Question

A radio wave has a frequency of \(3.8 \times 10^{10} \mathrm{~Hz}\). What is the energy (in \(J\) ) of one photon of this radiation?

Short answer

The energy of one photon is approximately \( 2.52 \times 10^{-23} \text{ J} \).

Step-by-step solution

- Identify the Given Information

The problem states that the frequency of the radio wave is \(f = 3.8 \times 10^{10} \text{ Hz}\).

- Use the Energy Formula

To find the energy of a photon, use the formula \(E = h \times f\), where \(h\) is Planck's constant, and \(f\) is the frequency of the wave. Planck's constant \(h\) is approximately \(6.626 \times 10^{-34} \text{ J} \text{ s}\).

- Substitute the Values into the Formula

Plug the given frequency and Planck's constant into the formula: \(E = (6.626 \times 10^{-34} \text{ J} \text{ s}) \times (3.8 \times 10^{10} \text{ Hz})\).

- Perform the Calculation

Now calculate the energy: \[ E = 6.626 \times 10^{-34} \times 3.8 \times 10^{10} = 2.51788 \times 10^{-23} \text{ J} \]

Key concepts

Planck's constant

Planck's constant is a fundamental constant in physics, represented by the symbol \(h\). It has a value of approximately \(6.626 \times 10^{-34} \text{ J} \text{ s}\). This constant is crucial because it relates the energy of a photon to its frequency. This relationship is given by the equation \(E = h \times f\), where \(E\) is the energy, \(h\) is Planck's constant, and \(f\) is the frequency. Planck's constant is named after Max Planck, the physicist who introduced it in 1900. This discovery was foundational in the development of quantum mechanics.

Frequency-Energy Relationship

The frequency-energy relationship is described by the equation \(E = h \times f\). This equation signifies that the energy of a photon is directly proportional to its frequency. Here, \(E\) stands for energy, \(h\) is Planck's constant, and \(f\) represents frequency. For instance, if the frequency of a photon increases, its energy also increases proportionally. This relationship is vital in various scientific fields, especially in understanding electromagnetic radiation. The direct proportionality helps in calculating the photon energy for different waves, such as radio waves, visible light, and X-rays.

Radio Waves

Radio waves are a type of electromagnetic radiation with frequencies that range from about \(3 \text{ kHz}\) to \(300 \text{ GHz}\). They have the longest wavelengths in the electromagnetic spectrum. Despite their long wavelengths and low energy compared to other types of electromagnetic waves, such as visible light or X-rays, radio waves are essential for communication technologies, including radio broadcasting, television, and cell phones. In the given problem, we used the radio wave's frequency to calculate the energy of a single photon by applying the formula \(E = h \times f\). This showcases how even low-energy radio waves still follow the same fundamental principles of quantum mechanics.