Calculate the wavelengths of (a) a 1.0 -MHz AM radio wave, (b) a channel 9 TV signal \((190 \mathrm{MHz}),(\mathrm{c})\) a police radar \((10 \mathrm{GHz})\) (d) infrared radiation from a hot stove \(\left(4 \times 10^{13} \mathrm{Hz}\right),\) (e) green light \(\left(6.0 \times 10^{14} \mathrm{Hz}\right),\) and (f) \(1.0 \times 10^{18} \mathrm{Hz}\) X rays. All are electromagnetic waves that propagate at \(3.0 \times 10^{8} \mathrm{m} / \mathrm{s}\)

Understanding how electromagnetic waves propagate is vital for comprehending how different frequencies are associated with different types of electromagnetic radiation. Electromagnetic waves are ripples in the electromagnetic field, carrying energy through space away from their source. They do not require a medium to travel through, meaning they can move through the vacuum of space.

Common examples of electromagnetic waves include light, radio waves, and X-rays. All of these waves travel at the speed of light in a vacuum, which is about 299,792,458 meters per second (approximately 3.0 x 10⁸ m/s). The energy carried by electromagnetic waves is used in various applications from broadcasting communications to medical imaging technologies.

The relationship between frequency and wavelength for electromagnetic waves is crucial in physics, especially when analyzing the electromagnetic spectrum. The formula that embodies this relationship is given by the equation \( c = f \lambda \), where \( c \) represents the speed of light in a vacuum, \( f \) is the frequency, and \( \lambda \) is the wavelength of the wave.

The higher the frequency of the wave, the shorter the wavelength. Conversely, the lower the frequency, the longer the wavelength. This inverse relationship allows us to calculate one if the other is known, and it holds true across the entire electromagnetic spectrum, from low-frequency radio waves to high-frequency gamma rays.

In physics, the speed of light in a vacuum is commonly denoted as \( c \) and is one of the most fundamental constants. The speed of light is approximately 3.0 x 10⁸ meters per second. This remarkable speed is the cosmic speed limit, meaning nothing in the universe can travel faster than light.

This constant isn't just the speed at which light travels; it's also the speed at which all electromagnetic radiation propagates in a vacuum, including radio waves, microwaves, infrared light, visible light, ultraviolet light, X-rays, and gamma rays. Whenever calculating frequencies and wavelengths of electromagnetic waves, the speed of light is a critical component of the calculation.

In physics, using standard units of measurement is essential for accurate scientific communication. The International System of Units (SI) is the standard used worldwide in science and industry. For our discussion on frequency and wavelength, the key SI unit is Hertz (Hz), which measures frequency, or the number of cycles per second.

When calculating wavelengths, converting to the proper SI units simplifies the mathematics and helps avoid mistakes. For example, 1 Megahertz (MHz) is equivalent to 1.0 x 10⁶ Hz, and 1 Gigahertz (GHz) is equal to 1.0 x 10⁹ Hz. This conversion is critical when inputting frequency values into the \( c = f \lambda \) formula to ensure that units match, allowing for the calculation of the correct wavelength in meters (m).