IIT JEE Physics Formula Revision 40. Electromagentic Waves

Electromagnetic waves formula revision

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.


2. Maxwell generalised Ampere’s law to

∫B.dl = µ₀(i + i_d)


i_d = ε₀*(d Φ_E/dt)

Where

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

3. 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 = -dΦ_B/dt

Ampere’s law

∮B.dl = µ₀(i + i_d)


i_d = ε₀*(d Φ_E/dt)

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

4. 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.


5. Total energy of the electromagnetic wave 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

u_av = ½ ε₀E₀² = B₀²/2µ₀

6. intensity of electromagnetic wave is per unit area per unit time I = U/AΔt = u_avc.

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

I = ½ ε₀E₀²c


7. 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

where U = energy contained in the portion of the wave.