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### Transcript of Electromagnetic (EM) Waves

Electromagnetic (EM) WavesHarmonic 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:
• 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 and the 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
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 the Medium 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 the Medium 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=
, 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 = √2 / √1 .
Reflection and Transmission Coefficient (TM)
rTM and tTM can be simplified into: