Electromagnetic (EM) Waves

Short review on calculus vector

OutlineA. Various formulations of the Maxwell equation:

1. In a vacuum

2. In a vacuum without source charge

3. In a medium

4. In a dielectric medium without source charge

5. In a conductive medium

B. Wave equation in various situations and the solutions

C. Polarization

D. Continuity condition and the conservation of charges

E. Waves in the boundary between mediums

F. Waves in a conductive medium

Maxwell Equation in a vacuum

• How EM wave is generated:

– Accelerating electric charge(s)

– Orbital electron shifts

• Maxwell Equation (IS) in a vacuum

Gauss Law, ρ : charge density

Faraday-Lenz Law

impossibility of monopole (magnetic Gaussian law)

Ampere-Maxwell law. J is the current density

Maxwell Equation: in a vacuum butfree of sources

• If no source charge ρ=0 and current density J=0, we have :

Gauss law without source charge

Faraday Lenz Law

Magnetic Gauss law; no monoploe

Ampere-Maxwell Law without J

Maxwell equation in a medium

• In medium the electric and magnetic field may induce polarization and magnetization.

• To take into account the medium property, we use constitutive relationship:

• D : displacement field, P : polarization (electric dipole moment per volume), and for linear material:

• as such :

• Where ε=(1+ χe ) ε0 is the permittivity of the material.

• Magnetic field B is related to magnetization M by :

Maxwell Equation in a medium

• For a linear isotropic material:

• Hence with µ=(1+ χm ) µ0 the medium permeability we have

• Maxwell equation in a medium:

The term ∂D/ ∂t :displacement current

Related to free charge

Related to free current

Continuity equationFor a volume V enclosed by a surface S, contains a total charge of Q with current density pointing outwards from S of J. Inside V we have no source charge/well so that:

Surface:S

J: charge density

Charges: Q

Volume:VRate of decay of Q in V per unit time

Rate of the outflow of Q from the surface S per unit time=

With charge density per volume ρ, : and with Gauss law”

also known as the continuity equation

EM waves in vacuum without sources

• EM wave equation in a vacuum is given as:

• Using identity :

• So:

• In vacuum (no charges and free current):

• With c2 =1/µ0ε0

• Similarly, we get for the field B:

EM waves in vacuum without sources

EM waves in dielectric materials

• In dielectric medium, there is no free current, but there might still be free charges:

• Using Maxwell Equation:

𝛻 ∙ 𝑫 = 𝜌𝑓𝑟𝑒𝑒 𝛻 ∙ 𝑩 = 0

𝛻 × 𝑯 = 𝑱𝑓𝑟𝑒𝑒 + 𝜕𝑫/𝜕𝑡 𝛻 × 𝑬 = 𝜕𝑩/𝜕𝑡

• In case of no free current Jfree =0, and no free charge 𝜌𝑓𝑟𝑒𝑒 =

0 and ε, µ = constant, non conductive medium, we can rewrite the Maxwell equation as (using the E and H fields):

EM waves in dielectric material without sources

• So, EM wave equation in a homogenous non-conducting material (dielectric):

• With v2 = 1/µε, if one of the equation can be solved, the other can be solved conversely.

• e.g, E is known, then H can be computed from Maxwell eq:

The solution of EM wave in a vacuum without sources

• General solution is in the form of a plane wave:

• E(r,t)= E0 f(ωt-k.r)

• B(r,t)= B0 f(ωt-k.r)

• f(r,t) must be differentiable up to the second order with respect to r dan t.

• The Simplest example of solutions is a plane monochromatic harmonic wave with a fixed amplitude:

• E(r,t)= E0 sin(ωt-k.r) B(r,t)= B0 sin(ωt-k.r), or

• E(r,t)= E0 cos(ωt-k.r) B(r,t)= B0 cos(ωt-k.r), or

• E(r,t)= E0 exp i(ωt-k.r) B(r,t)= B0 exp i(ωt-k.r)

Wavefront of a plane wave

• For solution in the complex representation, the physical quantity can be taken from real or imaginary parts.

• With E0 and B0 are the amplitudes, k: wave propagation vectors.

• Wavefront is defined at all times by k.r, for k.r=constant the position of r will be on the plane perpendicular to k.

k

r

k.r

Relationship between E,B and k

• Assuming : E(r,t)= E0 exp i(ωt-k.r) and B(r,t)= B0 exp i(ωt-k.r)

• Relationship between E and B derived from Maxwell equation:

So:

Where ω= kc (note the relative directions between k,B and E.)

Relationship between E,B and kThe last expression can be derived from the vector triple product:

Starting from:

taking k x (…..) we will get

But

𝛻 ∙ 𝑬 = 0 → 𝛻 ∙ 𝑬𝟎𝑒𝑖 𝜔𝑡−𝒌.𝒓 = 𝑬𝟎 ∙ 𝛻𝑒

𝑖 𝜔𝑡−𝒌.𝒓 = −𝑖𝒌. 𝑬 = 𝟎

Hence

Relationship between E and B

In case of plane wave in vacuum without source we have shown that :

𝒌 ∙ 𝑬 = 𝟎

This means E is orthogonal to k. In this case B is also orthogonal to k, since

Consequently E, B are k orthogonal to each other.

From relationship

And identity ax(bxc)= b(a.c)-c(a.b), we can also get

Directionality of E,B,and k

E

B

k

Simpleright-hand-rule representation

Wave Polarization

• Transversally Polarized Harmonic Plane Wave propagating along x3 is:

• Variable E01 and E01 are real, propagation direction is along x3. Polarization are determined by :

• 1. Amplitude ratio of E01/E02

• 2. Phase difference between the two amplitudes : ϕ= ϕ2-ϕ1

• Case 1: Linear Polarization

• If ϕ=0 or ±π, E02/E01 are random, the wave amplitude is then:

Linear Polarization

Case 1: Linear Polarization

Field B is obtained from:

x3

E02

x2

E01

x1

α

Tanα=E02/E01

Circular Polarization

Case 2: Circular Polarization

If ϕ= ±π/2, E02=E01 =E0, the amplitudes become:

Full expression of the wavefunction:

To observe the oscillation, take the real part, for x3=0

Counterclockwise phase rotation

Clockwise phase rotation

Circular Polarization

ωt

x2

x1

E+

E-

• If ϕ, E1 and E2 are random, we’ll get elliptical polarization.

• For x3=0, we get:

• With

• We’ll get:

Elliptical Polarization

Elliptical Polarization

Elliptical equation with skewed axis

E1

E01

-E02

-E01 α

E2

E02