Question

The antenna of a cell phone is a straight rod \(8.0 \mathrm{~cm}\) long. Calculate the operating frequency of the signal from this phone, assuming that the antenna length is \(\frac{1}{4}\) of the wavelength of the signal.

Short answer

Answer: The operating frequency of the cell phone is \(9.375 \times 10^8 \mathrm{~Hz}\).

Step-by-step solution

Determine the length of the wavelength

To find the full length of the wavelength, we can use the information given: \(\frac{1}{4} \mathrm{wavelength} = 8.0 \mathrm{~cm}\) Now, we can find the full wavelength by multiplying both sides of the equation by 4: \(\mathrm{wavelength} = 4 * 8.0 \mathrm{~cm} = 32 \mathrm{~cm}\)

Convert wavelength to meters

As we know that the speed of light is measured in meters per second (\(\mathrm{m/s}\)), we need to convert the wavelength to meters: \(\mathrm{wavelength} = 32 \mathrm{~cm} \frac{1 \mathrm{m}}{100 \mathrm{cm}} = 0.32 \mathrm{~m}\)

Use the speed of light to find the frequency

The speed of light (\(c\)) is approximately equal to \(3.0 \times 10^8 \mathrm{~m/s}\). The formula relating frequency, wavelength, and speed of light is: \(c = \mathrm{frequency} \times \mathrm{wavelength}\) We can rearrange the formula to find the frequency: \(\mathrm{frequency} = \frac{c}{\mathrm{wavelength}}\) Now, we can plug in the value for the speed of light and the wavelength: \(\mathrm{frequency} = \frac{3.0 \times 10^8 \mathrm{~m/s}}{0.32 \mathrm{~m}}\)

Calculate the frequency

Now we can calculate the frequency: \(\mathrm{frequency} = \frac{3.0 \times 10^8 \mathrm{~m/s}}{0.32 \mathrm{~m}} = 9.375 \times 10^8 \mathrm{~Hz}\) So, the operating frequency of the cell phone with the given antenna length is \(9.375 \times 10^8 \mathrm{~Hz}\).

Key concepts

Wavelength to Frequency Conversion

Understanding how to convert wavelength to frequency is essential when studying electromagnetic waves, as seen in the problem about cell phone antennas. Wavelength (\r\rightarrow\( \lambda \)) and frequency (\r\rightarrow\( f \)) are inversely related to each other through the speed of light (\r\rightarrow\( c \)), which can be expressed by the formula:\r\r\[ c = \lambda \times f \]\r\rThis means that if you know the wavelength and the speed of light, which is constant in a vacuum, you can determine the frequency with the rearranged formula:\r\r\[ f = \frac{c}{\lambda} \]\r\rIn our example, knowing that the antenna length is a quarter of the wavelength allows us to determine the full wavelength. By converting this into meters, the standard unit of measurement in physics, we are then set up to use the speed of light to find the frequency. It's a crucial skill in physics problem solving to be able to rearrange and apply formulas to given data.

Speed of Light

The speed of light (\r\rightarrow\( c \)) is a fundamental constant in physics, valued at approximately \( 3.0 \times 10^8 \) meters per second (m/s) in a vacuum. This constant is not only crucial for our understanding of the universe but also plays a pivotal role in practical applications such as calculating the operating frequency of cell phones based on their antenna length. The speed of light connects time and space and is the maximum speed at which all conventional matter and information in the universe can travel. Its role in the wavelength to frequency conversion offers a perfect example of how universal constants are applied in everyday technology.

Electromagnetic Spectrum

The electromagnetic spectrum portrays the range of all possible frequencies of electromagnetic radiation, from radio waves to gamma rays. Cell phone signals are a part of this spectrum and usually fall under the category of radio waves or microwaves. These signals possess wavelengths longer than infrared light, making them ideal for wireless communication. Understanding where a cell phone signal falls on the electromagnetic spectrum is important, as each part of the spectrum has its own characteristics and uses in technology. For instance, the frequency calculated for our cell phone antenna problem indicates the type of electromagnetic waves used and can help understand the strengths and limitations of these waves for communication purposes.

Physics Problem Solving

Effective physics problem solving often involves a step-by-step approach, much like the method used for calculating the operating frequency of a cell phone. It starts with identifying the given data and determining what needs to be found. In our textbook problem, the goal was to calculate frequency given the antenna length and knowing it corresponds to a quarter of the wavelength. The next step involves recalling relevant formulas and principles, such as the speed of light and its relationship to wavelength and frequency. Finally, calculations are performed, often converting units for consistency, and the formula is manipulated accordingly to yield the required result. This systematic approach not only simplifies complex problems but also enhances comprehension, making learning more effective for students.