Video Transcript
A wire that carries a constant
current of 0.15 amperes is formed into a solenoid with 11 turns per centimeter. Calculate the strength of the
magnetic field at the center of the solenoid. Give your answer in teslas
expressed in scientific notation to one decimal place. Use a value of four 𝜋 times 10 to
the negative seven tesla meters per ampere for 𝜇 naught.
This question is asking us about a
solenoid, which is a wire that’s shaped into a series of equally spaced loops or
turns as shown here. We’re told that the wire carries a
constant current of 0.15 amperes, which we’ve labeled as 𝐼. As a result of this current,
there’s a magnetic field inside of the solenoid, and the strength of this field,
which we’ll label as 𝐵, is exactly what we’re asked to find in this question. We can recall that the strength of
the magnetic field inside of a solenoid of total length 𝐿 that consists of 𝑁 turns
of wire and carries a current of 𝐼 is equal to a constant 𝜇 naught, the
permeability of free space, multiplied by the number of turns, capital 𝑁,
multiplied by the current 𝐼 divided by the solenoid’s length 𝐿.
On the right-hand side of the
equation, we know the current 𝐼 in the wire and we’re also given a value for the
constant 𝜇 naught. However, we don’t know the total
number of turns of wire, capital 𝑁, and we don’t know the solenoid’s length 𝐿. What we are told though is that the
solenoid has 11 turns of wire per centimeter. If we label the number of turns of
wire per unit length of the solenoid as lowercase 𝑛, then we can say that lowercase
𝑛 is equal to 11 centimeters to the negative one. Since the SI base unit of length is
not the centimeter but rather the meter, let’s convert this value for lowercase 𝑛
from centimeters to the negative one into meters to the negative one.
In order to do this, let’s recall
that one meter is equal to 100 centimeters. If we then divide both sides of
this relationship by one meter and by 100 centimeters so that on the left the
one-meter terms cancel out, while on the right the 100-centimeter terms cancel, we
see that one over 100 centimeters is equal to one over one meter. Since one over units of centimeters
is centimeters to the negative one and one over units of meters is meters to the
negative one, then we can rewrite this as one over 100 centimeters to the negative
one is equal to one meter to the negative one.
Finally, if we multiply both sides
of this by 100 so we can cancel the 100s on the left-hand side, we find that one
centimeter to the negative one is equal to 100 meters to the negative one. That means that to convert from
centimeters to the negative one to meters to the negative one, we multiply by a
factor of 100.
Multiplying our value of 11
centimeters to the negative one for lowercase 𝑛 by a factor of 100, we find that
lowercase 𝑛 is equal to 11 multiplied by 100 meters to the negative one. This works out as 1100 meters to
the negative one, or, in other words, there are 1100 turns of wire per meter of the
solenoid’s length. Now, the number of turns of wire
per unit length, which is this value for lowercase 𝑛, must be equal to the total
number of turns, capital 𝑁, divided by the solenoid’s length 𝐿.
Since in this equation here, we
don’t know the value for either capital 𝑁 or for 𝐿, but we do know lowercase 𝑛
and that lowercase 𝑛 is equal to capital 𝑁 divided by 𝐿, let’s use this
relationship in order to replace the capital 𝑁 divided by 𝐿 in the equation for
the magnetic field strength by lowercase 𝑛, the number of turns per unit
length. When we do this, we find that 𝐵,
the strength of the magnetic field inside of the solenoid, is equal to the
permeability of free space, 𝜇 naught, multiplied by the turns per unit length,
lowercase 𝑛, multiplied by the current 𝐼.
Let’s now clear some space on the
screen so that we can substitute our values into the right-hand side of this
equation. When we substitute in our values
for lowercase 𝑛 and 𝐼 along with the value we’re given for the constant 𝜇 naught
, we find that 𝐵 is equal to four 𝜋 times 10 to the negative seven tesla meters
per ampere, which is our value for the constant 𝜇 naught, multiplied by 1100 meters
to the negative one, which is the turns per unit length lowercase 𝑛, multiplied by
0.15 amperes, which is the current 𝐼 in the wire.
If we look at the units on the
right-hand side, we can see that the meters and the meters to the negative one
cancel each other out, and likewise the amperes and the per ampere also cancel. This just leaves us with teslas as
our units for 𝐵, the strength of the magnetic field. Evaluating this expression, we find
that 𝐵 is equal to 2.073 et cetera times 10 to the negative four tesla. The question wants our answer in
teslas expressed in scientific notation to one decimal place. We have the correct units, and our
answer is already in scientific notation, which means we just need to round the
value to one decimal place.
Rounding to one decimal place, we
get our final answer to the question. The strength of the magnetic field
at the center of the solenoid is equal to 2.1 times 10 to the negative four
tesla.
