Video Transcript
A laser emits light with a
wavelength of 200 nanometers. How many photons must be emitted by
the laser for the amount of energy emitted to be one joule? Use a value of 6.63 times 10 to the
negative 34 joule-seconds for the Planck constant and 3.00 times 10 to the eighth
meters per second for the speed of light in free space. Give your answer in scientific
notation to two decimal places.
This question is asking us about
the total energy of a collection of photons. In particular, we want to know how
many photons have a total energy of one joule if their wavelength is 200
nanometers. To get an idea of how to approach
this question, let’s draw a picture of the laser. Here is the laser, and here are the
photons. Note that the color that we have
used to represent these photons is not their true color. These photons have a wavelength of
200 nanometers, which is far in the ultraviolet almost bordering on X-rays. These photons are invisible. Anyway, each photon has an energy
which we’ll call 𝐸 sub p for energy of a photon.
This energy is the same for all of
the photons emitted by the laser because they are all identical. They all have a wavelength of 200
nanometers. We’ll also call the total energy of
all of the photons emitted by the laser 𝐸 sub t for total energy. As this drawing suggests, the total
energy is just the sum of the energies of each individual photon. And since each individual photon
has exactly the same energy, the total energy is just the energy of an individual
photon times the number of photons. Now, what we are looking for is the
number of photons. So let’s rearrange this equation by
dividing both sides by the energy of a single photon.
On the left-hand side, we have the
total energy divided by the energy of a single photon. On the right-hand side, 𝐸p divided
by 𝐸p is just one. And we’re left with the number of
photons, which is what we are looking for. If we let the capital letter N
represent the number of photons, then we can write N is equal to 𝐸t divided by
𝐸p. At this point, we know we are on
the right track because the right-hand side of this expression is an energy divided
by an energy, which is dimensionless, and the left-hand side is N, which is a
number, which also has no dimensions.
Additionally, we already have a
value for the total energy. It’s one joule. All that we need then is the energy
of a single photon. Recall that the energy of a photon
can be calculated as ℎ, the Planck constant, times 𝑐, the speed of light in free
space, divided by 𝜆, the wavelength of the photon. We are given a value for the Planck
constant and a value for the speed of light in free space and also a value for the
photon’s wavelength. So all we need to do is substitute
these values into our formula to find the energy of a single photon and then
substitute that value into our formula to find the total number of photons. We have that the energy of a single
photon is 6.63 times 10 to the negative 34 joule-seconds times 3.00 times 10 to the
eighth meters per second divided by 200 nanometers.
Now we are calculating an
energy. So the overall quantity must have
units appropriate for energy. We see that joules, which are units
of energy, appear on the right-hand side. But other units appear as well,
mainly seconds, meters per second, and nanometers. We need to make sure that all of
these other units combine to give us a dimensionless number and the overall units of
joules for energy. We can immediately eliminate
seconds because seconds per second is just one. But we are still left with meters
in the numerator and nanometers in the denominator. Meters and nanometers are both
units of length. So meters divided by nanometers is
dimensionless, which is what we need. But we also need to make sure that
we have the correct numerical factor.
Recall that one nanometer is 10 to
the negative nine meters. So we can replace 200 nanometers
with 200 times 10 to the negative nine meters. Now we have units of meters in both
the numerator and denominator. And we can safely say that meters
divided by meters is just one. Observe that in the process of
eliminating the dimensions of length from the numerator and denominator, we had to
introduce a factor of 10 to the negative nine to make the units agree. Anyway, the overall units are now
joules, which is exactly what we want when calculating energy.
Putting all these numbers into a
calculator, we find that the energy of a single photon is 9.945 times 10 to the
negative 19 joules. Now, all we need to do is
substitute this value for the energy of a single photon and one joule for the total
energy of all of the photons into our formula to find the total number of
photons. Let’s clear some space to do this
calculation.
The number of photons that must be
emitted so that the laser emits one joule of energy is one joule divided by 9.945
times 10 to the negative 19 joules. The first thing we notice is that
the numerator and denominator have the same units. So the overall quantity is
dimensionless, which is again exactly what we want because we are looking for a
number, and numbers have no dimensions. Putting this into a calculator, we
get 1.005 and several more decimal places times 10 to the 18. This number is already in
scientific notation. And rounding to two decimal places,
we get our final answer, 1.01 times 10 to the 18 photons.
