- Geometrical Optics :Fermat's Principle of Least Time
- Electromagnetic Field
- Electromagnetic Waves
- Interference and Diffraction
- Doppler Effect for Light

Electromagnetic radiation; light

From the special theory of relativity, it is known that the velocity of light is a universal constant. Hence, in describing radiation, we can dispense with frequency of radiation , and use only its wavelength to fully describe its properties. The wavelength of radiation has a vast range, from waves with infinitely long wavelengths to waves of infinitely short wavelength. Starting from long wavelengths, we have radio waves which have a wavelength of about 500 m, followed by short wave radio waves, and so on down to around 1 m wavelength for TV waves. We then have microwave radiation with wavelength around 1 cm, followed by infrared radiation down to m.

(6.2) |

Fermat's Principle provides an answer to the question of propagation of light in the presence of different media. The first guess one might have is that light should take the path that minimizes the

We restate Fermat's principle in a more mathematical form. Let be time
taken for the light ray to go from point to point by taking some path.
Then Fermat's principle states that

(6.4) |

(6.5) |

The relation of the angles made by light in the two media goes under the name of Snell's law.

Instead of plunging into the
details of the derivation, let us illustrate the problem with an
analogy. Since the velocity of light is faster in air than in
glass, an analogy of the propagation of light in air is a man
running. The analogy of light propagating in glass can be taken to
be that of a man swimming. Suppose a
lifeguard standing on the ground
is trying to reach a buoy floating in a swimming pool in the shortest
possible time. If he goes in a straight line, he will spend a
certain fraction of time
running on the ground, and the remaining fraction in swimming.
Since he swims slower than he runs, he should try run for a longer
time on the ground before diving into the pool. On the other hand, if he runs too
far on the ground, he will end up swimming more than he should. The exact point
where he should dive into the pool for minimizing the
time he takes to reach the buoy is the analogy of how light rays should bend as they
leave one medium and enter the other.

(6.6) |

(6.7) | |||

(6.8) |

For Path 1 to be the path of minimum time, Fermat's principle requires that the difference in time taken by light for Paths 1 and 2 should be zero, that is

(6.9) | |||

(6.10) |

From Figure 6.6 we have that

(6.11) | |||

(6.12) |

Hence, from equations above we have the condition for minimum time given by

(6.13) |

(6.14) |

Electromagnetic Waves

Intuitively speaking, if one attaches an electric charge
to the end of a stick, and shakes the stick back and forth with a
frequency , the charge will ``radiate off'' electromagnetic
waves of frequency . The energy in the waves comes from the
energy being expended in shaking the charge. The only reason that the
the radiation that we create by shaking a piece of stick
is not easily observable is because the
amplitude of the radiation is very low, making detection difficult.
A modern antennae is not much different from our intuitive picture, since in an
antennae charge is made to flow back and forth - with some frequency - along
the length of an antennae, and results in the emission of electromagnetic
radiation. Radio waves are generated in this manner, and
constitute the signals that are up picked by a radio receiver.
Electromagnetic radiation is the result of the simultaneous
oscillations of electric and magnetic fields. A changing electric
field creates a magnetic field, and in turn, a changing magnetic
field creates an electric field. It is this positive feedback
mechanism that sustains the propagation of electromagnetic wave over
billions of light years of distance. Since the
electric field and magnetic field are **
three-dimensional vectors**, the propagation of radiation is only
possible in three-dimensions. For simplicity, we study the electric
and magnetic fields far from the charges that have generated the
radiation.
Suppose the electromagnetic wave is propagating in vacuum along
the -direction. Since the electromagnetic radiation is a
transverse wave, the oscillations of the electric and magnetic
field are in the directions perpendicular to the direction of
propagation, and hence lie in the -plane. Let us denote the
unit vector in the -directions by
and
respectively. The simplest example of radiation is
given by the electric field always lying along the
-axis, and the magnetic field always lying along the
-axis. In other words, the and fields for radiation
are similar to the transverse wave given in
eq.(5.22), with the added property that all electromagnetic waves
propagate in vacuum at the speed of light. From eq.(6.1), we have

The propagation of the radiation given by eqs. (6.24) and (6.25) is shown in the Figure 6.8, with the direction of propagation being in the -direction, and the sinusodial oscillations of the and field drawn in the -planes. The energy, per unit volume, of the electric and magnetic fields at point , denoted by , is given by

To obtain the total energy contained in the electromagnetic field, we have to sum the contribution to energy from all the volume of space. To develop a physical sense of the energy of a field, one can imagine breaking up space into volume elements, say into little cubes of side and with elemental volume . In every volume of , one can think of the field in that volume as being a tiny harmonic oscillator that can each carry a tiny amount of energy equal to

(6.19) |

(6.20) |

Energy propagates along the direction of motion of the wave, which
in the case of the electromagnetic wave being considered is in the
-direction. The **intensity** of light is a measure of how
much radiant energy is received at a point, and as can be seen by
the formula above, is proportional to .
To understand the dimensions of the and , recall
is the permittivity constant
defined in eq.(3.98), and is a measure of the strength of electromagnetic
fields in vacuum. In a medium other than vacuum, the permittivity constant of
the medium would appear in the energy equation eq.(6.26). From eqs.(3.97)
and (3.98)
. Since is the
energy per unit volume,
, and hence from eq.(6.26)
we have

(6.21) | |||

(6.22) |

From a simple exercise in dimensional analysis, we have obtained an important result that the electric field is proportional to a force. The SI unit for electric field

From eq.(6.26) it also follows that

(6.24) | |||

(6.25) |

and hence the SI unit of is given by

Recall that an accelerated charge
generate electromagnetic radiation; we see from equation (6.37) that
electric and magnetic in turn react back on charged particles, and
exert a force on them. We will study these effects in some detail
in the chapter on electric and magnetic fields.
The momentum of the electromagnetic field is along its direction of propagation,
and the field momentum per unit volume is given by

(6.27) |

(6.28) | |||

(6.29) |

Although the electric field is in phase at the slits (since light is from a common source), the electric field at from and has a phase difference since light from and have traveled different distances to arrive at . The superposition of waves yields the total electric field at to be

and eq.(6.42) follows from eq.(5.61). The intensity of light received at the screen is given in Figure 6.10. We analyze phase to determine the pattern of intensity shown in Figure 6.10. Note that the phase is a result of the

(6.33) | |||

(6.34) |

Hence, from eq.(6.43)

The intensity at is given by

(6.36) |

Hence, from eqs.(6.46) and (6.48), the maxima's of intensity are given by

(6.38) | |||

(6.39) |

If light waves from and arrive at with a path difference of that is a multiple of wavelength ,then they are

(6.40) |

(6.41) | |||

(6.42) | |||

(6.43) |

Due to laws of special relativity, the velocity of the light source can never exceed velocity of light in vacuum, namely . However, in a medium light travels at a speed less than , and a source can then exceed light's velocity leading to a phenomenon similar to shock waves for sound, and is called Crenenkov radiation.