We present a study of atmospheric electricity using the chemistry-climate model SOCOL considering ionization by solar energetic particles during an extreme solar proton event (SPE), galactic cosmic rays (GCR), and terrestrial radon (Rn-222). We calculate the global distribution of the atmospheric conductivity and fair-weather downward current density (J_z) using atmospheric ionization rates from all sources. We found that J_z is enhanced (by more than 3.5 pA/m²) in radon source and polar regions. Contribution of Rn-222 is essential at middle and low latitudes/altitudes where GCR-induced air conductivity is reduced. The model results are in good agreement with the available observations. We also studied the effects of an extreme SPE, corresponding to the 774 AD event, on the atmospheric electricity and found that it would lead to a large increase of J_z on a global scale. The magnitude of the effects depends on location and can exceed background value more than 30 times over the high latitudes (a conservative upper bound). Such an assessment has been performed for the first time.