### Video Transcript

A circular loop of wire with a
radius of 9.5 centimeters carries a constant current of ๐ผ 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 ๐ผ, rounding your answer
to one decimal place. Use the value of four ๐ times 10
to the negative seventh tesla meters per ampere for ๐ nought.

So here, we have a circular loop of
wire with a radius weโve called ๐ given as 9.5 centimeters. And weโre told that the loop
carries a constant current of ๐ผ amperes. So ๐ผ 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 ๐ต, weโre told that itโs equal to 5.2 times 10 to the negative fifth
teslas.

Knowing all this, we want to
calculate the magnitude of the current ๐ผ. 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 nought, the permeability of free space, multiplied by the
current magnitude in the loop divided by two times the loopโs radius. Now, in our statement, weโre given
๐ nought, weโre given ๐, and weโre also given ๐ต, wanting to solve for the current
๐ผ.

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

Weโre just about ready to calculate
๐ผ. But before we do, letโs 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, letโs convert our
radius from centimeters into meters. 9.5 centimeters is 0.095
meters. So now, when we go ahead and
calculate ๐ผ, to one decimal place, we find a result of 7.9 amperes.

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