A long, straight wire has fixed negative charge with a linear charge density of magnitude 3.6nC/m . The wire is to be enclosed by a coaxial, thin-walled non-conducting cylindrical shell of radius 1.5 cm . The shell is to have positive charge on its outside surface with a surface charge density s that makes the net external electric field zero. Calculate s.
The surface density of the outside surface is .
a) The magnitude of linear density,
b) The wire has fixed negative charge as -q .
c) The radius of the cylindrical shell, r=1.5 cm
d) The net external field is zero.
Using the concept of the electric field of a Gaussian cylindrical surface, we can get the value of the linear density of the cylinder for the net external field to be zero. Now, using this linear density, we can get the surface density of the cylinder.
The magnitude of the electric field of a Gaussian cylindrical surface, (1)
The surface density of a cylinder,
The net electric field for r > R is given by using equation (1) as follows:
But, for the given net external field value to be zero, we get the linear charge density of the cylinder from the above equation as:
Now, substituting this value in equation (2), we can get the value of the surface charge density as follows:
Hence, the value of the surface charge density is .
A Gaussian surface in the form of a hemisphere of radius lies in a uniform electric field of magnitude . The surface encloses no net charge. At the (flat) base of the surface, the field is perpendicular to the surface and directed into the surface. What is the flux through
(a) the base and
(b) the curved portion of the surface?
A charge of uniform linear density 2.0nC/m is distributed along a long, thin, non-conducting rod. The rod is coaxial with a long conducting cylindrical shell (inner radius=5.0 cm , outer radius=10 cm ). The net charge on the shell is zero. (a) What is the magnitude of the electric field from the axis of the shell? What is the surface charge density on the (b) inner and (c) outer surface of the shell?
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 1970 s. 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 or greater, so that electrical breakdown and thus sparking could occur. (2) The energy of a spark was 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 . Assume that the powder and its charge were spread uniformly through the pipe with a volume charge density r.
(a) Using Gauss’ law, find an expression for the magnitude of the electric field 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 directed radially inward or outward?
(d) For (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? (The story continues with Problem 70 in Chapter 24.)
A square metal plate of edge length 8.0cm and negligible thickness has a total charge of . (a) Estimate the magnitude E of the electric field just off the center of the plate (at, say, a distance of 0.50mm from the center) by assuming that the charge is spread uniformly over the two faces of the plate. (b) Estimate E at a distance of 30m (large relative to the plate size) by assuming that the plate is a charged particle.
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