Retarded potentials

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Key focus: Understand retarded potentials – the basic building block for understanding antenna array patterns. Retarded potentials are potentials at an observation point when the quantities at the source are non-static (varies in both space and time)

The static case : potentials

The fundamental premise of understanding antenna radiation is to understand how a radiation source influences the propagation of travelling electromagnetic waves. Propagation of travelling waves are best described by electric and magnetic potentials along the propagation path.

In the static case, the electric field E, charge density ρ, current density J and the electric potential Φ , magnetic field B and magnetic potential A are all constant in time. That is, they are functions of radial distance r in the spherical co-ordinate system, but not functions of time t . The solution for Maxwell’s equations (in the spherical co-ordinate system) take the following equivalent form in terms of electric and magnetic potentials:

\[\begin{aligned} \Phi(r) & = \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(r’)}{R} d^3 r’ \quad \quad (1) \\ A(r) & = \frac{\mu}{4 \pi} \int_V \frac{J(r’)}{R} d^3 r’ \quad \quad (2) \end{aligned}\]
Potentials - solutions for Maxwell's equations for static case
Figure 1: Potentials – solutions for Maxwell’s equations for static case

The quantity R is the distance between the source and the point at which the corresponding fields are observed and d3 r’ is the volume element at the source point.

The non-static case : retarded potentials

Since electromagnetic radiation are produced by time-varying electric charges, we are interested in describing the potentials at a given the observation point for this non-static situation. Obviously for the non-static case, the electric field E , charge density ρ , current density J and the electric potential Φ , magnetic field B and magnetic potential A are functions of both radial distance and time. The corresponding formulas for the potentials are

\[\begin{aligned} \Phi(r, t) & = \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(r’, t – \frac{R}{c})}{R} d^3 r’ \quad \quad (3) \\ A(r, t) & = \frac{\mu}{4 \pi} \int_V \frac{J(r’, t – \frac{R}{c} )}{R} d^3 r’ \quad \quad (4) \end{aligned}\]

With c defined as the velocity of light, the factor t – R/c is the time delay between the emission of electromagnetic photon from the source and the time it gets observed by an observer. The electromagnetic field travels at certain velocity and hence the potentials at the observation point (due to the changing charge at source) are experienced after a certain time delay. Such potentials are called retarded potentials and the propagation delay t – R/c is called retarded time. Hence, we can say that the retarded potentials are related to electromagnetic fields of a current or charge distribution that vary in time.

Retarded potentials - solutions for Maxwell's equations for non-static case
Figure 2: Retarded potentials – solutions for Maxwell’s equations for non-static case

Sinusoidal time dependence of retarded potentials

In antenna theory, the antenna elements (source) are fed with sinusoidal waves. So, the next step is to express the retarded potentials at the observation point when all the quantities at source vary sinusoidally in time. Therefore, when the quantities are sinusoidal single-frequency waves, the shift property of Fourier transform can be applied.

\[\begin{aligned} \Phi(r, t) &= \Phi(r) e^{j \omega t} \\ \rho(r,t) &= \rho(r) e^{j \omega t} \\ A(r, t) &= A(r) e^{j \omega t} \\ J(r, t) &= J(r) e^{j \omega t} \end{aligned} \quad \quad (5)\]

Then, the retarded potential Φ(r,t) at the field observation point become,

\[\begin{aligned} \Phi(r, t) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(r’, t – \frac{R}{c})}{R} d^3 r’ \\ \Rightarrow \Phi(r) e^{j \omega t} & = \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(r’)e^{j \omega (t – \frac{R}{c})}}{R} d^3 r’\\ \Rightarrow \Phi(r) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(r’)e^{-j \omega \frac{R}{c}}}{R} d^3 r’\\ \Rightarrow \Phi(r) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(r’)e^{-j \frac{\omega}{c} R }}{R} d^3 r’ \\ \Rightarrow \Phi(r) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(r’)e^{-j k R }}{R} d^3 r’ \end{aligned}\]

The retarded potential A(r,t) can be derived in a similar manner. Therefore, the retarded potentials (equations (3) and (4) ) for the single frequency sinusoidal wave is given by

\[\boxed{ \begin{aligned} \Phi(r) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(r’)e^{-j k R }}{R} d^3 r’ \\ A(r) &= \frac{\mu}{4 \pi} \int_V \frac{J(r’)e^{-j k R }}{R} d^3 r’ \end{aligned} }\]

In these equations, the quantity k = ω/c = 2 π/λ is called the free-space wavenumber.

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