Electrostatic precipitators use electric forces to remove pollutant particles from smoke, in particular in the smokestacks of coal-burning power plants. One form of precipitator consists of a vertical, hollow, metal cylinder with a thin wire, insulated from the cylinder, running along its axis (Fig. P23.65). A large potential difference is established between the wire and the outer cylinder, with the wire at lower potential. This sets up a strong radial electric field directed inward. The field produces a region of ionized air near the wire. Smoke enters the precipitator at the bottom, ash and dust in it pick up electrons, and the charged pollutants are accelerated toward the outer cylinder wall by the electric field. Suppose the radius of the central wire is 90.0 µm, the radius of the cylinder is 14.0 cm, and a potential difference of 50.0 kV is established between the wire and the cylinder. Also assume that the wire and cylinder are both very long in comparison to the cylinder radius, so the results of 23.61 apply. (a) What is the magnitude of the electric field midway between the wire and the cylinder wall? (b) What magnitude of charge must a 30.0-µg ash particle have if the electric field computed in part (a) is to exert a force ten times the weight of the particle? 23.61 . CALC Coaxial Cylinders. A long metal cylinder with radius a is supported on an insulating stand on the axis of a long, hollow, metal tube with radius b. The positive charge per unit length on the inner cylinder is ?, and there is an equal negative charge per unit length on the outer cylinder. (a) Calculate the potential V(r) for (i) r < a; (ii) a < r < b; (iii) r > b. (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take V = 0 at r = b. (b) Show that the potential of the inner cylinder with respect to the outer is (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitude (d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?

Solution 67P We need to calculate the electric field midway between the wire and the cylinder wall. The radius of the cylinder = 14.0 cm = 0.14 m Radius of the central wire = 90.0 m = 90.0 × 10 6 m Potential difference = 50.0 kV = 50000 V The distance of midway between the wire and cylinder is = 0.07 m (a) The electric field can be calculated as follows, V 50000 E = ln (radius...