# WAVE EQUATION IN FREE SPACE OR LOSSLESS OR NON-CONDUCTING OR PERFECT DIELECTRIC MEDIUM

**WAVE EQUATION IN TERMS OF MAGNETIC FIELD INTENSITY,H**

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)-∇^{2}E] on left hand side of above equation, we get

∇ (∇ .E)-∇^{2}E= -μd/dt (∇*H) (2)

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

-∇^{2}E= – μεd/dt (dE/dt)

Or ∇^{2}E=με d ^{2 }E/dT^{2} (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)-∇^{2}H]

On left side of above equation ,we get

∇(∇.H)-∇^{2}H= ε d/dt(∇*E) (4)

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

-∇^{2}H= – μεd/dt(dH/dt)

Or ∇^{2}H=με d^{2}E/dt^{2} (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=μ_{0} and ε=ε_{0}, equations(3) and (5) will become

∇^{2} E=μ_{0}ε_{0} d^{2}E/dt^{2} (6(a))

And ∇^{2}H= μ_{0}ε_{0 }d^{2}H/dt^{2} (6(b))

_{ }