L Theory: The conventional description of the resistance of a wire is that it is given by RM\". = a, where L=wire length, A=wire crosssectional area, and 0 is the conductivity. The conductivity can be expressed in terms the electron mobility ,u, and the electron number density ne: CI = e Ne [.19. In a metallic wire, cris generally taken to be a constant, though there may be some temperature dependence as the mobility decreases with increasing thermal collisions. In an electrostatic discharge, the resistance varies in time as the degree of ionization increases. The high electric field accelerates any free electrons, and as they collide with atoms, the electron temperature rises. When the electron thermal energy (k5 T) is above a few eV, the tail of the energy distribution contains highenergy electrons that can create more electrons through impact ionization. The new electrons are also heated and create more electrons, and the process builds exponentially as an avalanche. A simple model ofthe nonlinear resistance can be constructed using energy conservation with a few major assumptions. 1. The spark channel can be thought of as uniform cylinder 2. the mobility is a constant 3.there is a fixed energy 'cost' Um to produce each new electron. Questions: Starting from the rate of change of the gas internal energy, a'Ugas dne J2 dt 'Ue dt 'J 5'3 show that the electron density can be obtained by integrating overJz: \"e = \":0 +17J'JZQH, Where neg is the initial electron density. Assuming the channel has a cross-sectional area A, calculate the time-dependent resistance as a function of an integral over the current (instead of current density) and of a? the initial resistance R0. In the final result, you should find that the resistance is actually independent of the area of the channel