Electromagnetic waves

1. The wave equation for light propagating in x-direction in vacuum may be written as
 E = E0 sin ω(t-x/c)
Where E is the sinusoidally varying electric field at the position x at time t.
c is the speed of light in vacuum.
The electric field is in the Y-Z plane. It is perpendicular to the direction of propagation of the wave.
There is a sinusoidally varying magnetic field associated with the electric field when light is propagating. This magnetic field is perpendicular to the direction of wave propagation and the electric field E.
B = B0 sin ω(t-x/c)
Such a combination of mutually perpendicular electric and magnetic fields constitute an electromagnetic wave in vacuum.

Maxwell developed the theory of electromagnetic wave.
 Maxwell generalised Ampere’s law to

B.dl = µ0(i + id)

id = ε0*(d ΦE/dt)


ΦE/ = the flux of the electric field through the area bounded by the closed curve along which the circulation of B is calculated.

Maxwell termed id as displacement current

i  is current due to flow of charges is often called conduction current.

Ampere’s law without Maxwell’s modification is not valid if the electric field at the surface varies with time. The book gives the example of the region between capacitor plates to prove this point.

Continuity of electric current

Consider closed surface enclosing a volume a conductor in the shape of a cylinder. If all the charge entering it is leaving it, the conduction is continuous. But if some charge is accumulated inside the volume, this continuity breaks.

However, it we consider both conduction current as well as displacement current, the total current is still continuous.

Maxwell’s Equations

Gauss’s laws for electricity and magnetism, Faraday’s law and Ampere’s are collectively known as Maxwell’s equations

Gauss’s law of electricity

E.ds = q/ ε0

Gauss’s law for magnetism

B.ds = 0

Faraday’s law

E.dl = -dΦB/dt

Ampere’s law

B.dl = µ0(i + id)

id = ε0*(d ΦE/dt)

These equations are satisfied by a plane electromagnetic wave given by
Ey = E = E0 sin ω(t-x/c)
Bz = B = B0 sin ω(t-x/c)

In proving these equations we come to the formula that

c = i/√(µ0ε0)
the value calculated from this expression comes out to be 2.99793*10^8 m/s which was same as the experimentally measured value of speed of light in vacuum. This also provides a confirmatory proof that light is an electromagnetic wave.

Energy Density and Intensity 
For the waves described by

Ey = E = E0 sin ω(t-x/c)
Bz = B = B0 sin ω(t-x/c)

In any small volume dV, the energy of the electric field is

UE = ½ ε0E²dV

And the energy of the magnetic field is
UB = B²dV/2µ0

Total energy is

U = ½ ε0E²dV + B²dV/2µ0

When we substitute the values of E and B in the above equation and take an average over a longer period of time

uav = ½ ε0E0² = B0²/2µ0


The energy crossing per unit area per unit time perpendicular to the direction of propagation is called the intensity of a wave.

Consider a cylindrical volume with area of cross section A and length cΔt along the X-axis (the direction of propagation of the light wave).

The energy contained in this cylinder U = uav(cΔt)A

As intensity if per unit area per unit time I = U/AΔt = uavc.

In terms of maximum electric field (substituting the value of uavc)

I = ½ ε0E0²c


The electromagnetic wave also carries linear momentum with it. The linear momentum carried by the portion of wave having energy U is given by

p = U/c

If the wave incident on a material surface is completely absorbed, it delivers energy U and momentum p = U/c to the surface. Hence electromagnetic waves incident on a surface exert a force on the surface.

Electromagnetic Spectrum

Maxwell’s equations are applicable for electromagneticwaves of all wavelengths.

Basic source of electromagnetic waves is an accelerated charge.

Radio waves (used in radio and TV communication) are produced by charges accelerating in AC circuits having an inductor and capacitor.

Microwaves ( used for radar communication and in cooking) are also produced by such electric circuits with oscillating current.

Infrared waves (used in physical therapy) are produced by the atoms and molecules of hot bodies.

Visible light is also emitted by atoms under suitable conditions.

Ultraviolet radiation is emitted by atoms due to acceleration of electrons in the atom (the mechanism is same for visible light emission also).

The sun emits large amount of UV radiation. UV radiation is harmful to humans if absorbed in large amount.

X-rays (used in medical diagnosis) are produced when fast moving electrons collide with a metal target and decelerate. They are also harmful to living tissues.

Gamma rays are radioactive emissions from nuclei and have the shorted wavelengths among electromagnetic waves.

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