AcademyElectromagnetic Waves
Academy
Sinusoidal Electromagnetic Waves
Level 1 - Physics topic page in Electromagnetic Waves.
Principle
A sinusoidal electromagnetic wave has electric and magnetic fields that oscillate together with the same phase, frequency, wavelength, and propagation speed.
Notation
\(E_0\)
electric-field amplitude
\(\mathrm{V\,m^{-1}}\)
\(B_0\)
magnetic-field amplitude
\(\mathrm{T}\)
\(\phi\)
phase
\(\mathrm{rad}\)
\(T\)
period
\(\mathrm{s}\)
\(\omega\)
angular frequency
\(\mathrm{rad\,s^{-1}}\)
Method
Derivation 1: Field forms
For a wave moving in the \(+x\) direction, choose \(\\vec E\) along \(y\) and \(\\vec B\) along \(z\).
Electric field
\[E_y(x,t)=E_0\cos(kx-\omega t+\phi)\]
Magnetic field
\[B_z(x,t)=B_0\cos(kx-\omega t+\phi)\]
Derivation 2: Amplitude relation
The field amplitudes are linked by the wave speed.
Vacuum relation
\[E_0=cB_0\]
Instantaneous relation
\[E=cB\]
Derivation 3: Periodic quantities
Use the same wave relationships as mechanical sinusoidal waves.
Period and frequency
\[T=\frac{1}{f}\]
Angular frequency
\[\omega=2\pi f\]
Wave number
\[k=\frac{2\pi}{\lambda}\]
Rules
Sinusoidal electric field
\[E_y=E_0\cos(kx-\omega t+\phi)\]
Sinusoidal magnetic field
\[B_z=B_0\cos(kx-\omega t+\phi)\]
Amplitude ratio
\[\frac{E_0}{B_0}=c\]
Wave speed
\[c=\lambda f=\frac{\omega}{k}\]
Examples
Question
A sinusoidal wave has
\[\lambda=600\,\mathrm{nm}\]
Find its frequency in vacuum.Answer
\[f=\frac{c}{\lambda}=\frac{3.0\times10^8}{600\times10^{-9}}=5.0\times10^{14}\,\mathrm{Hz}\]
Checks
- The electric and magnetic fields reach maxima and zeros together in a traveling sinusoidal plane wave.
- A negative sign in \(kx-\\omega t\) means travel in the \(+x\) direction.
- A positive sign in \(kx+\\omega t\) means travel in the \(-x\) direction.
- Use radians when working with \(k\), \(\\omega\), and phase.