AIEEE Questions: Photons

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Here is a fact: a photon has zero rest mass. Now picture one sodium atom, sitting completely at rest. It sends out a single photon of light, and the atom moves backwards, just like a gun that recoils when it fires. So how can something with zero rest mass give anything a kick? Hold that thought, because by the end of this video, you will calculate that kick yourself. Today we solve an A I triple E question on photons, and a harder one from the lesson on the photon. Questions like these do appear in JEE, so let's begin.

0:39The energy of one photon

So, first, what is a photon? In its interaction with matter, radiation behaves as if it were made of particles, and we call these particles photons. So a beam of light is really a stream of photons. With me? Now, how much energy does each photon carry? E equals h nu, which also equals h c over lambda. Here nu, the Greek letter that looks like a v, is the frequency, and lambda, another Greek letter, is the wavelength. And c is the speed of light. Okay? And h is the Planck constant: six point six three times ten to the power minus thirty four joule second. The J s on screen is that unit, joule second. So a higher frequency, or a shorter wavelength, means more energy in every photon. Right? Now let me ask you this. I keep the same frequency of light, but I make the beam much brighter. Does each photon now carry more energy? No, it does not, and this is a classic trap. All photons of a given frequency have the same energy, whatever the intensity of the beam. So a brighter beam has more photons, not more energetic ones. Clear?

1:53Zero rest mass, but real momentum

Now, back to our puzzle. Photons travel at the speed c in vacuum, and they have zero rest mass. Got it? So here is the question. Momentum is mass times velocity, and the rest mass of a photon is zero. So the momentum of a photon must be zero, right? No, no, wait, that is the trap! Every photon carries momentum, even with zero rest mass. p equals h nu over c, which equals h over lambda, which also equals E over c. Here p is the momentum, and E is the photon's energy from before. So the shorter the wavelength, the bigger the momentum. Remember p equals E over c, because it is the bridge between energy and momentum. Okay?

2:41Momentum is conserved, photons are not

Now, the rule behind most photon collision questions. In a collision between a photon and a particle, total energy and total momentum are both conserved. With me? But the number of photons need not be conserved. One photon may be absorbed, or a new one may be created, as when an atom sends one out. Even then, the total momentum still balances. Okay? Here is a quick check, from a solved example in the lesson on the photon. One photon of energy E hits a mirror and is reflected straight back. How much momentum does the mirror get? E over c, perhaps? Careful here, because the direction flips. The photon's momentum goes from E over c to minus E over c; the arrow means changes to. Right? So the change is two times E over c, and the mirror receives that, two times E over c, not just E over c. Momentum is a vector, so a reversal counts twice. Clear?

3:41Example 1: momentum of a photon of frequency nu (AIEEE 2007)

Now, Example one, the A I triple E question from two thousand seven. Here it is: a photon of frequency nu has a momentum associated with it. If c is the speed of light, what is this momentum? Pause the video and try it first. Done? Okay, let's check. First, the energy of the photon is h nu. Right? Now, which relation turns a photon's energy into its momentum? Yes, exactly, p equals E over c, the bridge we just learnt. So p equals h nu over c, and that is the answer. Easy, isn't it? Now a topper's habit: check the units. We have joule over metre per second, the m s to the power minus one. One joule is a kilogram metre squared per second squared, the k g m squared s to the power minus two on screen. Divide, and you get kilogram metre per second, the unit of momentum. So the answer makes sense. With me?

4:47Example 2: recoil speed of a sodium atom emitting a photon

And now Example two, the harder one, from the lesson on the photon in this same chapter. It is our puzzle from the start. One sodium atom, of mass three point eight two times ten to the power minus twenty six kilogram, is at rest and emits a photon of wavelength five hundred and eighty nine nanometres. On screen, k g is kilogram, and n m is nanometre. Find its speed of recoil. Pause the video and try it first. Done? Okay, let's check. Before the emission, the atom is at rest, so the total momentum is zero. Okay? Momentum is conserved. So after the photon leaves, what must the momentum of the atom be? Yes, exactly, it is equal in size to the photon's momentum, and opposite in direction, so the total stays zero. So m times v equals h over lambda, where m is the mass of the atom, and v is its recoil speed. Right? Now, the wavelength is five hundred and eighty nine nanometres. Can we put the number five hundred and eighty nine straight into the formula, for lambda? Oh no, careful, not straight in! It is in nanometres, and h is in joule second, so lambda must be in metres. Five hundred and eighty nine nanometres is five point eight nine times ten to the power minus seven metre. Clear? So v equals h over, m times lambda. That is six point six three times ten to the power minus thirty four, over three point eight two times ten to the power minus twenty six, times five point eight nine times ten to the power minus seven. Got it? The denominator is about two point two five times ten to the power minus thirty two. Divide, and v comes to about two point nine times ten to the power minus two metre per second. The wavy equals sign means approximately equal. Okay? That is about two point nine centimetres per second, the c m s to the power minus one. So there is our puzzle, solved. The photon has zero rest mass, yet it gives the atom a kick, a tiny one, but a real one, because it carries momentum h over lambda. Clear?

7:08Quick revision: photon energy, momentum and conservation

So, quick revision. Each photon has energy h nu, or h c over lambda, and brighter light just means more photons. Zero rest mass, yet the momentum is h nu over c, h over lambda, or E over c. And in collisions, reflection and emission, total momentum is conserved. Now go back to the lesson on the photon, and try its practice problems. You can do this. All the best!