Problem — Wavelength and FM antenna

Problem

An FM radio station broadcasts at frequency $f = 100\;\text{MHz}$. Calculate the wavelength of the signal in vacuum and the typical length of a dipole antenna, equal to $\lambda/2$.

Solution

Setting up. In vacuum, wavelength and frequency are related by the speed of light:

$\lambda = \frac{c}{f}$

with $f = 100\;\text{MHz} = 1{,}00\cdot 10^8\;\text{Hz}$.

Calculating the wavelength.

$\lambda = \frac{3{,}00\cdot 10^8}{1{,}00\cdot 10^8} = 3{,}0\;\text{m}$

Length of the dipole antenna ($\lambda/2$):

$\ell = \frac{\lambda}{2} = \frac{3{,}0}{2} = 1{,}5\;\text{m}$

$\ev{\lambda = 3{,}0\;\mathrm{m},\quad \ell = \lambda/2 = 1{,}5\;\mathrm{m}}$

An antenna about a metre and a half long is indeed typical for FM reception: its size is comparable with the wavelength, a necessary condition for efficient radiation/reception.

Connections

Topics: Electromagnetic waves Concepts: Electromagnetic wave