The combustion of fossil fuels produces micron-sized particles of soot, one of the major components of air pollution. The terminal speeds of these particles are extremely small, so they remain suspended in air for very long periods of time. Furthermore, very small particles almost always acquire small amounts of charge from cosmic rays and various atmospheric effects, so their motion is influenced not only by gravity but also by the earth’s weak electric field. Consider a small spherical particle of radius r, density ρ, and charge q. A small sphere moving with speed v experiences a drag force F_drag = 6πηrv, where η is the viscosity of the air. (This differs from the drag force you learned in Chapter 6 because there we considered macroscopic rather than microscopic objects.)
a. A particle falling at its terminal speed v_term is in equilibrium with no net force. Write Newton’s first law for this particle falling in the presence of a downward electric field of strength E, then solve to find an expression for v_term.
b. Soot is primarily carbon, and carbon in the form of graphite has a density of 2200 kg/m³. In the absence of an electric field, what is the terminal speed in mm/s of a 1.0-μm-diameter graphite particle? The viscosity of air at 20°C is 1.8 × 10^-5 kg/m s.
c. The earth’s electric field is typically (150 N/C, downward). In this field, what is the terminal speed in mm/s of a 1.0- μm-diameter graphite particle that has acquired 250 extra electrons?