### Video Transcript

What is the energy of a photon that
has a wavelength of 400 nanometers? Use 6.63 times 10 to the negative
34 joule-seconds for the value of the Planck constant and 3.00 times 10 to the
eighth meters per second for the value of the speed of light in free space. Give your answer in scientific
notation to two decimal places.

To answer this question, we will
need to relate the energy of a photon to its wavelength. To help us do this, we are given a
value for the Planck constant and also a value for the speed of light in free
space. With this information, the
relationship that we need is that 𝐸 is equal to ℎ𝑐 divided by 𝜆, where 𝐸 is the
energy of the photon, ℎ is the Planck constant, 𝑐 is the speed of light in free
space, and 𝜆 is the wavelength of the photon. This formula is physically
identical to expressing the energy as the Planck constant times the frequency of the
photon, with these two formulas being related by the fact that the speed of light is
equal to the frequency of a photon times its wavelength.

Anyway, before we substitute values
into our formula, we need to modify the wavelength of the photon slightly. This is because the Planck constant
with units of joule-seconds and the speed of light with units of meters per second
are both expressed in terms of basic SI units. However, nanometers are not basic
SI units. So we need to convert nanometers
into meters. One nanometer is 10 to the negative
nine meters. So the wavelength of our photon is
400 times 10 to the negative nine meters.

Alright, now all that’s left is to
substitute this value and the values that we are given for the Planck constant and
the speed of light. We have 6.63 times 10 to the
negative 34 joule-seconds times 3.00 times 10 to the eighth meters per second
divided by 400 times 10 to the negative nine meters. Let’s start by working out the
units. In the numerator, we have seconds
and also per second, but seconds per second is just one. And then we have meters in the
numerator and meters in the denominator. But meters divided by meters is
also one. So the overall units are just
joules. This tells us we’re on the right
track because joules are a unit of energy. When we evaluate the numerical
portion of this quantity, we get 4.9725 times 10 to the negative 19, and then the
units are joules. Rounding to two decimal places, our
final answer is 4.97 times 10 to the negative 19 joules.

Now, when it comes to photons, it
is often convenient to express their energies as electron volts. And 4.97 times 10 to the negative
19 joules is approximately 3.1 electron volts. The reason we mentioned this is
because if we look at the units for the Planck constant and for the speed of light,
we have joule-seconds and meters per second. The product of these joule-seconds
times meters per second is joules times meters, which is an energy times a
length. If we express energy in terms of
electron volts and length in terms of nanometers, it turns out that the Planck
constant times the speed of light is almost exactly 1240 electron volt
nanometers.

This means that as long as we
express the wavelength of the photon in nanometers, then the energy of the photon in
electron volts is very nearly 1240 divided by the wavelength. In fact, 1240 divided by 400 is
exactly equal to 3.1. And our true answer is less than a
tenth of a percent different from 3.1 electron volts. Indeed, in general, approximating
ℎ𝑐 as 1240 electron volt nanometers will give answers that are accurate to within a
small fraction of one percent.