Introduction to the Physics and Techniques of Remote Sensing. Jakob J. van Zyl

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Introduction to the Physics and Techniques of Remote Sensing - Jakob J. van Zyl


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vector

       H = magnetic vector

       B = induction vector

       μ0, ∈0 = permeability and permittivity of vacuum

       μr, ∈r = relative permeability and permittivity

       ∇. = divergence

       ∇x = curl

      Maxwell’s concept of electromagnetic waves is that a smooth wave motion exists in the magnetic and electric force fields. In any region where there is a temporal change of the electric field, a magnetic field appears automatically in that same region as a conjugal partner and vice‐versa. This is expressed by the above coupled equations.

      2.1.3 Wave Equation and Solution

      In homogeneous, isotropic, and nonmagnetic media, Maxwell’s equations can be combined to derive the wave equation:

      (2.7)equation

      where ∇2 is the Laplacian. In the case of a sinusoidal field:

      (2.8)equation

      where

      (2.9)equation

      Usually μr = 1 and ∈r varies from 1 to 80 and is a function of the frequency. The solution for the above differential equation is given by:

      where A is the wave amplitude, ω is the angular frequency, ϕ is the phase, and k is the wave vector in the propagation medium (imagesspeed of light in vacuum). The wave frequency ν is defined as ν = ω/2π.

      2.1.4 Quantum Properties of Electromagnetic Radiation

      The electromagnetic energy can be presented in a quantized form as bursts of radiation with a quantized radiant energy Q, which is proportional to the frequency ν:

      (2.11)equation

      where h = Planck’s constant = 6.626 × 10−34 joule second. The radiant energy carried by the wave is not delivered to a receiver as if it is spread evenly over the wave, as Maxwell had visualized, but is delivered on a probabilistic basis. The probability that a wave train will make full delivery of its radiant energy at some place along the wave is proportional to the flux density of the wave at that place. If a very large number of wave trains are coexistent, then the overall average effect follows Maxwell’s equations.

      2.1.5 Polarization

      An electromagnetic wave consists of a coupled electric and magnetic force field. In free space, these two fields are at right angles to each other and transverse to the direction of propagation. The direction and magnitude of only one of the fields (usually the electric field) is sufficient to completely specify the direction and magnitude of the other field using Maxwell’s equations.

      The polarization of the electromagnetic wave is contained in the elements of the vector amplitude A of the electric field in Equation (2.10). For a transverse electromagnetic wave, this vector is orthogonal to the direction in which the wave is propagating, and therefore we can completely describe the amplitude of the electric field by writing A as a two‐dimensional complex vector: