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### **Introduction** * **Electromagnetic waves** are disturbances that propagate through space by the interaction of electric and magnetic fields. * They can transport energy, momentum and angular momentum away from a source. * Electromagnetic waves include radio waves, microwaves, infrared,...

### **Introduction** * **Electromagnetic waves** are disturbances that propagate through space by the interaction of electric and magnetic fields. * They can transport energy, momentum and angular momentum away from a source. * Electromagnetic waves include radio waves, microwaves, infrared, light, ultraviolet, X-rays, and gamma rays. ### **Properties of Electromagnetic Waves** * Electromagnetic waves have the following properties: * They are transverse waves (i.e., the electric and magnetic fields are perpendicular to the direction of propagation). * They travel at the speed of light, $c = 2.998 \times 10^8 m/s$ in a vacuum. * They do not require a medium to propagate. * They can be polarized (i.e., the electric field can be oriented in a specific direction). ### **Mathematical Description** * A plane electromagnetic wave traveling in the z-direction can be described by the following equations: $\qquad \vec{E}(z,t) = E_0 \cos(kz - \omega t) \hat{x}$ $\qquad \vec{B}(z,t) = B_0 \cos(kz - \omega t) \hat{y}$ * Where: * $\vec{E}$ is the electric field * $\vec{B}$ is the magnetic field * $E_0$ is the amplitude of the electric field * $B_0$ is the amplitude of the magnetic field * $k$ is the wave number, $k = \frac{2\pi}{\lambda}$ * $\omega$ is the angular frequency, $\omega = 2\pi f$ * $\lambda$ is the wavelength * $f$ is the frequency * The electric and magnetic fields are related by: $\qquad E_0 = cB_0$ * The speed of light, frequency, and wavelength are related by: $\qquad c = f\lambda$ ### **Electromagnetic Spectrum** * The electromagnetic spectrum is the range of all possible frequencies of electromagnetic radiation. * The electromagnetic spectrum is typically divided into the following regions: | Region | Wavelength (m) | Frequency (Hz) | | ------------- | --------------- | --------------- | | Radio waves | $> 10^{-1}$ | $< 3 \times 10^9$ | | Microwaves | $10^{-3}-10^{-1}$ | $3 \times 10^{9} - 3 \times 10^{11}$ | | Infrared | $7 \times 10^{-7} - 10^{-3}$ | $3 \times 10^{11} - 4.3 \times 10^{14}$ | | Visible light | $4 \times 10^{-7} - 7 \times 10^{-7}$ | $4.3 \times 10^{14} - 7.5 \times 10^{14}$ | | Ultraviolet | $10^{-8} - 4 \times 10^{-7}$ | $7.5 \times 10^{14} - 3 \times 10^{16}$ | | X-rays | $10^{-11} - 10^{-8}$ | $3 \times 10^{16} - 3 \times 10^{19}$ | | Gamma rays | $< 10^{-11}$ | $> 3 \times 10^{19}$ | ### **Energy and Momentum** * Electromagnetic waves carry energy and momentum. * The energy density (energy per unit volume) of an electromagnetic wave is given by: $\qquad u = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0}B^2$ * Where: * $\epsilon_0$ is the permittivity of free space * $\mu_0$ is the permeability of free space * Since $E = cB$ and $c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$, the energy density can also be written as: $\qquad u = \epsilon_0 E^2 = \frac{B^2}{\mu_0}$ * The average energy density of an electromagnetic wave is given by: $\qquad u_{avg} = \frac{1}{2}\epsilon_0 E_0^2 = \frac{B_0^2}{2\mu_0}$ * The intensity (power per unit area) of an electromagnetic wave is given by: $\qquad I = cu_{avg} = \frac{c\epsilon_0}{2} E_0^2 = \frac{cB_0^2}{2\mu_0}$ * Electromagnetic waves also carry momentum. The momentum density (momentum per unit volume) of an electromagnetic wave is given by: $\qquad \vec{p} = \frac{\vec{S}}{c^2}$ * Where $\vec{S}$ is the Poynting vector, which represents the direction and rate of energy flow of the electromagnetic wave. The Poynting vector is given by: $\qquad \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}$ * The magnitude of the Poynting vector is the intensity of the electromagnetic wave: $\qquad S = \frac{E_0 B_0}{\mu_0}$ * When an electromagnetic wave is absorbed by a surface, it exerts a radiation pressure on the surface. The radiation pressure is given by: $\qquad P_{rad} = \frac{I}{c}$ (if the wave is totally absorbed) $\qquad P_{rad} = \frac{2I}{c}$ (if the wave is totally reflected)