Chapter 23, Problem 60P

The chocolate crumb mystery. Explosions ignited by electrostatic discharges (sparks) constitute a serious danger in facilities handling grain or powder. Such an explosion occurred in chocolate crumb powder at a biscuit factory in the 1970s. Workers usually emptied newly delivered sacks of the powder into a loading bin, from which it was blown through electrically grounded plastic pipes to a silo for storage. Somewhere along this route, two conditions for an explosion were met: (1) The magnitude of an electric field became 3.0 × 10⁶ N/C or greater, so that electrical breakdown and thus sparking could occur. (2) The energy of a spark was 150 mJ or greater so that it could ignite the powder explosively. Let us check for the first condition in the powder flow through the plastic pipes.

Suppose a stream of negatively charged powder was blown through a cylindrical pipe of radius R = 5.0 cm. Assume that the powder and its charge were spread uniformly through the pipe with a volume charge density ρ. (a) using Gauss’ law find an expression for the magnitude of the electric field $\stackrel{\to }{E}$ in the pipe as a function of radial distance r from the pipe center, (b) Does E increase or decrease with increasing r? (c) Is $\stackrel{\to }{E}$ directed radially inward or outward? (d) For ρ = 1.1 × 10⁻³ C/m³ (a typical value at the factory), find the maximum E and determine where that maximum field occurs. (e) Could sparking occur, anti if so, where? (The story continues with Problem 70 in Chapter 24.)