Question

Compare the energies and frequencies of two photons, one with a short wavelength and the other with a long wavelength.

Short answer

Answer: A photon with a shorter wavelength will have a higher energy and frequency than a photon with a longer wavelength. Conversely, a photon with a longer wavelength will have a lower energy and frequency than a photon with a shorter wavelength.

Step-by-step solution

Identify the Given Parameters

In this problem, we are comparing two photons: one with a short wavelength and the other with a long wavelength. We can denote the short wavelength as "λ_s," and the long wavelength as "λ_l." We know that the speed of light (c) is 3.00 × 10^8 m/s, and the value of Planck's constant (h) is 6.626 x 10^-34 Js.

Relate Wavelength to Frequency Using the Speed of Light

The speed of light is related to the wavelength and frequency of a photon by the equation c = λ * ν, where c is the speed of light, λ is the wavelength, and ν is the frequency. We can use this equation to find the ratio between the frequencies of the short and long wavelength photons: ν_s/ν_l = λ_l/λ_s

Determine the Ratio of Frequencies

From Step 2, we obtained the equation: ν_s/ν_l = λ_l/λ_s Now, we want to express the frequencies of the short and long wavelength photons in terms of their wavelengths: ν_s = (λ_l/λ_s) * ν_l

Calculate the Ratio of Energies

The energy of a photon is given by E = h * ν, where E is the energy, h is Planck's constant and ν is the frequency. Using this equation, we can determine the ratio of the energies of the short and long wavelength photons: E_s/E_l = (h * ν_s) / (h * ν_l) From Step 3, we can substitute the expression for ν_s: E_s/E_l = (h * (λ_l/λ_s) * ν_l) / (h * ν_l) Simplifying the equation, we get: E_s/E_l = λ_l/λ_s

Interpret the Results

From the above analysis, we find that the ratio of the energies of the short and long wavelength photons is the inverse of the ratio of their wavelengths. So a photon with a shorter wavelength will have a higher energy and frequency than a photon with a longer wavelength. Conversely, a photon with a longer wavelength will have a lower energy and frequency than a photon with a shorter wavelength.