Photon A has twice the frequency of photon B. What is the ratio of the energy of photon A to the energy of photon B?
To answer this question, we will need to know the relationship between a photon’s frequency and its energy. To answer this question, it is sufficient to know that the energy of a photon is directly proportional to its frequency. We can therefore write that 𝐸, the energy of the photon, is equal to ℎ times 𝑓, where ℎ is some constant and 𝑓 is the frequency. Now it turns out that this constant ℎ is the Planck constant. But even if we didn’t know that, we could still write that energy is some constant times frequency.
Now, the quantity that we’re looking for is the ratio of the energy of the two photons. If we use the subscript A and B to denote the two photons, we are looking for 𝐸 sub A divided by 𝐸 sub B. From the question, we know the ratio of the frequencies of these two photons. So let’s use our directly proportional relationship to substitute in for energy as a function of frequency. We have that the ratio of the energies is equal to the constant ℎ times the frequency of photon A divided by the constant ℎ times the frequency of photon B.
We now see why all we need to know is that energy and frequency are directly proportional. We don’t need to know that that constant of proportionality happens to be the Planck constant. In a ratio, we have ℎ divided by ℎ. Since a constant of proportionality cannot be zero, ℎ divided by ℎ is just one. So the actual value of ℎ does not affect the value of this fraction. This leaves us with the fact that the ratio of the energies of two photons is exactly equal to the ratio of their frequencies. In the question, we are directly told the ratio of the frequency of photon A to the frequency of photon B. The ratio is two. So since the frequency of photon A is twice that of photon B, the ratio of the energy of photon A to the energy of photon B is also two.