Electromagnetic (EM) Waves

Electromagnetic (EM) Waves

Short review on calculus vector

Outline A. 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 but free 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+ χ e ) ε 0 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+ χ m ) µ 0 the medium permeability we have Maxwell equation in a medium: Related to free charge Related to free current The term D/ t :displacement current

Continuity equation For 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: Volume:V Charges: Q J: charge density Rate of decay of Q in V per unit time = Rate of the outflow of Q from the surface S per unit time Surface:S 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):

EM waves in vacuum without sources With c 2 =1/µ 0 ε 0 Similarly, we get for the field B:

EM waves in dielectric materials In dielectric medium, there is no free current, but there might still be free charges: Using Maxwell Equation: D = ρ free B = 0 H = J free + D/ t E = B/ t In case of no free current J free =0, and no free charge ρ free = 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 v 2 = 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)= E 0 f(ωt-k.r) B(r,t)= B 0 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)= E 0 sin(ωt-k.r) B(r,t)= B 0 sin(ωt-k.r), or E(r,t)= E 0 cos(ωt-k.r) B(r,t)= B 0 cos(ωt-k.r), or E(r,t)= E 0 exp i(ωt-k.r) B(r,t)= B 0 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 E 0 and B 0 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. r k.r k

Relationship between E,B and k Assuming : E(r,t)= E 0 exp i(ωt-k.r) and B(r,t)= B 0 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 k The last expression can be derived from the vector triple product: Starting from: taking k x (..) we will get But E = 0 E 0 e i ωt k.r = E 0 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 k Simple right-hand-rule representation B

Wave Polarization Transversally Polarized Harmonic Plane Wave propagating along x 3 is: Variable E 01 and E 01 are real, propagation direction is along x3. Polarization are determined by : 1. Amplitude ratio of E 01 /E 02 2. Phase difference between the two amplitudes : ϕ= ϕ 2 -ϕ 1 Case If 1: Linear Polarization ϕ=0 or ±π, E 02 /E 01 are random, the wave amplitude is then:

Linear Polarization Case 1: Linear Polarization Field B is obtained from: x 1 E 01 Tanα=E 02 /E 01 α x 3 E 02 x 2

Circular Polarization Case 2: Circular Polarization If ϕ= ±π/2, E 02 =E 01 =E 0, the amplitudes become: Full expression of the wavefunction: To observe the oscillation, take the real part, for x 3 =0 Counterclockwise phase rotation Clockwise phase rotation

Circular Polarization x 2 E + ωt x 1 E -

Elliptical Polarization If ϕ, E 1 and E 2 are random, we ll get elliptical polarization. For x 3 =0, we get: With We ll get:

Elliptical Polarization Elliptical equation with skewed axis E 2 E 02 -E 01 α E 01 E 1 -E 02