Question

How is the energy carried per photon of light related to the wavelength of the light? Does short-wavelength light carry more energy or less energy than long- wavelength light?

Short answer

The energy carried per photon of light is inversely proportional to its wavelength, as given by the equation \(E = \dfrac{hc}{\lambda}\), where \(E\) is the photon's energy, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength. Therefore, short-wavelength light carries more energy than long-wavelength light.

Step-by-step solution

Understand and recall Planck's equation

Planck's equation helps us find the relationship between the energy of a photon and its wavelength. This equation is given by the formula \(E = \dfrac{hc}{\lambda}\), where \(E\) is the energy of the photon, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} Js\)), \(c\) is the speed of light (\(3 \times 10^8 m/s\)), and \(\lambda\) is the wavelength of the light.

Analyze the relationship between energy and wavelength

Now that we have the formula \(E = \dfrac{hc}{\lambda}\), let's analyze how the energy carried per photon is related to the wavelength of the light. Since \(h\) and \(c\) are constants, the energy of the photon is inversely proportional to the wavelength (\(E \propto \dfrac{1}{\lambda}\)).

Determine if short-wavelength light carries more or less energy than long-wavelength light

As we established that the energy is inversely proportional to the wavelength, we can now answer the question about whether short-wavelength light carries more energy or less energy than long-wavelength light. Since \(E \propto \dfrac{1}{\lambda}\), we can conclude that as the wavelength (\(\lambda\)) decreases, the energy (\(E\)) increases. Therefore, short-wavelength light carries more energy than long-wavelength light. In conclusion, the energy carried per photon of light is inversely proportional to the wavelength of the light, and short-wavelength light carries more energy than long-wavelength light.