The current through a circular loop of conducting wire is 6.0 amps. And the magnitude of the magnetic field at the center of the loop is 2.0 times 10 to the negative fourth tesla. What is the radius of the loop.
We can call the current through the loop, 6.0 amps, 𝐼. And the magnetic field at the center of the loop, 2.0 times 10 to the negative fourth tesla, we’ll call 𝐵. We want to solve for the loop radius which we’ll call capital 𝑅.
If we draw this current-carrying loop, we’re told the current 𝐼 that runs through the loop as well as the magnitude of the magnetic field at the loop center and we want to solve for 𝑅, the radius of the loop. To do that, we can recall a mathematical relationship that connects 𝐵, 𝐼, and 𝑅.
For a circular wire, like we have in this example, and the magnetic field in the center, 𝐵 is equal to the permeability of free space, 𝜇 naught, times the current running through the loop, 𝐼, all over two times the loop’s radius, 𝑅. 𝜇 naught is a constant whose value we’ll assume is equal to exactly 1.26 times 10 to the negative sixth teslas meters per ampere.
When we rearrange this equation to solve for radius 𝑅, we find it’s equal to 𝜇 naught times 𝐼 over two times 𝐵. Knowing all three of these values, we can plug in and solve for 𝑅. When we do and enter these values on our calculator, we find that, to two significant figures, 𝑅 is 0.019 meters. That’s the radius of this current-carrying loop.