Question Video: Calculating the Current Carried by a Circular Loop Physics

A circular loop of wire with a radius of 9.5 cm carries a constant current of ๐ผ A. The strength of the magnetic field produced by the current is 5.2 ร— 10โปโต T at the center of the loop. Calculate ๐ผ, rounding your answer to 1 decimal place. Use a value of 4๐œ‹ ร— 10โปโท T.m/A for ๐œ‡โ‚€.

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Video Transcript

A circular loop of wire with a radius of 9.5 centimeters carries a constant current of ๐ผ amperes. The strength of the magnetic field produced by the current is 5.2 times 10 to the negative fifth teslas at the center of the loop. Calculate ๐ผ, rounding your answer to one decimal place. Use the value of four ๐œ‹ times 10 to the negative seventh tesla meters per ampere for ๐œ‡ nought.

So here, we have a circular loop of wire with a radius weโ€™ve called ๐‘Ÿ given as 9.5 centimeters. And weโ€™re told that the loop carries a constant current of ๐ผ amperes. So ๐ผ is some pure number. And this is a current value expressed in amperes. Due to this current, a magnetic field is produced at the center of this circular loop. If we call the magnitude of that field ๐ต, weโ€™re told that itโ€™s equal to 5.2 times 10 to the negative fifth teslas.

Knowing all this, we want to calculate the magnitude of the current ๐ผ. To do this, we can recall that the magnetic field magnitude at the center of a current-carrying circular loop is equal to this constant ๐œ‡ nought, the permeability of free space, multiplied by the current magnitude in the loop divided by two times the loopโ€™s radius. Now, in our statement, weโ€™re given ๐œ‡ nought, weโ€™re given ๐‘Ÿ, and weโ€™re also given ๐ต, wanting to solve for the current ๐ผ.

Now, if we multiply both sides of this expression by two ๐‘Ÿ divided by ๐œ‡ nought, then over on the right-hand side, the factors of two cancel and the factors of ๐‘Ÿ and ๐œ‡ nought, leaving us just with the current ๐ผ. So two times ๐‘Ÿ times ๐ต over ๐œ‡ nought equals ๐ผ. And if we substitute in the values weโ€™re given for ๐ต, ๐‘Ÿ, and ๐œ‡ nought, then we come up with this expression here.

Weโ€™re just about ready to calculate ๐ผ. But before we do, letโ€™s change the units of our radius, which are in centimeters, so that they match the SI base units in the rest of our expression. In other words, letโ€™s convert our radius from centimeters into meters. 9.5 centimeters is 0.095 meters. So now, when we go ahead and calculate ๐ผ, to one decimal place, we find a result of 7.9 amperes.

For our final answer, though, weโ€™ll just box the number portion of this quantity because recall that our problem statement tells us we have a constant current of ๐ผ amperes. So to solve for ๐ผ, we just want a number. And that result is 7.9.

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