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