Question

Microwave radiation has a wavelength on the order of \(1.0 \mathrm{cm} .\) Calculate the frequency and the energy of a single photon of this radiation.

Short answer

The frequency of a single photon of microwave radiation with a wavelength of \(1.0 \mathrm{cm}\) is \(3 \times 10^{10} \mathrm{Hz}\), and its energy is approximately \(1.989 \times 10^{-23} \mathrm{J}\).

Step-by-step solution

Calculate the frequency of a single photon

To calculate the frequency, we will use the formula \(v = \frac{c}{\lambda}\). We know that the speed of light \(c = 3 \times 10^8 \mathrm{m/s}\) and the given wavelength \(\lambda = 1.0 \mathrm{cm} = 0.01 \mathrm{m}\). Now, we can find the frequency: \(v = \dfrac{3 \times 10^8 \mathrm{m/s}}{0.01 \mathrm{m}} = 3 \times 10^{10} \mathrm{Hz}\) The frequency of a single photon of microwave radiation is \(3 \times 10^{10} \mathrm{Hz}\).

Calculate the energy of a single photon

Now that we know the frequency, we can calculate the energy of a single photon using the formula \(E = h \times v\), where \(h\) is the Planck's constant, \(h = 6.63 \times 10^{-34} \mathrm{J \cdot s}\), and \(v\) is the frequency we calculated in step 1. \(E = (6.63 \times 10^{-34} \mathrm{J \cdot s}) \times (3 \times 10^{10} \mathrm{Hz})\) \(E = 1.989 \times 10^{-23} \mathrm{J}\) The energy of a single photon of microwave radiation is approximately \(1.989 \times 10^{-23} \mathrm{J}\).

Key concepts

Understanding the Electromagnetic Spectrum

The electromagnetic spectrum encompasses a range of radiation types, each characterized by its wavelength and frequency. At one end of the spectrum, we find long wavelength, low-frequency radio waves, while at the other end, we encounter high-frequency, short-wavelength gamma rays. Microwave radiation, which is the focus of our exercise, falls between radio and infrared waves on this spectrum.

Imagine the spectrum as a vast array of colors, each representing a different type of radiation. Microwaves are like the 'green' in this array – distinct, with their own set of properties. Specifically, microwaves have wavelengths ranging from one millimeter to one meter, making them ideal for various applications such as satellite communications and, most famously, cooking in microwave ovens.

• Microwave radiation has wavelengths approximately in the range of 1 mm to 1 m.
• It is used in various technologies, including communication and cooking.
• The energy of microwave photons can influence matter at the molecular level, as seen in microwave heating.

Understanding where microwaves sit on the electromagnetic spectrum helps us appreciate their unique frequency and energy characteristics, relevant to the exercise at hand.

The Role of Planck's Constant

A pivotal breakthrough in understanding photon energy came from Max Planck, who introduced a fundamental constant in physics known as Planck's constant, symbolized by 'h'. This constant is essential when examining the energy of photons, the quanta of light in the electromagnetic spectrum.

Planck's constant has a value of approximately \(h = 6.63 \times 10^{-34} \mathrm{J \cdot s}\), which might seem insignificant at a glance. However, it is crucial for establishing the relationship between the energy of a photon and its frequency, as depicted in our exercise. In simple terms, the higher the frequency of the photon, the more energy it carries, and Planck's constant acts as the scale factor in this relationship.

• Planck's constant provides a bridge between the energy and frequency of a photon.
• As frequency increases, so does the energy of a photon, proportional to Planck's constant.

The ability to use Planck’s constant to calculate photon energy allows us to delve into a more profound understanding of how light interacts with matter – a principle that underpins technologies such as lasers, solar cells, and even our microwave ovens.

Wave Frequency Calculation

The frequency of a wave can be intuitively understood as the number of wave cycles that pass a given point per second. It is denoted in Hertz (Hz), with one Hz corresponding to one cycle per second. When it comes to radiation like microwaves, the frequency determines how energetic the photons are.

To compute the frequency of a photon, given its wavelength, we use the simple yet profound formula \(v = \frac{c}{\lambda}\), where \(v\) is the frequency, \(c\) is the speed of light (approximately \(3 \times 10^{8} \mathrm{m/s}\), a universal constant), and \(\lambda\) is the wavelength of the photon. In our exercise, the quick conversion of the wavelength from centimeters to meters and application of the formula yielded the microwave's frequency.

• Frequency is measured in Hertz (Hz) or cycles per second.
• The speed of light is a key part of the equation used to calculate wave frequency from wavelength.
• A precise wave frequency calculation is vital for determining the photon's energy.

This way of calculating frequency not only aids in academic exercises but also has practical implications in designing and understanding technological devices that operate based on electromagnetic waves.