Given that E E0 cosomegat betazax Vm in free space determi

Given that E = E_0 cos(omegat - betaz)a_x V/m in free space, determine D, H, and B.

Solution

An oscillating electric charge creates oscillating variation  
of the electric (and also magnetic) field.  
The oscillations spread away as an electromagnetic wave
propagating at the speed of light.

In a uniform dielectric medium with permittivity e = e_re₀ and permeability m = m₀, Maxwell\'s equations

for the vector fields of electric intensity E and magnetic intensity H

Both solutions are valid under the condition

=

_    1 _
sqrt ( me )

These two equations represent plane wavefronts (x-y plane) moving in the z direction with speed  v = w/b = 1/sqrt (me) = c/n , where c is the speed of light in free space and n = sqrt (e_r) is the refractive index of the dielectric medium. There is sinusoidal variation of magnitude of the E and H vector fields which are aligned at the right angles to each other, and the direction of propagation.
Time Domain
If we are at any fixed location z = const., as the E (and H) wave passes by, we register magnitude variations with frequency f [Hz].
w = 2pf = 2p/T is the well known angular frequency in [rad/sec] and T is the period in [sec].

Spatial Domain
If we stop the motion (stop the time), i.e., t = const. the variation of the E (or H) waveform along the z-axis depends on the constant b
which can be expressed as b = 2pf_s = 2p/l, where l is the wavelength (spatial period) in [m] and f_s represents the spatial frequency in [cycles/m

It can be easily shown by substituting the E and H solutions into the first Maxwell equation that the relationship
between the E and H vector fields is

In free space the intrinsic impedance h₀= sqrt (m₀/e₀) = m₀c = 377 W

E = E0 cos (wt - bz) ax

and

H = H0 cos (wt - bz) ay