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  Propagation of plane wave in Isotropic Media, Physics tutorial

Introduction:

There are numerous types of waves in nature, like sound, light, heat, electromagnetic waves, mechanical waves etc. Such waves are either transverse or longitudinal. Movement of wave from one point to another with respect to time is termed as propagation.

Plane waves refer to waves whose wavefront are parallel to each other. Plane electromagnetic wave that is polarized would have electric field vector oscillating in specific direction and Magnetic field (B) oscillating perpendicularly to it. Direction of propagation is then normal to both directions of E and B. If E and B vectors oscillate randomly {with both perpendicular to each other} electromagnetic wave is said to be unpolarised or randomly polarized. The electromagnetic wave can be linearly polarized (E vector oscillating in straight line) or circularly polarized (E vector oscillating or circular path) or elliptically polarized {E vector oscillating in elliptical path). Isotropic medium is medium having similar property in all direction. Unbounded isotropic medium is thus unconfined medium having similar properties in all direction. Simple example is free space.

Propagation of plane wave in unbounded isotropic media:

The wave may be considered as plane wave far away from its source of radiation. This applies to wavefronts of different shapes. Wavefronts of the plane wave are parallel to each other. It is essential to state that line normal to wavefronts or planes is known as a ray. The ray signifies direction of propagation.

There are several kinds of waves like sound wave, hydromagnetic wave, electromagnetic wave etc. Their properties comprise (i) transfer of energy from one place to another. (ii) Exhibition of diffraction effect and (iii) obeying principle of superposition.

Electromagnetic wave which is produced by accelerated charged particles. In neighborhood of electric charge is electric field, E. As charge moves (oscillates), both electric field and magnetic field exist in neighborhood. The electromagnetic wave is then propagated. Electromagnetic wave exists because of variation in electric field with time, generating magnetic field (i.e. at high frequency) and varying magnetic field generating electric field (faraday's law) that process is repeated constantly.

The electric field, E, is represented by

E = uxE0exp[jw(t - z/v)]

This equation implies that E oscillates along x-axis whereas wave propagates along z-axis. Velocity of wave, v = w/k, where w is angular frequency and k is wave number. In free space, v = 2.998 × 108ms-1 (to 3 d.p). Eo is amplitude or peak value of varying electric field. Magnetic field, B that oscillates along y-axis is represented by

B = uyB0exp[jw(t - z/v)]

Where Bo is amplitude or peak value of magnetic field. Polarized plane wave has its field of oscillation changing with time in the specified direction while for unpolarized plane wave the direction of oscillation of field change randomly with time. Specified direction of oscillation of field could be rectilinear, circular or elliptical in which case rectilinear, circular and elliptical polarization result. Circular polarization of plane wave will be represented by

E = uxE0exp[jw(t - z/v)]+uyE0exp[jw(t - z/v + p/2)]

This is because for circular polarization two components of equal amplitude should be perpendicular and have phase difference of p/2 between them. If amplitude of components are unequal elliptical polarization results.

2348_Propagation of plane wave.jpg

The relationship between the electric field, E, and the magnetic field, B, is obtained from Faraday's law as follows.

Using differential form of Faraday's law i.e.

∇xE = -dB/dt

or

uydEx/dz -uz(dEx/dy)= -dB/dt

As simple case of Ex constant in x-y plane is considered, d/dy = 0

The 2nd term on LHS = 0 as uy(dEx/dy) = 0, thus

-uyj(w/v)E0[jw(t-z/v)]= -dB/dt

Integrating with respect to t provides,

B= uy(1/v)E0exp[jw(t-z/v)]

Comparing equations shows that amplitude, B0, of magnetic field equals E0/ v. Wave equations of electric and magnetic fields of which equations are solutions are attained as follows:

Maxwell's equations in free space in differential form are

∇.E = Ρ/ε0.............................. Eq.i

Where Ρ ≡ charge density and εo ≡ permittivity of free space

∇.B = 0.............................. Eq.ii

∇xE = -dB/dt.............................. Eq.iii

∇xB = ε0μ0(dE/dt) + μ0j.............................. Eq.iv

Where j ≡ conduction current density and μ0 ≡ permeability of free space, (iv) equation is the Ampere's law modified by addition of displacement current, ε0(dE/dt) when electric field differs rapidly.

Outside the region of changing charge and current distribution, Maxwell's equations given above i.e. equations (i) to (iv) becomes

∇xE = 0

∇xB = 0

∇xE = -dB/dt

∇xB = ε0μ0dE/dt

By taking curl of equation i.e. ∇x∇xE = -d/dt(∇xB),

∇x∇xE = ∇(∇.E) - ∇2E

Replacing for ∇x∇xE we have

∇(∇.E)-∇2E = -d/dt(∇xB)

Replacing for ∇.E = 0 from equation and for ∇xB = ε0μ0 (dE/dt) gives

-∇2E = -d/dt(ε0μ0dE/dt) or ∇2E = ε0μ0(d2E/dt2)

or

∇2E = (1/c2)(d2E/dt2)

Where c = 1/√ε0μ0

By taking curl of ∇x∇xB = ε0μ0(d/dt)(∇xE)

∇(∇.B)-∇2B = -d/dt(∇xE)

Using equations:

-∇2B = -ε0μ0(d2B/dt2) or ∇2B = ε0μ0(d2B/dt2)

∇2B = (1/c2)(d2B/dt2)

∇2E = (1/c2)(d2E/dt2) This is wave equations of electric fields and

∇2B = (1/c2)(d2B/dt2) this is wave equations of magnetic fields.

Propagation of electromagnetic wave in an isotropic medium:

Case 1: Propagation of electromagnetic wave in the isotropic insulating medium.

Assume electromagnetic wave travels in the isotropic insulating medium and that relative permittivity and relative permeability of medium are εr and μr respectively. Equation becomes

∇2E = εμ(d2E/dt2) or ∇2E = (1/v2)(d2E/dt2)

Where ε, permittivity of medium is product of εo and εr i.e. ε = εoεr and μ, permeability of medium is product of μo and μr i.e. μ = μoμr, i.e. the velocity of wave in medium. Equation becomes

∇2B = (1/v2)(d2B/dt2)

√εrμr = c/v

But refractive index, n = c/v thus, n = √εrμr and n is refractive index of medium. Both relative permittivity and relative permeability are known to differ with frequency for dispersive medium implying that refractive index of dispersive medium differs with frequency.

Case II: Propagation of electromagnetic wave in the conducting medium.

For propagation of electromagnetic wave in conducting medium, modified Ampere's law can be written as:

∇xH = jf + dD/dt

Where H - the magnetic intensity - equals B/μoμr (μr being the relative permeability of medium) and D -electric displacement - equals εoεrE (εr being relative permittivity of the medium).

(i) H = B/μoμr in absence of magnetization current and D = εoεrE in absence of polarization charges otherwise H = B/(μoμr)-M, where M = magnetization, vector quantity and D = εoεrE - P (P = polarization, a vector quantity). Equation can be written as:

∇xB/(μoμr) = jf + εoεr(dE/dt)

From ohm's law i.e. I = V/R

Then equation can be written as

∇x(B/μoμr) = σE + εoεr(dE/dt)

By taking curl

∇x∇xE = -d/dt(∇xB)

∇2E = d2E/dz2

This is because electric field, E is constant in x-y plane at fixed z-coordinate its amplitude though decreases exponentially with increase in z. Electrical conductivity σ>>ωεoεr. Thus equation can be written as:

d2E/dz2 = μoμrσ(dE/dt)

Further solving α=β = √μoμrσω/2

Reciprocal of α = √2/μoμrσω is referred to as skin depth, δ and it estimates how rapidly wave is attenuated. Using μr≈1, δ≈√2/μoσω. When ω is high, δ is very small.

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