Question

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

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

Step-by-step solution

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)

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 \(10^9\): $$ f = 2.0 \times 10^9\mathrm{Hz} $$

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: $$ \lambda = 1.5 \times 10^{-1}\mathrm{m} $$

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.