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

Ultraviolet radiation would yield more electrical energy on a per-photon basis than infrared radiation, assuming equal efficiency of conversion. This is because ultraviolet radiation has a higher frequency and shorter wavelength compared to infrared radiation, resulting in higher energy per photon based on Planck's equation.

Step-by-step solution

Understand the Planck's Equation

Planck's equation relates the energy of a photon (E) to its frequency (v) through the proportionality constant Planck's constant (h). The equation is given by: \[ E = h \times v\] where h = 6.63 × 10^{-34} Js (Planck's constant).

Understand the relation between frequency, wavelength, and speed of light

The frequency (v) of a photon can also be expressed in terms of its speed (c) and wavelength (λ) using the formula: \[ v = \cfrac{c}{\lambda}\] Knowing the numerical values of speed of light (c), we can calculate and compare the frequencies for infrared and ultraviolet radiation based on their wavelengths.

Compare the Energy of Infrared and Ultraviolet Photons

Now that we have the formulas connecting the photon's energy, frequency, and the speed of light, the comparison between the energy of infrared and ultraviolet photons becomes straightforward. 1. Infrared (IR) radiation has a longer wavelength (typically 700 nm to 1 mm) while Ultraviolet (UV) radiation has a shorter wavelength (typically 10 nm to 400 nm). 2. Shorter wavelengths result in higher frequencies and vice versa - this means that the UV radiation has higher frequency compared to the IR radiation. 3. Using Planck's equation, as the frequency of the photons increases, so does the energy of the photons. 4. Since UV photons have a higher frequency, they have more energy per photon compared to IR photons. From this analysis, we can conclude that ultraviolet radiation would yield more electrical energy on a per-photon basis than infrared radiation, assuming equal efficiency of conversion.

Key concepts

Planck's Equation

To understand how the energy of photons is calculated, we must first look at Planck's Equation. This equation provides a simple relationship between the energy of a photon (E) and its frequency (ν), using Planck's constant (h) as a factor. The formula is as follows: \[ E = h \times u \]

Here, Planck's constant (6.63 \times 10^{-34} Js) plays a vital role in connecting frequency to energy. When you examine this equation, you'll notice that higher frequency results in higher photon energy. This foundational concept is important to understand why different types of electromagnetic radiation, such as infrared and ultraviolet, yield different energy levels.

Ultraviolet vs Infrared

The electromagnetic spectrum is broad, but let's focus on ultraviolet (UV) and infrared (IR) radiation. These two types of radiation are on opposite ends of the visible light spectrum, each with distinct characteristics.

- **Infrared Radiation (IR):** - Wavelength: 700 nm to 1 mm - Typically associated with heat - Lower frequency and energy compared to UV - Often used in remote controls and thermal imaging
- **Ultraviolet Radiation (UV):** - Wavelength: 10 nm to 400 nm - Higher frequency and energy than IR - Can cause sunburn, but also used for sterilization and fluorescence

The smaller the wavelength, the higher the frequency, which means ultraviolet photons carry more energy per photon than infrared photons. This characteristic is directly related to a photon's ability to generate electrical energy when used in photovoltaic cells.

Photovoltaic Cells

Photovoltaic cells are devices that convert light into electric energy through the photovoltaic effect, a method for generating electric power by directly converting solar radiation into electricity using semiconductors.

- Photovoltaic cells depend on the energy per photon to convert light energy effectively. - Ultraviolet light, having more energy per photon, can potentially generate more electrical energy than infrared light, assuming all other conditions such as conversion efficiency are the same.
The efficiency of photovoltaic cells largely depends on the material used and the spectrum of light they're exposed to. Thus, ultraviolet light, with its higher energy, can be more efficient in generating electricity per photon.

Frequency and Wavelength

Frequency and wavelength are core concepts when discussing photon energy. Understanding their relationship helps in comprehending how different light types provide varying energy levels.

- **Speed of Light (c):** Constant at approximately 3 \times 10^8 m/s.- **Frequency (ν):** Number of wave cycles per second, inversely proportional to wavelength.- **Wavelength (λ):** Distance between identical points in the adjacent cycles of a waveform.

The relationship between frequency and wavelength is given by the equation:\[ u = \cfrac{c}{\lambda} \]With a shorter wavelength, the frequency is higher because waves cycle more times per second. This is why UV light, with its shorter wavelength compared to IR, has higher frequency and energy. Grasping these relationships allows us to understand why the energy yielded by photons in different parts of the electromagnetic spectrum differs.