The chocolate crumb mystery this story begins with Problem 56 in Chapter 23 and continues through Chapters 24 and 25. The chocolate crumb powder moved to the silo through a pipe of radius R with uniform speed v and uniform charge density p.

(a) Find an expression for the current i (the rate at which charge on the powder moved) through a perpendicular cross section of the pipe.

(b) Evaluate i for the conditions at the factory: pipe radius R = 5.0 cm, speed v = 2.0 m/s, and charge density p = 1.1 x 10-3 C/m3. If the powder were to flow through a change V in electric potential, its energy could be transferred to a spark at the rate P = iV.

(c) Could there be such a transfer within the pipe due to the radial potential difference discussed in Problem 68 of Chapter 24? As the powder flowed from the pipe into the silo, the electric potential of the powder changed. The magnitude of that change was at least equal to the radial potential difference within the pipe (as evaluated in Problem 68 of Chapter 2)

(d) Assuming that value for the potential difference and using the current found in (b) above, find the rate at which energy could have been transferred from the powder to a spark as the powder exited the pipe.

(e) If a spark did occur at the exit and lasted for 0.20 s (a reasonable expectation), how much energy would have been transferred to the spark? Recall from Problem 56 in Chapter 23 that a minimum energy transfer of 150mJ is needed to cause an explosion.

(f) Where did the powder explosion most likely occur: in the powder cloud at the unloading bin (Problem 56 of Chapter 25), within the pipe, or at the exit of the pipe into the silo?