Video Transcript
A circular loop of wire with a
radius of 9.5 centimeters carries a constant current of 𝐼 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 𝐼, rounding your answer
to one decimal place. Use the value of four 𝜋 times 10
to the negative seventh tesla meters per ampere for 𝜇 nought.
So here, we have a circular loop of
wire with a radius we’ve called 𝑟 given as 9.5 centimeters. And we’re told that the loop
carries a constant current of 𝐼 amperes. So 𝐼 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 𝐵, we’re told that it’s equal to 5.2 times 10 to the negative fifth
teslas.
Knowing all this, we want to
calculate the magnitude of the current 𝐼. 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 𝜇 nought, the permeability of free space, multiplied by the
current magnitude in the loop divided by two times the loop’s radius. Now, in our statement, we’re given
𝜇 nought, we’re given 𝑟, and we’re also given 𝐵, wanting to solve for the current
𝐼.
Now, if we multiply both sides of
this expression by two 𝑟 divided by 𝜇 nought, then over on the right-hand side,
the factors of two cancel and the factors of 𝑟 and 𝜇 nought, leaving us just with
the current 𝐼. So two times 𝑟 times 𝐵 over 𝜇
nought equals 𝐼. And if we substitute in the values
we’re given for 𝐵, 𝑟, and 𝜇 nought, then we come up with this expression
here.
We’re just about ready to calculate
𝐼. But before we do, let’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, let’s convert our
radius from centimeters into meters. 9.5 centimeters is 0.095
meters. So now, when we go ahead and
calculate 𝐼, to one decimal place, we find a result of 7.9 amperes.
For our final answer, though, we’ll
just box the number portion of this quantity because recall that our problem
statement tells us we have a constant current of 𝐼 amperes. So to solve for 𝐼, we just want a
number. And that result is 7.9.
