Poynting vector and radiation

Energy in electromagnetic fields

Electric and magnetic fields store energy. The electric field energy density in vacuum is

u_E=rac{1}{2}epsilon_0E^2

The magnetic field energy density is

u_B=rac{B^2}{2mu_0}

The total electromagnetic energy density is

u=rac{1}{2}epsilon_0E^2+rac{B^2}{2mu_0}

For an electromagnetic plane wave in vacuum, the electric and magnetic contributions are equal on average and instantaneously.

The Poynting vector

The Poynting vector describes electromagnetic energy flow:

ec{S}=rac{1}{mu_0}ec{E} imesec{B}

Its direction is the direction energy travels. Its magnitude has units of power per area, or intensity:

$W/m^2$

For a plane wave, ec{E} and ec{B} are perpendicular, and ec{S} points in the propagation direction.

Intensity of an electromagnetic wave

The instantaneous energy flow varies in time for a sinusoidal wave. The average intensity is the time average of the Poynting vector magnitude:

angle$$ For a sinusoidal plane wave in vacuum, $$I=rac{1}{2}cepsilon_0E_0^2$$ Using $E_0=cB_0$, this can also be written in terms of magnetic field amplitude: $$I=rac{1}{2}rac{c}{mu_0}B_0^2$$ ## Radiation pressure Electromagnetic waves carry momentum as well as energy. When light is absorbed or reflected by a surface, it exerts pressure. For complete absorption at normal incidence, $$p_{rad}=rac{I}{c}$$ For perfect reflection, $$p_{rad}=rac{2I}{c}$$ Radiation pressure is usually small in everyday situations but important in astronomy, laser physics, optical tweezers, and solar sails. ## Point source spreading If a source radiates power $P$ uniformly in all directions, intensity at distance $r$ is $$I=rac{P}{4pi r^2}$$ This inverse-square law follows from energy spreading over the surface of a sphere. Real antennas and light sources may radiate directionally, producing different angular intensity patterns. ## Accelerating charges radiate Electromagnetic radiation is produced by accelerating charges. A charge oscillating in an antenna emits radio waves. Electrons changing motion in atoms emit light. Thermal motion of charges in matter produces thermal radiation. A charge moving at constant velocity does not radiate in the same way; acceleration is essential. ## Antennas An antenna converts electrical signals into electromagnetic waves and vice versa. In a transmitting antenna, charges oscillate, producing time-varying fields that propagate outward. In a receiving antenna, an incoming electromagnetic wave drives charges, inducing an electrical signal. Antenna size is often related to wavelength. Efficient antennas commonly have dimensions comparable to a significant fraction of $lambda$. ## Conservation of electromagnetic energy Energy conservation for fields and charges is expressed by Poynting's theorem. In simplified meaning, it says that decrease of field energy in a region equals energy flowing out plus work done on charges. The Poynting vector is the field energy flux. This field-energy view is essential in understanding circuits too: energy often flows through electromagnetic fields around wires, not inside electrons as little packets of stored energy. ## The big idea Electromagnetic fields carry energy, momentum, and power flow. The Poynting vector $ec{S}=ec{E} imesec{B}/mu_0$ gives the direction and rate of energy transport. Radiation pressure, inverse-square spreading, antennas, and light intensity all follow from the reality of electromagnetic field energy.