Video Transcript
Light from two lasers is directed at a 100 percent reflective surface. Laser A produces light with a wavelength of 500 nanometers, and laser B produces light with a wavelength of 700 nanometers. Both lasers produce the same number of photons per second. Which laser produces light that exerts a greater force on the surface?
Let’s say that these are our two lasers, laser A and B. Laser A, we’re told, emits light with a wavelength of 500 nanometers, which to our eyes would appear green. Laser B, on the other hand, produces light with a wavelength of 700 nanometers, which to our eyes would look red. We’re told that light from both of these lasers lands on a 100 percent reflective surface. This means that the surface doesn’t absorb any light but rather reflects all of it back. We want to know which of these two lasers exerts a greater force on this surface.
To understand why a laser light would exert a force on a surface, it will help to think of these beams as comprised of particles. Specifically, the beams are made of photons. The photons in each beam have a wavelength. We’ll call them 𝜆 sub A and 𝜆 sub B. And by the de Broglie relationship, the wavelength of a photon in each beam is equal to the Planck constant divided by the momentum 𝑃 of that photon. We can write then that the wavelength of a photon in beam A is equal to the Planck constant divided by the momentum of that photon and that, similarly, 𝜆 B is equal to ℎ divided by 𝑃 sub B. Since the Plank constant is a constant, we know that for a given photon the wavelength of that photon is inversely proportional to its momentum.
The conclusion we can draw from that is the photon with the smaller wavelength is actually the one that has the greater momentum. What we’ve called 𝜆 sub A is 500 nanometers, and 𝜆 sub B is 700 nanometers. And because 500 is smaller than 700, we know that the momentum of a photon in beam A, 𝑃 sub A, is greater than the momentum of a photon in beam B, 𝑃 sub B. Our question though, rather than asking about momentum, asks about force. Force and momentum are connected though through this mathematical relationship. When an object experiences a change in momentum, Δ𝑃, over a change in time, Δ𝑡, then the ratio Δ𝑃 to Δ𝑡 is equal to the force exerted by that object.
Let’s imagine that we could look at our two beams of laser light so that we could see individual photons from each beam being incident on our surface. We’ve seen so far that because the wavelength of this photon is shorter than the wavelength of this one, its momentum is greater. Both photons, each with its respective momentum, will be incident on the surface and then be reflected back, each with the same magnitude of momentum it had before. We can say that these two collisions happen over the same amount of time Δ𝑡. And so the photon that experiences the greater change in momentum, Δ𝑃, will be the one that had the greater momentum going into the collision in the first place. That, we know, is the photon in beam A.
A photon in the green beam of laser light will experience a greater change in momentum through a collision with this reflective surface. And therefore, it will exert a larger force on that surface than the photon in beam B. All this tells us that when it comes to the laser that produces light that exerts a greater force on the surface, our answer is laser A. This is the laser whose photons possess greater momentum and, therefore, exert greater force on the surface.
