Maxwell's Equations

From the experimental results on the search of Electromagnetism Maxwell had ultimately been able to create 4 golden rules of Electromagnetism.
They are :-

(i) ∇.E = ρ/ϵ 

(ii) ∇.B = 0 

(iii) ∇×E = -∂B/ ∂t 

(iv) ∇×B = μ(J + ϵ.∂E/ ∂t) 

E and B are electric and magnetic fields respectively at a point and ϵ,μ are electric and magnetic permittivity respectively. J is the electric current density. 

Notice that there is no mention of the reference frame in the above equations because these are true for any inertial reference frame. 

Now in free space J = 0 and ρ = 0 . So 

∇×B = μ₀ϵ₀.∂E/ ∂t  

=> ∇×∇×E = -  μ₀ϵ₀.∂²E/ ∂²t 

=> ∇(∇.E) - ∇²E   = - (1/c²) ∂²E/ ∂²t  

=> ∇²E - (1/c²) ∂²E/ ∂²t = 0 [ As ρ = 0 so from (i) ∇.E = 0 ]

=>( ∇² - 1/c² . ∂²/ ∂²t ) E = 0 

Similarly we can get ( ∇² - 1/c² . ∂²/ ∂²t ) B = 0 

For plane wave along +ve x axis E(x,t) = f(x - ct).