Question

It is possible to convert radiant energy into electrical energy using photovoltaic cells. Assuming equal efficiency of conversion, would infrared or ultraviolet radiation yield more electrical energy on a per-photon basis?

Short answer

The energy of a single photon is inversely proportional to its wavelength. Since ultraviolet radiation has shorter wavelengths than infrared radiation, ultraviolet photons will have more energy than infrared photons. Therefore, ultraviolet radiation would yield more electrical energy on a per-photon basis, assuming equal efficiency in photovoltaic cell conversion.

Step-by-step solution

Recall the relationship between energy, frequency, and wavelength

The energy of a single photon (E) can be calculated using the formula: \[ E = h\nu \] where \(h\) is Planck's constant (\(6.63 \times 10^{-34} Js\)) and \(\nu\) is the frequency of the photon. We can also relate the energy to the wavelength (\(\lambda\)) using the speed of light (c) relationship: \[\nu = \cfrac{c}{\lambda} \]

Substitute the wavelength-energy relationship into the energy-frequency relationship

Now that we know both relationships, we can replace \(\nu\) in our energy equation with the wavelength-energy relationship: \[E = h \cdot \cfrac{c}{\lambda} \]

Compare infrared and ultraviolet wavelengths

For infrared radiation, the range of wavelengths is about 700 nm to 1 mm, while ultraviolet radiation has wavelengths ranging from 10 nm to 400 nm. Since the energy of a photon is inversely proportional to its wavelength, a photon with a shorter wavelength will have higher energy than one with a longer wavelength.

Determine which radiation has more energy per photon

As ultraviolet radiation has shorter wavelengths than infrared radiation, we can conclude that ultraviolet radiation photons will have more energy than those of infrared radiation. Therefore, ultraviolet radiation would yield more electrical energy on a per-photon basis, assuming equal efficiency in photovoltaic cell conversion.