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Chapter 29: Problem 36

You're engineering a new cell phone, and you'd like to incorporate the antenna entirely within the phone, which is \(9 \mathrm{cm}\) long when closed. The antenna is to be a quarter-wavelength long-a common design for vertically oriented antennas. If the cell-phone frequency is \(2.4 \mathrm{GHz}\), will the antenna fit?

Short Answer

Expert verified
Yes, the antenna will fit inside the phone comfortably as it is approximately 3.125 cm, which is less than the 9 cm length of the phone.

Step by step solution

01

Define known quantities

The frequency \(F\) of the cell-phone is known and is equal to \(2.4 \mathrm{GHz}\) or \(2.4 \times 10^9 \mathrm{Hz}\). The maximum possible length for the antenna is given as \(\frac{\lambda}{4}\), where \(\lambda\) is the wavelength. The length should be equal to or less than \(9 \mathrm{cm}\) or \(0.09 \mathrm{m}\). The speed of light \(c\) is \(3 \times 10^8 \mathrm{ms^{-1}}\). Remember the relation between the speed of light, the frequency, and the wavelength - \(c = \lambda F\).
02

Calculate the Wavelength

From the formula \(c = \lambda F\), rearrange it to calculate the wavelength, we get \(\lambda = \frac{c}{F}\). Substituting the known values, we get \(\lambda = \frac{3 \times 10^8}{2.4 \times 10^9}\) which simplifies to approximately \(\lambda = 0.125 \mathrm{m}\).
03

Determine if the antenna will fit

Now we check if the quarter-wavelength fits inside the phone. We calculate this by \(\frac{\lambda}{4}\), which gives us \(\frac{0.125}{4} = 0.03125 \mathrm{m}\) or \(3.125 \mathrm{cm}\). Comparing with the length of the phone, we can see that \(3.125 \mathrm{cm}\) is less than \(9 \mathrm{cm}\). Thus, the antenna will fit inside the phone.

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