is just what happens in our
particular scenario, with our plane moving perpendicularly to the magnetic
field. The potential difference induced
across the wingtips is equal to 𝑙 times 𝑣 times 𝐵 times the sin of 90 degrees or,
since the sin of 90 degrees is one, 𝑙 times 𝑣 times 𝐵. We’re given values for all three of
these variables.
But notice that our magnetic field
has units of microtesla. Before we calculate the potential
difference, we’d like to change this over into units simply of teslas. The conversion between these units
is that one million microteslas is equal to one tesla. To convert microteslas to teslas
then, we would divide by one million. Perhaps the simplest way to solve
for the equivalent value of magnetic field in teslas is to multiply by 10 to the
negative six. That means we have thirty-five one
one millionth of a tesla.
Our units are now all in order, and
we’re ready to calculate the potential difference. Rounding to two significant figures
and note that leading zeros are never significant in a number, the potential
difference we calculate is 0.063 volts. That’s the potential difference
induced across the aircraft’s wingtips. Notice that this potential
difference is much smaller than even the typical potential difference supplied by a
standard cell or battery. In other words, it would be
difficult to make the potential difference generated by this aircraft’s motion
through a magnetic field useful. To make it so, we might need to
significantly increase the aircraft speed or significantly increase its
wingspan.
