A laser emits four times 10 to the 20th
photons, each with a frequency of six times 10 to the 14th hertz. What is the total energy radiated by the
laser? Use a value of 6.63 times 10 to the
negative 34th joules seconds for the Planck constant. Give your answer in joules to three
Okay, so, in this exercise, we have a
laser. Let’s say that this is our laser. And we’re told that, over some amount of
time, this laser emits four times 10 to the 20th photons. This, of course, is a huge number over a
1,000,000,000 billions of photons. And we’re told that each of these photons
has a frequency — we’ll call that 𝑓 — of six times 10 to the 14th hertz. Based on this, we want to calculate the
total energy radiated by the laser.
The way we can think of doing this is by
calculating first the energy of one of these four times 10 to the 20th photons and then
multiplying that amount by the total number of photons we have. So, let’s start by calculating the energy
of one of these photons. We can recall that the energy of an
individual photon — we can call it 𝐸 sub 𝑝 — is equal to a constant value, known as
Planck’s constant, times the frequency of that photon. And we’re told that we can use a value of
6.63 times 10 to the negative 34th joule seconds for that constant.
So then, the energy of one of the photons
emitted by our laser is equal to Planck’s constant multiplied by the frequency of that
photon. And so, the total amount of energy
radiated by the laser — we can call it 𝐸 — is equal to the total number of photons, four
times 10 to the 20th, times the energy of a single photon. Before we multiply these three numbers
together, let’s take a quick look at the units involved.
In Planck’s constant, we have joules
times seconds. And in frequency, we have a unit of
hertz. We know that our hertz though indicates a
number of cycles completed per second, which is equivalent to the unit of one over
seconds. Written this way, we can see that the
unit of seconds in our Planck’s constant will cancel out with the units of one over seconds
in our photon frequency.
This means we’ll be left simply with
units of joules when we calculate this figure. And that’s perfect because we’re
calculating an energy. When we compute this product to three
significant figures, the result is 159 joules. That’s the total amount of energy
radiated by the laser.