(a) Dipole representation along fuselage from nose to tail and (b) dipole representation along wings [12]. 

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... fl ashes originate by a complex process from cloud-based charge centres [1,2]. Positive charges accu- mulate in the upper region of a thunder cloud and negative charges in the lower region. The strong electric fi elds produced initiate electric breakdown. Scienti fi c methods are available to trace the lightning channel leader initiated by aircraft during landing or take-off under a thundercloud at very low altitudes [2]. The great current magnitudes and rapid rise times of cloud- to-ground fl ashes are the most hazardous for an air- plane [3]. Since an aircraft can become a part of the nat- ural lightning discharge process, the direct and indirect effects due to lightning strikes are recognized as a threat to fl ight safety. Thus, it is vital for the aircraft industry to restructure aircraft design and properly protect and shield its electronic devices. The SAE [4,5] has speci fi ed the idealized waveforms ’ component of current and potential for quali fi cation tests. A full-scale vehicle lightning-induced coupling test is reported in [5]. However, because of the cost and personal bodily risks for such physical aircraft-lightning tests, computer simulation studies are preferable when it comes to studying lightning-aircraft electrodynamics. Different kinds of geometrical and electrical models have been proposed for aircraft representation and computer simulation under various conditions and parameters [6 À 11]. These models are used to simulate and to fi nd the airframe resonances, dynamic currents and charges on the aircraft for studying aircraft-lightning electrodynamics. An electric circuit model which yields results validated when compared to laboratory tests of model aircraft has been recently presented [12] which we extend here through a simple matrix formulation. The aircraft geometry used in this research simulation model is the Airbus A380 passenger aircraft with a fuselage approximately 72-m long, and 6-m high, the same as in [12]. Following the description and methodology in [12], the aircraft ’ s body is subdivided into a number of dipoles, each directed along the z -axis and placed along the y -axis with induced positive charge at the top pole and equivalent negative charge at the bottom of the aircraft fuselage as shown in Figure 1 [12]. The radome, wings and the tail of the aircraft are the most prominent edges to get struck by lightning strikes due to more charges accumulating at these edges. These aircraft charges initiate the top and bottom leaders from these points. The mathematical dipole model for a metal aircraft with a single charge is used to determine the capacitance of the aircraft skin using the potential coef fi cients of the dipoles. The vertical and horizontal fi elds at a dipole due to other dipoles have been taken into account for the calculation. Note that it is possible to account for the conical radome, sharp tail and wings by using dipoles that gradually decrease in distance d of charge separation. As noted in [12], the dipole model proposed and veri fi ed in that paper is a very powerful tool for minute representation of different shapes of aircraft frame to determine the best geometrical shape and fuselage material for reduced electric stress. And because that model has been validated in the laboratory with aircraft models, we feel con fi dent extending its use here in this paper. In this paper a new matrix formulation of that dipole electrostatics of aircraft is presented. This matrix formulation of the thundercloud-induced electric charges on aircraft and ground yields a solu- tion of the capacitive elements of the aircraft. The computed capacitive elements are readily used in the aircraft-lightning electrodynamics during the lightning return stroke phase of the electrodynamics or the severe transient interaction phase. The aircraft body, from radome to tail, is divided into 12 segments. Each wing is divided into 11 segments. Once the capacitance for each segment is calculated, the per unit length capacitance is calculated for each region of the segment. When the per unit length capacitance was not signi fi cantly altered the number of dipoles was deemed suf fi cient. The pre-strike dipole modelling of electrostatic charges on an aircraft gives a succinct representation of the distribution of charge build up on the aircraft surfaces and can be used for post-strike analysis since the capacitances will not change. The method makes use of elementary theory of electrostatic induction on the distribution of charges within an object that occurs as a reaction to the presence of a nearby charge. The analogy is applied to an aircraft as it goes through a charged electric storm causing migration of polarized charges on the surface with positive charges on the top. Aircraft build up static charges just by virtue of fl ying through the atmo- sphere [13]. However, the breakdown of the static charges occurs as the aircraft enters a charged electric storm. The pre-breakdown charges and capacitances are determined here based on the dipole model. The dipole model incorporates the real geometrical dimensions of an aircraft with surface charge distribution represented by diploes of various separation distances placed along the top and bottom radome, wings, fuselage, and tail end of the aircraft (Figure 2). The cloud charge and its image charge are taken into account as the two charges highly in fl uence the overall electric fi eld on the surface of an aircraft. The cloud charge is determined based on the cylindrical Gaussian surface. The surface charge layer on the aircraft surface is modelled as a line charge with an electric dipole moment per unit area. The fi eld of an electric dipole on the top and bottom of an aircraft surface is obtained by representing an aircraft as a fl oating electrode [14] iso- lated in space and charged to a speci fi c voltage. The fl oating electrode is discretized into fi nite lengths placed on the top and bottom of the aircraft thus forming a series of line charges with an electric dipole moment per unit area. The aircraft dipole model is shown in Figure 2. Note that any number of dipoles is suf fi cient to compute the capacitance of an aircraft. However, to accurately represent the aircraft geometry, more dipoles are required. The cloud charge is assumed to be 1000 m above ground and its image charge 1000 m below ground. Thus, the earth is assumed to be a perfect conductor for the electrostatic computation, its effect being more signi fi cant when the aircraft is close to ground, say at a height of 200 m. The equations for the geometrical and physical properties of the cylindrical Gaussian surface are expressed elegantly in terms of the generalized matrix of the potential coef fi cient functions. We may use the fact that the potential from a discretized conductor due to the charge on itself is mainly from the middle and therefore can be approximated as from a fl at circular disk-like conductor [14]. The potential V everywhere on the 4 discretized pieces at charges q 1 to q 4 of the aircraft and V Cl on the cloud, then will be each due to itself and the other 4 charges. For q 1 (with 3 simi- lar equations for q 2 to q 4 ), we have 3 : 52 1 1 ...