Expressions for fields in terms of potentials where is the electric field intensity, is the magnetic flux density, and is the magnetic vector potential, ϕ is the electric scalar potential.

Four scenarios to be considered:

The magnetic vector potential at point P due to a differential current element: Differential current element at z'=0 in free space; field point at z>0 The magnetic flux density due to a z- directed current element has only the ϕ - component ( ϕ here is the azimuth, not to be confused with the scalar potential): Since R=(r 2 +z 2 ) 1/2

Cont. The spatial derivative can be converted to the time derivative Thus, Noting that r/R = sin(180 o – θ) = sinθ, we can also write

Cont. Lorentz condition : Since the gradient operator (differentiation with respect to spatial coordinates) and integration over time are independent, we can write In the following, we will obtain equations for, and where -z/R 3 i the derivative of 1/R with respect to z.

Cont. Due to the cylindrical symmetry of the problem, is independent of azimuth and has only radial (r) and vertical (z) components, so that the gradient operator can be written as Thus,

Cont. Expand the integrand: r-component 1st term 2nd term z-component 1st term

Cont. Convert the spatial derivatives to time derivatives: Expand the second order derivatives:

Cont.

Thus, the r- and z-components of become: r-component z-component

Cont. Finally, the time derivative of is where μ 0 is replaced with The electric field intensity expression has three terms and is given by

Cont. The two components of can be expressed as follows: In the dE z equation, the geometrical factors in the three terms were obtained as follows: 1st term (electrostatic) 2nd term (induction) 3rd term (radiation)

Elevated differential current element (z'>0) and its image

Cont.

Elevated differential current element above ground and its image; field point on the ground surface

(A3.16) Cont. (A3.17)

Vertical lightning channel above ground; field point on the ground surface

Cont. Integrating Eq. A3.16 over the radiating channel length H(t) we get Alternatively, we can write: where the following relations were used:

Cont. Integrating Eq. A3.17 over the radiating channel length H(t) we get Alternatively, using we can write

Figure: Geometry used in deriving equations for electric and magnetic fields at point P on earth (assumed to be perfectly conducting) a horizontal distance r from the vertical lightning return ‑ stroke channel extending upward with speed v f. Adapted from Thotappillil et al. (1997).

Figure. Illustration of the reversal distance for the electrostatic and the induction field components (dipole technique; see Section 5.3). Inset shows the direction of the radiation component of electric field vector for different combinations of the charge polarity and the direction of its motion (also the direction for all three components when α< 35.3°). The direction of the electric field vector refers to the initial half cycle in the case of bipolar waveforms. Adapted from Nag and Rakov (2010).

Figure. Current waveforms recommended by the IEC Lightning Protection Standard (2006) for (a) first and (b) subsequent return strokes. The risetime was measured between the 10% and 90% of peak value levels on the front part of the waveform. The rate-of-rise is the ratio of 0.8 of the peak value and 10-to-90% risetime. The time to half-peak value was measured between the peak and half-peak value on the tail part of the waveform. Drawing by Vijaya B. Somu and Potao Sun.

Table. Comparison of electric field components (at ground level) based on dipole (e.g., Uman, 1987) and monopole (Thomson, 1999) techniques for a differential current element Idz' at height z' above ground. φ is the scalar potential (different in the two techniques) and Ā is the vector potential; ; α=sin- 1(z'/R(z')).

Table. Approximate expressions for the electromagnetic fields based on the TL model with an arbitrary return-stroke speed v for near and far ranges, converging to exact expressions as v approaches c. Adapted from Chen et al. (2015).

Field equations (5.7) (5.9) The height H(t) can be found from the equation below. (5.8)

A current discontinuity at the moving front gives rise to an additional term in each Eqs. 5.7 and 5.8: (5.10) (5.11) The radiation field component: (5.12)

From Eq. 5.9, noting that for the TL model vf = v and that at far ranges R ≈ r, we have (5.13) (5.14) Thus, Eq becomes:

(5.15) Normally I(0, 0) = 0, so that