Electromagnetic (EM) Waves

Part-2

Harmonic plane wave solution for EM in non-conductive material

Similar to general solution for EM in vacuum, for non conductive medium with no sources we’ll get similar solution.General solution for the plane wave:

E(r,t)= E0 f(ωt-k.r)H(r,t)= H0 f(ωt-k.r)

With f(r,t) is well defined up to its second order derivative.Relationship between E and B is also applicable for E and H in this case:

Harmonic plane wave solution for EM in non-conductive material

• For monochromatic plane wave we’ll use:

• E(r,t)= E0 expi(ωt-k.r) dan H(r,t)= H0 expi(ωt-k.r)

• hence:

• as before:

• and

• Such as:

• However since here ω=kv, where v=1/√µε, we get:

Harmonic plane wave solution for EM in non-conductive material

Non-conductive material ( ε,µ≠0 , σ=0) , assume the solution is in the form of a monochromatic plane wave propagating along Z. The fields E and H are then not functions of x and y, so E=E(z,t) and H=H(z,t) while k is also parallel to Z.

If E is polarized along x , we get:

Ey and Ez =0. The expression for the field H can be obtained from:

Density of energy flow andthe Poynting Vector

• For fields E and B we will get energy density:

• Which can be simplified for linear isotropic medium:

• If the wave is a harmonic plane wave, this further simplifies to

Energy Density Flux and the Poynting Vector

• In a space without source charge, the rate of change in the energy density is given by:

• Rewritten with the help of Maxwell equation:

• ExH is defined as the Poynting Vector N = ExH, which expresses the energy density flowing through the defined surface per unit time.

• So it implies the continuity equation:

Density of energy flow and the Poynting Vector

• N in its integral form is:

• By Gauss divergence theorem, U denotes the total energy inside the volume V

• The left hand side denotes the rate of energy change inside the volume while the right hand side denotes the total energy which escape per unit time from the surface S which envelops V

• The Poynting Vector N denotes the energy density per unit area flowing through the surface.

Density of energy flow and the Poynting Vector

• For harmonic plane wave, we can simplify N :

• Average N value with respect to time:

• Where H*is the complex conjugate of H.

• For harmonic plane wave, the second expression above can be simplified to:

Note the disappearance of <> on the right hand side! This form still applies for non-monochromatic wave.

Wave’s Impedance

• Using dimensional analysis, we deduce that the dimension of [E ]= volt/m and [H] = amp/m. As such the impedance for an EM wave can be written as :

• Impedance for vacuum η0 ≈ 377 ohm.

Reflection and Transmission on the interface between mediums

1. The law of reflection and transmission directionality

Consider a linearly polarized plane wave from a homogeneous dielectric medium (µ1 ε1 ) to the other medium (µ2 ε2)

o

E1

H1 k1

µ2 ε2

µ1 ε1

Θ’1θ1

θ2

TM

case

Reflection and Transmission at theMedium Boundary

Let the incident wave is a plane wave

So the reflected and refracted wave are of the forms:

Apply the continuity condition on the boundary at every points.

Reflection and Transmission at theMedium Boundary

On the interface: incidence field + reflected field= transmitted field

1. time continuity constraint gives : ω1 = ω’1 = ω2 = ω

2. Continuity condition along all points on r, gives:

This last condition means :

(i) incident plane = reflected plane = refracted plane

(ii) k1 sin θ1 = k’1 sin θ’1 = k2 sin θ2 (see to previous slide)

Reflection and Transmission at theMedium Boundary

as such with kn = ω/vn and defining n = c/vn we get

a. k1 = k’1 thus θ1 = θ’1 →incident angle=reflected angle

b. k1 sin θ1 = k2 sin θ2 → n1 sin θ1 = n2 sin θ2 (Snell’s Law)

2. Relationship between fields on the interface

on the interface :

tangential component of the incident wave + reflected wave = refracted wave

Reflection and Transmission : TM -Case

For TM (Transverse Magnetic) - that is the H field perpendicular to the incident plane (see the previous picture), the continuity boundary condition means:

(i) H1 + H’1 = H2

(ii) (E1 - E’1 )cosθ1 = E2 cosθ2

Equation (i) can be converted to E as H= 𝜖

𝜇𝐸, and with help of

Snell’s law:

(ia) ε1 (E1 + E’1 ) sin θ1 = ε2 sin θ2

Equation ii) and (ia) describes the relationship between amplitudes of fields on the interface.

Reflection and Transmission Coefficient (TM)

Reflection coefficient is defined as (eliminate E2 ):

Transmission coefficient is defined as (eliminate E’1 ):

For non-magnetic dielectric material µ1 ≈ µ2 ≈ µ0, which means

the Snell’s law becomes : sin θ1 /sin θ2 = √𝜀2 / √𝜀1 .

Reflection and Transmission Coefficient (TM)

rTM and tTM can be simplified into:

Alternatively using Snell’s Law : n1 sinθ1 = n2 sinθ2 , those can be rewritten as: