Appendix 5

Electric Dipole Formulas

A5.1. Complete formulas of the electric dipole

Let us consider a short wire of longitudinal dimension ΔL where a uniform and sine wave current I of ω angular frequency flows. An observer P is located at a distance r from the short wire in order to meet the r >> ΔL condition (Fresnel region).

With the assumptions previously established and taking into account the fact that the distance r is attached to a spherical coordinate system, whose origin coincides with the center of the wire, it is an electric dipole, which is represented with the notation conventions from Figure A5.1. Solving the wave equation in order to determine the radiated electromagnetic fields from this electric dipole, leads to an electric field vector, including two projections according to the polar angle and θ the radial direction r respectively. A magnetic field vector with symmetry of revolution including only one component is attached to the previous electric field vector. The magnetic field vector is directed according to the azimuth angle φ [DEM 05]:

[A5.1]

We find, in this relation, unit vectors which were omitted in Figure A5.1.

Figure A5.1. The electric dipole in the spherical coordinate system

Under the previous assumptions, we seek E_θ, E_r and H_φ in the analytical expressions below:

[A5.2]

[A5.3]

[A5.4]

We find in these formulas the wave number in free space k, whose expressions linked to the ω angular frequency or to the λ wavelength will be recalled below.

[A5.5]

In this formula c, the speed of light in vacuum, appears.

A5.2. Near-field formulas of the electric dipole

From the previous relationships, we take approximated formulas only valid for the distances r that are much lower than the λ wavelength. Under these conditions and taking into account equation [A5.5], the kr product being much lower than one, the use of the first term of the series expansion of equations [A5.2] to [A5.4] leads to the near-field formulae of E_θ, E_r and H_φ, i.e.:

[A5.6]

[A5.7]

[A5.8]

The p parameter is similar to the dipolar moment found in electrostatic theory:

[A5.9]

In equation [A5.9], q then represents the electric charges of a displacement current with an amplitude strictly similar to that of the I flowing current on the short wire in Figure A5.1.

A5.3. Far-field formulas of the electric dipole

The far-field of the electric dipole is produced at a distance r that is much longer than the wavelength. Under these conditions, the kr product takes a value that is much higher than one, which suggests use of the asymptomatic forms of equations [A5.2] to [A5.4]. In that case, we can show that the radial component of the electric field vanishes.

Consequently, only E_θ and H_φ remain:

[A5.10]

[A5.11]

The remaining electric and magnetic field projections obey to the same law with the angular frequency ω and the distance r of the observer. The ratio of E_θ and H_φ corresponds to the impedance of the Z_w plane wave, which is recalled below:

[A5.12]

A5.4. Bibliography

[DEM 05] DEMOULIN B., Enseignement élémentaire sur la propagation des ondes, Volume II, Course Notes, ESEA bachelor, Lille 1 University, September 2005.