Chapter 32: Electromagnetic Waves

Speed of electromagnetic waves in a vacuum

c=1/√(ε₀µ₀)

Sinusoidal EM plane wave, propagating in the +x-direction

E(vec)(x,t)=j-hat*Emax*cos(kx-ωt) B(vec)(x,t)=k-hat*Bmax*cos(kx-ωt) k is the wave number, ω is the angular frequency Emax=cBmax here

Wave form equations for standing EM waves

E(x,t)=-2Emax*sinkx*sinωt B(x,t)=-2Bmax*coskx*cosωt

Electromagnetic wave in a vacuum

E=cB Magnitude of the electric field is equal to the magnitude of the magnetic-field times the speed of light in a vacuum.

Electromagnetic spectrum

Encompasses electromagnetic waves of all frequencies and wavelengths.

Maxwell's equations

Gauss's law: (closed surface)∫E(vec) dot dA(vec)=q/ε₀ Gauss's law for magnetism: (closed surface)∫B(vec) dot dA(vec)=0 Faraday's law: (closed surface)∫E(vec) dot dl(vec)=-dΦ/dt Ampere's law with displacement current: (closed surface)∫B(vec) dot dl(vec)=µ₀(i+ε₀dΦ/dt)

Intensity of a sinusoidal EM wave in vacuum

I=Sav=Emax*Bmax/2µ₀=Emax²/2µ₀c=½√(ε₀/µ₀)*Emax²=½ε₀cEmax²

Intensity

Magnitude of the average value of S(vec) at a point, with the same unit of intensity W/m² (power over area)

Nodal planes

Planes which are the equivalent of the nodes of a standing wave on a string. The positions depend on the equations, using k=2π/λ, and these occur when sinkx=0 for E

Index of refraction

Ratio of speed c in a vacuum to the speed v in material, represented with n=c/v=√(KKm)

Poynting vector in a vacuum

S(vec)=E(vec)xB(vec)*1/µ₀ This is the vector quantity that describes both the magnitude and direction of the energy flow rate (same as direction of propagation)

Antinodal planes

The planes where the magnitude of the wave is the maximum possible value. The positions depend on the equations, using k=2π/λ, and these occur when sinkx=±1 for E

Standing wave

The superposition of an incident wave and a reflected wave (a reflected wave occuring due to the surface of a conductor or a dielectric)

Electromagnetic radiation

Used interchangeably with electromagnetic waves, because it is also a disturbance.

Electromagnetic wave

When either an electric or magnetic field is changing with time, a field of the other kind is induced in adjacent regions of space. There is an electromagnetic disturbance, consisting of time-varying electric and magnetic fields, that can propagate through space from one region to another, even when there is no matter in the intervening region. This disturbance will have the properties of a wave.

Transverse

When the wave is perpendicular to the direction or propagation (both E and B)

Speed of light

c=3*10⁸ m/s

Key properties of EM waves

1. The wave is transverse, both E and B are perpendicular to the direction of propagation, which is the direction of E x B 2. There is a definite ratio between E and B: E=cB 3. The wave travels in vacuum with a definite and unchanging speed. 4. Unlike mechanical waves, which need the particles of a medium such as air to transmit a wave, electromagnetic waves require no medium.

Flow rate of EM momentum

1/A*dp/dt=S/c=EB/µ₀c A is the area in m², dp/dt is the momentum transferred per unit time.

Polarization

A property of EM waves.

Linearly polarized

A wave that is always parallel to a certain axis. (e.g. any wave travelling in the x-direction can be represented as a superposition of waves linearly polarized in the y and z directions.

Magnetic-field magnitude in a vacuum

B=ε₀µ₀cE

What is the relation in direction between magnetic and electric field waves?

Considering the direction of propagation to be on the x-axis, B is on the y-axis, and E on the z-axis. According to Faraday's law, in an electromagnetic wave, E and B must be mutually perpendicular.

Speed of EM waves in a dielectric

v=1/√(εµ)=1/√(KKm)*1/√(ε₀µ₀)=c/√(KKm) ε is permitivity and µ is permeability, K is the dielectric constant, Km is the relative permeability, ε₀ is the electric constant, µ₀ is the magnetic constant.

Wavelength-frequency relationship

v=λƒ, with v being the velocity, and this being the speed of light.