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 x 106 N/C or greater, so that electrical breakdown and thus sparking could occur. (2) The energy of a spark was 150mJ 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 p.

(a) Using Gauss' law, find an expression for the magnitude of the electric field 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 E directed radially inward or outward?

(d) For p = l.1 x 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, and if so, where?