We have calculated the resistivity due to a sequence of fluctuating weak double layers aligned parallel to the ambient magnetic field line. The average response of an electron drifting through a one‐dimensional randomly oriented array of WDLs is studied using a test particle approach. The average is taken over the randomly fluctuating values of the electric field associated with the double layers. Based on our calculations, we estimate that at 350 eV electron energy the thickness of the visual auroral arc is ≃ 2.5 km and that of the auroral fine structure as ≃ 250 m when mapped down to the ionosphere. The significance of our calculations is discussed in the context of magnetosphere‐ionosphere coupling.