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 + 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.
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
∮E.dl = -dΦB/dt
∮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)
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
uav = ½ ε0E0² = B0²/2µ0
6. intensity of electromagnetic wave is per unit area per unit time I = U/AΔt = uavc.
In terms of maximum electric field (substituting the value of uavc)
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.