Light as Electromagnetic Waves

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This is a topic from Higher Physics 1B

Contents

Introduction

In 1865, James Clark Maxwell provided a theory which showed the relationship between electric and magnetic phenomena, from which he derived the idea that light is an electromagnetic wave (a form of energy which propagates independently of a medium, with electric and magnetic components oscillating at right angles to one another and mutually orthogonal to the direction of propagation). The relationship between light and the phenomena of electricity and magnetism can be seen in the following expression which relates the constants of all three:

Screen Shot 2012-09-10 at 11.15.19 AM.png

Where μo represents the permeability of free space

εo represents the permittivity of free space
c represents the speed of light (approximately 3 x 108 ms-1)

Revision of Mathematical Modelling of Wave Forms

  • Period (T) is the time taken in seconds for the full cycle of a wave form
  • Frequency (f) is the number of whole cycles of a wave form per second
  • Angular frequency is the rotational analog of frequency:
ω = 2πf
Where w represents angular frequency in radians per second (rad/s)
f represents frequency in hertz (Hz)
  • Wavelength (λ) is the length in metres of one cycle of a wave form
  • Angular wave number is given by:
k = 2π/λ
Where k represents angular wave number in radians per metre (rad/m)
λ represents wavelength in metres
  • The above concepts are related in the following equation in which c represents the speed of light:
c = fλ = ω/k

Optics Terminology and EM Wave Properties

  • A ray is a line along which a light wave travels
  • A linearly polarised wave is one in which the electric and magnetic fields are restricted ro being parralel to a pair of perpendicular axes
  • A collection of waves is called a plane wave
  • A surface connecting points of equal phase on all waves is known as a wave front, analogous to an equipotent surface in electrostatics
  • The magnitudes of the electric field (E) and and magnetic field (B) of an EM wave in empty space are related by the expression c = E/B and so c = fλ = ω/k = E/B
  • EM waves obey the superposition principle
  • The electric and magnetic components of an EM wave are modelled mathematically by
E = EmaxCos(kx - wt) = EmaxCos2π(x/λ - t/T)
B = BmaxCos(kx - wt) = BmaxCos2π(x/λ - t/T)
c = E/B [From Maxwell's equations]

Poynting Vectors and Energy

Electromagnetic waves carry energy which they can transfer to objects in their path as they propagate through space. The rate of flow of energy in an EM wave is a vector called the Poynting vector, and is defined as follows:

Screen Shot 2012-09-10 at 12.21.50 PM.png

Where S represents the Poynting vector, and its magnitude is reflective of the rate at which energy flows through a unit surface area perpendicular to the direction of wave propagation, in (J/s)/m2 = W/m2


Because c = E/B, for the case where E and B are perpendicular such that the crossproduct E x B = EB the equation can be written as:

S = EB / μo = E2 / μoc = cB2 / μo

The direction of the Poynting vector is seen in the following diagram:

Screen Shot 2012-09-10 at 12.22.37 PM.png

Intensity

The wave intensity (I) is the time average of S (the Poynting vector) over at least one cycle.

  • Consider the product EB = EmaxBmaxCos2(kx - wt)
  • The time average of Cos2(kx - wt) is 1/2
  • Thus the intensity is given as follows:
I = EmaxBmax / 2μo

and is subject to the same iterations using c / E/B as the Poynting vector equations

Energy Density

Energy density (u) is the energy per unit volume.

  • Previously in the course we learned that for electric and magnetic fields, uE = εoE2/2 and uB = B2/2μo
  • For EM waves however the electric and magnetic fields are related by c = E/B and c = (μoεo)-1/2
  • Thus the magnetic and electric energy densities are equal:
Screen Shot 2012-09-10 at 1.15.39 PM.png
  • Thus the total instantaneous energy density in an EM wave is the sum of the component energy densities:
u = uE + uB = εoE2 = B2 / μo
  • The average of the energy density over at least one cycle of the wave form can be calculated by taking the average of the component fields, and thus the following equation can be formed to relate the Poynting vector, the intensity and the energy density:
I = Savg = cuavg

Momentum and Pressure

While light can be modelled in terms of waves, we also know that it is made up of particles or 'packets' of energy known as photons, and as such have mass, and momentum.

  • Momentum has the unit kgms-1 and so given that the unit of energy (the joule) equates to kgm2s-2, momentum for light can be expressed as follows:
p = U / c

Where p represents momentum in kgms-1

U represents energy transported to a surface in time interval Δt for a perfectly absorbing surface
c represents the speed of light
  • When this momentum is absorbed by a surface, pressure is exerted on the surface which can be related to momentum:
P = F/A = (1/A)(dp/dt) = (1/cA)(dU/dt)
  • The rate at which energy is transferred to a perfectly absorbing surface would by definition be given by (dU/dt)/A, which is incidentally the definition of the magnitude of the Poynting vector, and so:
P = S / c

Where P represents pressure in Pascales (Pa) or atmospheres (atm)

S represents the Poynting vector in W/m2
c represents the speed of light


As we have derived, for a perfectly absorbing surface, p = U/c and P = S/c. For a perfectly reflecting surface however there an equal amount of momentum in the light striking the surface and the light reflected and so p = 2U/c and it follows that P = 2S/c. In reality surfaces have a reflectivity between these two extremes and so the magnitude of the momentum and pressure of incident light waves is between the two sets of values.

The EM Spectrum

  • EM waves come in all combinations of frequency-wavelength combinations given by c = fλ
  • There are no clear cut divisions between types of EM waves in the spectrum, they can be grouped based on their properties and uses
  • The groups from longest to shortest wavelength (and so smallest to highest frequency) are as follows:
    • Radio waves
    • Microwaves
    • Infrared
    • Visible light
    • Ultraviolet
    • X-ray
    • Gamma ray
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