Free space or non-conducting or lossless or in general perfect dielectric medium has following characteristics:

(a)    No condition current i.e σ=0,thus J=0( J=σE)

(b)   No charges (i.e ρ=0)

Therefore for the above cases, Maxwell’s equations will become

∇.D=0 or ∇.E=0                   (ρ=0) (1(a))

∇.B=0 or ∇.E=0                              (1(b))

∇  x E= -dB/dt or ∇ x E= -μdH/dt          (1(c))

∇ x H=d D/dt or ∇ x H = εdE/dt         (J=0)(1(d))

Now taking curl of third Maxwell’s equation (1 c) ,we get

∇ x(∇ xE)=- μd/dt (∇ x H)

Applying standard vector identity ,that is [∇ *(∇*E)=∇(∇.E)-∇²E] on left hand side of above equation, we get

∇ (∇ .E)-∇²E= -μd/dt (∇*H)                                                        (2)

Substituting equations (1a) and (1d) in equations (2) ,we get

-∇²E= – μεd/dt (dE/dt)

Or                               ∇²E=με d ² E/dT² (3)

Equation (5) is the required wave equation in terms of electric field intensity , E for free space . This is the law that E must obey.

WAVE EQUATION IN TERMS OF MAGNETIC FIELD INTENSITY,H

Take curl of fourth Maxwell’s equation(1d) ,we get

∇*(∇*H)=ε d/dt(∇*E)

Applying standard vector identity that is

[∇*(∇*H)=∇ (∇.H)-∇²H]

On left side of above equation ,we get

∇(∇.H)-∇²H= ε d/dt(∇*E)                                          (4)

Substituting equations (1b) and (1c) in equation(4) ,we get

-∇²H= – μεd/dt(dH/dt)

Or                       ∇²H=με d²E/dt² (5)

Equations (5) is the required wave equation in terms of magnetic field intensity, H and this is the law that H must obey

If μ=μ₀ and ε=ε₀, equations(3) and (5) will become

∇² E=μ₀ε₀ d²E/dt² (6(a))

And                         ∇²H=    μ₀ε₀ d²H/dt² (6(b))

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