Video Transcript
A long wire is carrying a
direct current. As a result, a magnetic field
of 8.0 times 10 to the negative fifth teslas can be measured at a perpendicular
distance of 13 centimeters from the wire. What would the strength of the
magnetic field be at a perpendicular distance of 26 centimeters from the
wire? Give your answer in scientific
notation to one decimal place.
Alright, so let’s say that this
is our long wire that’s carrying a direct current. As a result of this current,
we’re told that if we go out a perpendicular distance from the wire of 13
centimeters, then we can measure a magnetic field strength of 8.0 times 10 to
the negative fifth teslas. Building on this, our question
asks, if we were to travel out from our wire a perpendicular distance of 26
centimeters, then what would be the strength of the magnetic field at that
distance from the wire?
So then we know the strength of
the magnetic field created by this wire a distance of 13 centimeters away from
it. We can call that field strength
𝐵 13. And it’s equal to 8.0 times 10
to the negative fifth teslas. And what we want to do is solve
for the field strength a distance of 26 centimeters from the wire, what we’ll
call 𝐵 26.
Now, because these two magnetic
field strengths, 𝐵 13 and 𝐵 26, are due to the same current in the same wire,
we can be helped by recalling that, in general, the magnetic field created by a
current-carrying straight wire is equal to the constant 𝜇 naught, the
permeability of free space, times the current in the wire divided by two times
𝜋 times the distance from the wire at which we’re measuring the field. For our purposes, the most
interesting aspect of this equation is how the magnetic field strength 𝐵 varies
with the distance from the wire 𝑑.
We can see that 𝐵 varies as
one over 𝑑 or, in other words, is inversely proportional to it. This means, for example, that
if we were to double our perpendicular distance from a given wire, then we would
cut in half the strength of the magnetic field at that point. And in fact, that’s exactly
what we’re doing here in this example. We started out a perpendicular
distance of 13 centimeters from our wire. We can consider that our
original distance. And then we doubled that to 26
centimeters. By this relationship here, we
can see that when we double the distance 𝑑, we cut 𝐵 in half.
And so, given that this is the
magnetic field strength a distance of 13 centimeters from our wire, we can
expect that 𝐵 26 will be one-half of this. And the reason we know that is
that 𝐵 26 is the strength of the magnetic field at twice the distance from the
wire, where 𝐵 13 was measured. In doubling our distance, we’ve
halved our field strength. And so, in scientific notation,
to one decimal place, 𝐵 26 is 4.0 times 10 to the negative fifth teslas. This is the strength of the
magnetic field a distance of 26 centimeters from the wire.
