13.4 Energy and momentum of electromagnetic waves

Electromagnetic waves pack a punch, carrying both energy and momentum as they zip through space. They're like invisible couriers, delivering power and exerting pressure on everything they touch. Understanding their behavior is key to grasping how light and radio waves work.

This section dives into the nitty-gritty of electromagnetic wave energy and momentum. We'll explore the Poynting vector, energy density, and radiation pressure. These concepts help explain phenomena from solar sails to comet tails, showing how waves shape our universe.

Energy and Intensity of Electromagnetic Waves

Poynting Vector and Energy Density

• The Poynting vector $\vec{S}$ represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic wave
  • Mathematically defined as the cross product of the electric field $\vec{E}$ and the magnetic field $\vec{H}$: $\vec{S} = \vec{E} \times \vec{H}$
  • Points in the direction of wave propagation and is perpendicular to both the electric and magnetic fields
• Energy density $u$ is the amount of energy stored in the electromagnetic fields per unit volume
  • Consists of two components: electric field energy density $u_E = \frac{1}{2}\epsilon_0E^2$ and magnetic field energy density $u_B = \frac{1}{2\mu_0}B^2$
  • The total energy density is the sum of these two components: $u = u_E + u_B = \frac{1}{2}\epsilon_0E^2 + \frac{1}{2\mu_0}B^2$

Energy Flux and Intensity

• Energy flux, also known as the Poynting vector magnitude $|\vec{S}|$, quantifies the rate at which energy flows through a unit area perpendicular to the direction of propagation
  • For a plane electromagnetic wave, the energy flux is given by $|\vec{S}| = \frac{1}{\mu_0}EB$, where $E$ and $B$ are the amplitudes of the electric and magnetic fields, respectively
• Intensity $I$ is the time-averaged energy flux of an electromagnetic wave
  • Defined as the average of the Poynting vector magnitude over one period of oscillation: $I = \langle |\vec{S}| \rangle = \frac{1}{2}\sqrt{\frac{\epsilon_0}{\mu_0}}E_0^2$, where $E_0$ is the peak amplitude of the electric field
  • Intensity decreases with the square of the distance from the source (inverse-square law) in the far-field region

Power Transmitted by Electromagnetic Waves

• The power $P$ transmitted by an electromagnetic wave through a surface is the integral of the Poynting vector over that surface: $P = \int \vec{S} \cdot d\vec{A}$
  • For a uniform plane wave propagating through a surface of area $A$ perpendicular to the direction of propagation, the power is simply the product of intensity and area: $P = IA$
• The power radiated by an isotropic point source (radiating equally in all directions) can be found using the intensity at a distance $r$: $P = 4\pi r^2 I$
  • This relation is useful for determining the power output of electromagnetic sources like antennas or light bulbs

Momentum and Pressure of Electromagnetic Waves

Momentum Density and Radiation Pressure

• Electromagnetic waves carry momentum, with the momentum density $\vec{g}$ given by the Poynting vector divided by the square of the speed of light: $\vec{g} = \frac{\vec{S}}{c^2}$
  • The direction of the momentum density is the same as the direction of wave propagation
• Radiation pressure $p$ is the force per unit area exerted by an electromagnetic wave on a surface that absorbs or reflects the wave
  • For a plane wave normally incident on a perfectly absorbing surface, the radiation pressure is equal to the energy density of the wave: $p = u = \frac{1}{2}\epsilon_0E^2 + \frac{1}{2\mu_0}B^2$
  • For a perfectly reflecting surface, the radiation pressure is twice the energy density: $p = 2u$
• Examples of radiation pressure include:
  • Solar sails, which use the radiation pressure from sunlight to propel spacecraft
  • Comet tails, which are formed by the radiation pressure from the Sun pushing dust particles away from the comet nucleus