Video: Calculating the Strength of the Magnetic Field at the Center of a Circular Loop

A circular loop of wire carries a constant current of 0.9 A. The radius of the loop is 13 mm. Calculate the strength of the magnetic field at the center of the loop. Give your answer in teslas expressed in scientific notation to 1 decimal place. Use a value of 4𝜋 × 10⁻⁷ T.m/A for 𝜇₀.

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Video Transcript

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.

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