Question

An FM radio station broadcasts at 99.5 MHz. Calculate the wavelength of the corresponding radio waves.

Short answer

The wavelength of the radio waves broadcast by the FM radio station at 99.5 MHz is approximately 3.015 meters.

Step-by-step solution

Convert frequency to hertz

The frequency is given as 99.5 MHz. To convert this to Hz, we'll multiply by 1,000,000 (since 1 MHz = 1,000,000 Hz). \(99.5 MHz × 1,000,000 Hz/MHz = 99,500,000 Hz\)

Calculate the wavelength

Now that we have the frequency in Hz, we can use the wavelength equation mentioned earlier: Wavelength (λ) = Speed of light (c) / frequency (f) Fill in the values: \(λ = \frac{3.0 × 10^8 m/s}{99,500,000 Hz}\)

Solve for the wavelength

Divide the speed of light by the frequency: \(λ = \frac{3.0 × 10^8 m/s}{99,500,000 Hz} = \frac{3.0 × 10^8 m/s}{9.95 × 10^7 Hz}\) \(λ = 3.015 \times 10^{7/9} m/s = 3.015 m\) The wavelength of the radio waves broadcast by the FM radio station at 99.5 MHz is approximately 3.015 meters.

Key concepts

Frequency Conversion

In the world of radio waves and electromagnetic phenomena, understanding frequency conversion is essential. Frequency is how often a wave repeats itself in a second, and it's commonly measured in Hertz (Hz). For radio frequencies, we often use megahertz (MHz), where 1 MHz equals 1,000,000 Hz. For instance, a radio station frequency of 99.5 MHz is equal to 99,500,000 Hz when converted to Hertz. This conversion is important because calculations of wavelength require the frequency in basic unit Hertz for precision. To convert MHz to Hz, always multiply the number of megahertz by 1,000,000. This step sets the stage for further calculations, such as determining the wavelength of radio waves.

Speed of Light

The speed of light is a fundamental constant in physics, represented by the symbol c, and is approximately 3.0 × 10^8 meters per second (m/s). It is the fastest speed at which information or energy can travel through space. The speed of light not only defines the tempo of electromagnetic wave transmission but is also crucial in equations relating frequency and wavelength. When calculating wavelengths of radio waves, which are a type of electromagnetic wave, knowing the speed of light allows us to use the formula: \[ \lambda = \frac{c}{f} \] Here, \( \lambda \) is the wavelength, \( c \) is the speed of light, and \( f \) is the frequency. Utilizing this constant helps us effortlessly move between aspects of wave calculations.

Radio Waves

Radio waves are a type of electromagnetic radiation with wavelengths that can range from about one millimeter to hundreds of kilometers. They are used in various forms of communication technology, such as AM and FM radio broadcasting, television broadcasting, and satellite transmissions. These waves are produced by the oscillation of electric charges, typically broadcasted through antennas. Their primary role is to carry information across distances without needing wires. When dealing with radio waves, typical frequencies range from 3 kHz to 300 GHz, which include the FM radio band used by everyday broadcast radio. Calculating the wavelength of a specific radio wave, like those broadcasted at 99.5 MHz, helps optimize antenna design and ensures proper signal reception and transmission.

Hertz

Hertz (Hz) is the standard unit of frequency, describing the number of cycles per second of a wave. Named after the physicist Heinrich Hertz, this unit simplifies the measurement of waves in physics, especially for electromagnetic waves such as radio waves. In the realm of electromagnetic waves, higher Hertz values indicate higher frequencies. For example, a wave with a frequency of 99,500,000 Hz means it cycles 99.5 million times in one second. Recognizing and converting frequency to Hertz is critical for calculating other characteristics of a wave, like its wavelength. This understanding allows us to better analyze and manipulate wave properties in real-world applications.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ordered by frequency or wavelength. This spectrum includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays, each with varying frequencies and wavelengths. Radio waves, located at the lower frequency end of the spectrum, particularly interest us for communication due to their longer wavelengths. These longer wavelengths enable long-distance transmission and are less susceptible to obstructions. Understanding a wave's position in the electromagnetic spectrum reveals its potential applications and transmission properties. In practice, calculating the wavelength of a station's broadcast helps place it on the spectrum accurately and ensures it fits within the correct band for communication purposes.