A long, straight cable in an
industrial power plant carries a direct current of 100 amperes. Calculate the strength of the
resulting magnetic field at a perpendicular distance of 0.06 meters from this
cable. Use four 𝜋 times 10 to the
negative seventh tesla meters per ampere for the value of 𝜇 naught. Give your answer in scientific
notation to two decimal places.
Alright, so in this example, we
have a long, straight wire. We can imagine it going on even
beyond the length that we’ve drawn here. And we’re told that this wire
carries a current 𝐼 of 100 amperes. That’s a lot of current, but then
again this is an industrial power plant. Our problem statement tells us that
if we go a perpendicular distance of 0.06 meters from the wire, and we’ll call that
distance 𝑑, then we want to solve for the strength of the magnetic field at that
distance. We can refer to that magnetic field
as capital 𝐵.
And we can recall a mathematical
relationship for the strength of the magnetic field a perpendicular distance 𝑑 from
a current-carrying wire. It’s equal to a constant 𝜇 naught,
the permeability of free space, multiplied by the current in the wire divided by two
times 𝜋 times the distance from the wire 𝑑. Since our problem statement tells
us 𝐼 and 𝑑 as well as the value to use for 𝜇 naught, we can substitute those
given values into this equation.
Here, we’ve used four 𝜋 times 10
to the negative seventh tesla meters per ampere for 𝜇 naught, 100 amperes for the
current 𝐼, and 0.06 meters for the distance 𝑑. All the units in this expression
are already in the form we’d like them. And we can see that when we
calculate this fraction, the units of meters in numerator and denominator will
cancel out, as will the units of amperes. We’ll be left with an answer in
units of teslas, which is good because we’re calculating a magnetic field
strength. When we compute 𝐵 and we give our
answer in scientific notation, keeping two decimal places, we find that it’s equal
to 3.33 times 10 to the negative fourth teslas. That’s the strength of the magnetic
field this perpendicular distance away from the wire.