Question

A radio can tune to any station in the 7. \(5 \mathrm{MHz}\) to \(12 \mathrm{MHz}\) band. What is the corresponding wave length band? (A) \(400 \mathrm{~m}-250 \mathrm{~m}\) (B) \(40 \mathrm{~m}-25 \mathrm{~m}\) (C) \(4 \mathrm{~m}-2.5 \mathrm{~m}\) (D) None of these

Short answer

The corresponding wavelength band for the given radio frequency range of 7.5 MHz to 12 MHz is 40 m - 25 m. Thus, the correct answer is option (B).

Step-by-step solution

Convert frequencies to hertz

To begin, we convert the given frequencies from MHz to Hz: \(7.5~MHz = 7.5 \times 10^6~Hz\) \(12~MHz = 12 \times 10^6~Hz\) The given frequency range is now 7.5 x 10^6 Hz to 12 x 10^6 Hz.

Find the lower limit of the wavelength band

Using the given formula, we find the longest wavelength corresponding to the lowest frequency: \(\lambda_1 = \frac{c}{f_1}\) \(\lambda_1 = \frac{3 \times 10^8~m/s}{7.5 \times 10^6~Hz}\) \(\lambda_1 = 40~m\) So, the lower limit of the wavelength band is 40 meters.

Find the upper limit of the wavelength band

Next, we find the shortest wavelength corresponding to the highest frequency: \(\lambda_2 = \frac{c}{f_2}\) \(\lambda_2 = \frac{3 \times 10^8~m/s}{12 \times 10^6~Hz}\) \(\lambda_2 = 25~m\) So, the upper limit of the wavelength band is 25 meters. The corresponding wavelength band for the given radio frequency is 40 m - 25 m, which corresponds to option (B).

Key concepts

Wavelength Calculation

When we talk about wavelength calculation, we are looking to find the distance between repeating units of a wave, often measured in meters. This is important in understanding how waves such as radio waves travel through space. To calculate the wavelength, we use a simple formula:\[\lambda = \frac{c}{f}\]where:

• \(\lambda\) (lambda) is the wavelength, measured in meters.
• \(c\) is the speed of light in vacuum, approximately \(3 \times 10^8\) meters per second.
• \(f\) is the frequency of the wave, measured in hertz (Hz).

Once you know the frequency of a wave, plugging that number into the formula gives you the corresponding wavelength. In the case of our radio frequency range, calculating the wavelength helps define the extent of the electromagnetic reach.

Frequency to Wavelength Conversion

Converting frequency to wavelength is a common task in physics, especially when dealing with the electromagnetic spectrum. The frequency of a wave is how often the wave passes a point in a certain period, typically one second. To convert frequency (\(f\)) to wavelength (\(\lambda\)), we utilize the same fundamental formula:\[\lambda = \frac{c}{f}\]

Why Convert Frequency to Wavelength?

• Wavelength provides a more visual understanding of the size of the wave.
• It is crucial for understanding wave interference, diffraction, and other wave behaviors.

For example, with a frequency of 7.5 MHz (megahertz), converted to 7.5 million hertz, the wavelength is found to be 40 meters. Similarly, 12 MHz translates to 25 meters. This conversion is crucial in tuning devices like radios to capture and use specific waves effectively.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation. Radiation is energy that travels and spreads out as it goes. Radio waves, which were central to our exercise, are just one type of electromagnetic radiation.

Understanding the Spectrum

The spectrum ranges from low energy waves, like radio waves, to high energy waves, like gamma rays. Each type of radiation has a different wavelength.

• Radio Waves: These have the longest wavelengths, which can be longer than a football field or as short as a piece of paper.
• Microwaves: Commonly used for cooking and certain communications.
• Visible Light: The only part of the spectrum that we can see with the naked eye.
• X-Rays and Gamma Rays: These are high-energy waves, often used in medicine and various technological applications.

Radio waves, with their long wavelengths, can travel long distances and penetrate through various materials, making them ideal for communication. Understanding where radio frequencies fall in the electromagnetic spectrum helps in designing and understanding devices like radios and mobile phones.