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Propagation of Electromagnetic Waves in Free Space

For free space              p =0

J =0

B =µ0 H

D =ε0 E

 

Maxwell’s equation for free space.

I                       div  =p

Putting                        D =ε0 E

And                              P=0

 

in above equation we getting

div (ε0 E) =0

ε0 div E=0

 

or                                 div E =0

 

II.                                div B =0

Putting                       B  =µ 0 H

Or                                 div µ0  H =0

µ0  div H =0

div H =0

 

III .                              curl E =- Db /dt

Putting                        B =µ0 H in above relation

Curl E =- d (µ0 Dh/dt)

 

IV .                              curl H =J +Dh /dt

Putting                                    D =ε0 E .J =0

Curl H =0+d (ε0 De/dt)

 

If we take curl of equation (3), we get

 

Curl (curl E ) = -µ0 d/dt (curl H)

 

From equation (4), we get

[Gradient (div E ) -∆2 E] =- µ0ε0 d2 E/dt2

 

From equation (1)                   div E =0

So the equation (4) become

2 E =µ 0 ε0 d2 E/dt2

 

                                E  = EX+ EY +EZ

Then L.H.S is

The wave equation is

 

D2y/dx2=1/u2 (d2y/dt2)

 

where v = velocity of wave

x = path of propagation wave

 

On comprising the equation (8) and (9), we get

 

V2 1/µ0 ε0 and v =1  √µ0 ε0

 

Since a time varying electric field travels in free space with the velocity

 

v =1  √µ0 ε0

for magnetic field

 

µ0 = 4π x 10- 7 Henry I farad

ε0= 8.8542 x 10-12 Farad/met

v = 2.9999 x 108 met/sec.

 

Similarly taking curl of equation (4), we getting

 

curl (curl H) = ε0· d/ dt curl E [From equation (3)]

 

 

(grad (div H) =∆2 H ) = – ε0 µ0 d2 H /dt2

 

H = HX +HY +HZ

2 H =µ0 ε0 d2H /dt2

The wave equation is

 

D2y/dx2 = – 1/u2 d2y/dt2

On comprising equations (10) and (11), and taking

µ0 = 4π x 10-7 Henry /met

ε0 = 8.8542 x 10-12 Farad /met

1/ v20 ε0

v=1/√µ0 ε0 = 2.9999 x 10 8 met / sec