A circular loop of wire carries a
constant current of 0.9 amperes. The radius of the loop is 13
millimeters. Calculate the strength of the
magnetic field at the center of the loop. Give your answer in teslas
expressed in scientific notation to one decimal place. Use a value of four 𝜋 times 10 to
the negative seventh tesla meters per ampere for 𝜇 nought.
In this exercise, we have a
circular loop of wire. And we’re told that it carries a
constant current, we’ll call it 𝐼, of 0.9 amperes. Along with this, the radius of this
circular loop, we can call it 𝑟, is given as 13 millimeters. What we want to do is calculate the
strength or the magnitude of the magnetic field at the center of the loop. On our sketch, that’s right here at
this point. We’ll call that magnetic field
magnitude at that point capital 𝐵. And to solve for it, we can recall
this relationship that that magnetic field strength at the center of a
current-carrying circular loop equals a constant called the permeability of free
space, that’s 𝜇 nought, times the current in the loop, 𝐼, all divided by two times
the loop’s radius.
Since we’re told the specific value
to use for this constant 𝜇 nought, and we also know the current 𝐼 as well as the
radius 𝑟, we’re ready to substitute in to solve for 𝐵. Before we calculate this fraction,
though, we’ll want to make one change to it. So long as the radius of our
circular loop is in units of millimeters, it doesn’t match the other SI base units
in this expression, for example, the unit of meters in 𝜇 nought. So let’s convert the radius of our
circle from millimeters into meters.
To do that, we can recall that 1000
millimeters equals one meter, which means that 13 millimeters equals 0.013
meters. See that we’ve moved our decimal
point three spots to the left. Now, we are ready to calculate the
field strength 𝐵. And when we do, to one decimal
place, we get a result of 4.3 times 10 to the negative fifth teslas. This is the magnitude or the
strength of the magnetic field at the very center of our circular loop.