Question

Police often monitor traffic with "K-band" radar guns, which operate in the microwave region at \(22.235 \mathrm{GHz}\) ( \(1 \mathrm{GHz}=10^{9} \mathrm{~Hz}\) ). Find the wavelength (in \(\mathrm{nm}\) and \(\dot{\mathrm{A}}\) ) of this radiation.

Short answer

The wavelength is approximately 1.35 x 10^7 nm or 1.35 x 10^8 Å.

Step-by-step solution

Understand the given frequency

The problem states that the frequency of the radar gun operation is 22.235 GHz. Note that 1 GHz = 10^9 Hz. Therefore, the frequency (f) is 22.235 x 10^9 Hz.

Use the speed of light equation

The relationship between the speed of light (c), wavelength (\textbackslash\textlambda), and frequency (f) is given by the equation: \[ c = \textbackslash\textlambda \times f \] where the speed of light (c) is approximately 3.00 x 10^8 m/s.

Solve for wavelength in meters

Rearrange the equation to solve for wavelength (\textbackslash\textlambda): \[ \textbackslash\textlambda = \frac{c}{f} \] Substituting the given values: \[ \textbackslash\textlambda = \frac{3.00 \times 10^8 \text{ m/s}}{22.235 \times 10^9 \text{ Hz}} \] \[ \textbackslash\textlambda \textbackslash approx 1.35 \times 10^{-2} \text{ m} \]

Convert wavelength from meters to nanometers

Since 1 meter = 10^9 nanometers (nm), we multiply the wavelength in meters by 10^9 to convert it to nanometers: \[ \textbackslash\textlambda = 1.35 \times 10^{-2} \text{ m} \times 10^9 \text{ nm/m} = 1.35 \times 10^7 \text{ nm} \]

Convert wavelength from nanometers to angstroms

Since 1 nm = 10 \textbackslash A, we multiply the wavelength in nanometers by 10 to convert it to angstroms: \[ \textbackslash\textlambda = 1.35 \times 10^7 \text{ nm} \times 10 = 1.35 \times 10^8 \textbackslash A \]

Key concepts

frequency

Frequency is a fundamental concept in understanding waves, including light and microwaves. It tells us how many wave cycles pass a point per second and is measured in Hertz (Hz). In the radar gun exercise, the frequency is given as 22.235 GHz. Remember that 1 GHz is equal to 1 billion (10^9) Hz. Thus, the frequency in this problem equals 22.235 x 10^9 Hz. Frequency is crucial because it determines the energy of the electromagnetic wave and is inversely related to wavelength: higher frequencies mean shorter wavelengths.

speed of light

The speed of light (c) is a constant that represents how fast light travels in a vacuum. It is approximately 3.00 x 10^8 meters per second (m/s). The speed of light is fundamental in relating frequency and wavelength through the equation: \[ c = \lambda \times f \] In this equation, c is the speed of light, \(\lambda\) is the wavelength, and f is the frequency. By rearranging this formula, you can solve for either wavelength or frequency if the other is known. This relationship is crucial in calculating the wavelength of electromagnetic waves such as those from a radar gun.

wavelength conversion

Converting wavelengths into various units is important for different scientific contexts. Initially, we calculate the wavelength using the speed of light formula, which often gives us the result in meters. For instance, in the step-by-step solution, the wavelength was first found to be approximately 1.35 x 10^-2 meters. To convert this to nanometers, multiply by 10^9 (since 1 meter = 10^9 nanometers). \(1.35 \times 10^{-2} \text{ m} \times 10^{9} \text{ nm/m} = 1.35 \times 10^7 \text{ nm}\) To further convert nanometers to angstroms: \(1 \text{ nm} = 10 \text{ Å}\), so: \(1.35 \times 10^7 \text{ nm} \times 10 = 1.35 \times 10^8 \text{ Å}\).

microwave region

The microwave region is a part of the electromagnetic spectrum that lies between radio waves and infrared waves. Microwaves have wavelengths ranging from about 1 millimeter to 1 meter and frequencies between 300 GHz and 300 MHz (0.3 GHz). They are widely used in technology. For example, they are used in radar guns for traffic monitoring, in microwave ovens for heating food, and for various communication technologies. In the exercise, the radar gun operates at a frequency of 22.235 GHz, placing it clearly in the microwave region of the spectrum. Understanding this region helps in recognizing the unique applications of microwaves in everyday life.