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Chapter 22: Problem 65

The antenna on a cordless phone radiates microwaves at a frequency of $2.0 \mathrm{GHz}$. What is the maximum length of the antenna if it is not to exceed half of a wavelength?

Short Answer

Expert verified
Answer: The maximum length of the antenna should not exceed 7.5 cm.

Step by step solution

01

Identify the equation to find the wavelength

To find the wavelength, we can use the formula: $$ \lambda = \frac{c}{f} $$ Where: - λ is the wavelength - c is the speed of light \((3.0 \times 10^8\mathrm{m/s})\) - f is the frequency (2.0 GHz)
02

Convert the frequency to Hertz

The frequency is given in gigahertz and needs to be converted to hertz. To do this, multiply the given frequency by \(10^9\): $$ f = 2.0 \times 10^9\mathrm{Hz} $$
03

Calculate the wavelength

Now, we can use the formula to find the wavelength: $$ \lambda = \frac{c}{f} = \frac{3.0 \times 10^8\mathrm{m/s}}{2.0 \times 10^9\mathrm{Hz}} $$ Calculate the wavelength: $$ \lambda = 1.5 \times 10^{-1}\mathrm{m} $$
04

Determine the maximum length of the antenna

The maximum length of the antenna should not exceed half a wavelength. So, to find this length, we divide the wavelength by 2: $$ L_{max} = \frac{\lambda}{2} = \frac{1.5 \times 10^{-1}\mathrm{m}}{2} $$ Calculate the maximum length: $$ L_{max} = 7.5 \times 10^{-2}\mathrm{m} $$ The maximum length of the antenna should not exceed \(7.5 \times 10^{-2}\mathrm{m}\) or 7.5 cm.

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Most popular questions from this chapter

In order to study the structure of a crystalline solid, you want to illuminate it with EM radiation whose wavelength is the same as the spacing of the atoms in the crystal \((0.20 \mathrm{nm}) .\) (a) What is the frequency of the EM radiation? (b) In what part of the EM spectrum (radio, visible, etc. \()\) does it lie?
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