Video Transcript
A thin, circular coil of wire with
radius 𝑟 and 𝑁 turns carries a constant current. The strength of the magnetic field
at the center of the coil is measured to be 2.3 times 10 to the negative four
teslas. Some time later, two 𝑁 turns of
wire are added to the coil. The current passing through the
coil remains the same. Calculate the strength of the
magnetic field at the center of the coil after the loops of wire are added. Give your answer in teslas
expressed in scientific notation to one decimal place.
This question is asking us to
calculate the magnetic field strength at the center of a circular coil of wire after
more turns have been added to the wire.
Let’s start by thinking about the
coil’s initial setup, before the extra turns are added. We need to recall the formula for
the magnetic field strength at the center of the coil of wire. The magnetic field strength, 𝐵, at
the center of a coil of wire with 𝑁 turns is equal to 𝜇 naught 𝑁𝐼 divided by two
𝑟, where 𝐼 is the current in the wire and 𝑟 is the radius of the coil.
Now, in this question, we’re not
given any numerical values for 𝑁, 𝐼, or 𝑟. All we know is that 𝐵 has a value
of 2.3 times 10 to the negative four teslas. So, how can we use this information
to calculate the strength of the magnetic field at the center of the coil after more
turns are added? Let’s clear some space on screen
and go back to the formula for magnetic field strength and rewrite it so that the
relationship between the magnetic field strength and the number of turns is
clearer.
The magnetic field strength is
equal to 𝜇 naught 𝐼 divided by two 𝑟 all multiplied by 𝑁. In other words, 𝐵 is proportional
to 𝑁, and our constant of proportionality is equal to 𝜇 naught 𝐼 divided by two
𝑟. So we don’t get confused between
our magnetic fields, we’ll call the field strength at the center of the coil after
more turns have been added 𝐵 two. To find 𝐵 two, we use the exact
same formula as before, except we now need to substitute in a greater number of
turns. Since the coil started out with 𝑁
turns of wire and then two 𝑁 more turns were added, the total number of turns is
now equal to three 𝑁.
We’re told the current passing
through the coil doesn’t change, and we can assume the radius of the coil stays the
same, too. So, substituting this into our
formula, we find that 𝐵 two is equal to 𝜇 naught 𝐼 divided by two 𝑟 all
multiplied by three 𝑁. Let’s rewrite this slightly and
move this factor of three to the front.
If we now compare our expressions
for 𝐵 and 𝐵 two, we might notice that they’re very similar. In fact, these factors here are
exactly the same as our expression for the initial magnetic field 𝐵. So, we can replace these terms by
subbing 𝐵 into this equation. We’ve now found that after the
extra turns of wire are added, the magnetic field strength, 𝐵 two, is equal to
three times the initial magnetic field strength, 𝐵.
Since we know that 𝐵 has a value
of 2.3 times 10 to the negative four teslas, all we need to do is substitute this in
and calculate 𝐵 two. This gives us a value of 6.9 times
10 to the negative four teslas. This is already in the form
required by the question, scientific notation to one decimal place. So our work here is done. The strength of the magnetic field
at the center of the coil after the loops of wire are added is equal to 6.9 times 10
to the negative four teslas.
