Chapter 23

An electric field given by \(\vec{E}=4.0 \mathrm{i}-3.0\left(y^{2}+2.0\right) \hat{\mathrm{j}} \quad\) pierces a Gaussian cube of edge length \(2.0 \mathrm{~m}\) and positioned as shown in Fig. 23-7. (The magnitude \(E\) is in newtons per coulomb and the position \(x\) is in meters.) What is the electric flux through the (a) top face, (b) bottom face, (c) left face, and (d) back face? (e) What is the net electric flux through the cube?

In Fig. \(23-32,\) a butterfly net is in a uniform electric field of magnitude \(E=3.0 \mathrm{mN} / \mathrm{C}\). The rim, a circle of radius \(a=11 \mathrm{~cm},\) is aligned perpendicular to the field. The net contains no net charge. Find the electric flux through the netting.

A particle of charge \(1.8 \mu \mathrm{C}\) is at the center of a Gaussian cube \(55 \mathrm{~cm}\) on edge. What is the net electric flux through the surface?

When a shower is turned on in a closed bathroom, the splashing of the water on the bare tub can fill the room's air with negatively charged ions and produce an electric field in the air as great as \(1000 \mathrm{~N} / \mathrm{C}\). Consider a bathroom with dimensions \(2.5 \mathrm{~m} \times 3.0 \mathrm{~m} \times 2.0 \mathrm{~m} .\) Along the ceiling, floor, and four walls, approximate the electric field in the air as being directed perpendicular to the surface and as having a uniform magnitude of \(600 \mathrm{~N} / \mathrm{C}\). Also, treat those surfaces as forming a closed Gaussian surface around the room's air. What are (a) the volume charge density \(\rho\) and (b) the number of excess elementary charges \(e\) per cubic meter in the room's air?

Fig. 23-31 shows a Gaussian surface in the shape of a cube with edge length \(1.40 \mathrm{~m}\). What are (a) the net flux \(\Phi\) through the surface and (b) the net charge \(q_{\mathrm{cnc}}\) enclosed by the surface if \(\vec{E}=(3.00 y \hat{j}) \mathrm{N} / \mathrm{C},\) with \(y\) in meters? What are (c) \(\Phi\) and (d) \(q_{\text {cnc }}\) if \(\vec{E}=[-4.00 \hat{\mathrm{i}}+(6.00+3.00 y) \mathrm{j}] \mathrm{N} / \mathrm{C} ?\)

Figure 23-34 shows a closed Gaussian surface in the shape of a cube of edge length \(2.00 \mathrm{~m} .\) It lies in a region where the nonuniform electric field is given by \(\vec{E}=(3.00 x+\) 4.00)\(\hat{\mathrm{i}}+6.00 \mathrm{j}+7.00 \hat{\mathrm{k}} \mathrm{N} / \mathrm{C},\) with \(x\) in meters. What is the net charge contained by the cube?

Go Figure \(23-35\) shows a closed Gaussian surface in the shape of a cube of edge length \(2.00 \mathrm{~m},\) with one corner at \(x_{1}=5.00 \mathrm{~m}\), \(y_{1}=4.00 \mathrm{~m} .\) The cube lies in a region where the electric field vector is given by \(\vec{E}=-3.00 \mathrm{i}-4.00 y^{2} \mathrm{j}+3.00 \mathrm{k} \mathrm{N} / \mathrm{C},\) with \(y\) in meters. What is the net charge contained by the cube?

The electric field in a certain region of Earth's atmosphere is directed vertically down. At an altitude of \(300 \mathrm{~m}\) the field has magnitude \(60.0 \mathrm{~N} / \mathrm{C} ;\) at an altitude of \(200 \mathrm{~m},\) the magnitude is \(100 \mathrm{~N} / \mathrm{C} .\) Find the net amount of charge contained in a cube \(100 \mathrm{~m}\) on edge, with horizontal faces at altitudes of 200 and \(300 \mathrm{~m}\).

A particle of charge \(+q\) is placed at one corner of a Gaussian cube. What multiple of \(q / \varepsilon_{0}\) gives the flux through (a) each cube face forming that corner and (b) each of the other cube faces?

A uniformly charged conducting sphere of \(1.2 \mathrm{~m}\) diameter has surface charge density \(8.1 \mu \mathrm{C} / \mathrm{m}^{2}\). Find (a) the net charge on the sphere and (b) the total electric flux leaving the surface.

The electric field just above the surface of the charged conducting drum of a photocopying machine has a magnitude \(E\) of \(2.3 \times 10^{5} \mathrm{~N} / \mathrm{C} .\) What is the surface charge density on the drum?

Space vehicles traveling through Earth's radiation belts can intercept a significant number of electrons. The resulting charge buildup can damage electronic components and disrupt operations. Suppose a spherical metal satellite \(1.3 \mathrm{~m}\) in diameter accumulates \(2.4 \mu \mathrm{C}\) of charge in one orbital revolution. (a) Find the resulting surface charge density. (b) Calculate the magnitude of the electric field just outside the surface of the satellite, due to the surface charge.

An isolated conductor has net charge \(+10 \times 10^{-6} \mathrm{C}\) and a cavity with a particle of charge \(q=+3.0 \times 10^{-6} \mathrm{C}\). What is the charge on (a) the cavity wall and (b) the outer surface?

An electron is released \(9.0 \mathrm{~cm}\) from a very long nonconducting rod with a uniform \(6.0 \mu \mathrm{C} / \mathrm{m}\). What is the magnitude of the electron's initial acceleration?

(a) The drum of a photocopying machine has a length of \(42 \mathrm{~cm}\) and a diameter of \(12 \mathrm{~cm} .\) The electric field just above the drum's surface is \(2.3 \times 10^{5} \mathrm{~N} / \mathrm{C}\). What is the total charge on the drum? (b) The manufacturer wishes to produce a desktop version of the machine. This requires reducing the drum length to \(28 \mathrm{~cm}\) and the diameter to \(8.0 \mathrm{~cm} .\) The electric field at the drum surface must not change. What must be the charge on this new drum?

An infinite line of charge produces a field of magnitude \(4.5 \times 10^{4} \mathrm{~N} / \mathrm{C}\) at distance \(2.0 \mathrm{~m}\). Find the linear charge density.

A long, straight wire has fixed negative charge with a linear charge density of magnitude \(3.6 \mathrm{nC} / \mathrm{m}\). The wire is to be enclosed by a coaxial, thin-walled nonconducting cylindrical shell of radius \(1.5 \mathrm{~cm}\). The shell is to have positive charge on its outside surface with a surface charge density \(\sigma\) that makes the net external electric field zero. Calculate \(\sigma\).

A charge of uniform linear density \(2.0 \mathrm{nC} / \mathrm{m}\) is distributed along a long, thin, nonconducting rod. The rod is coaxial with a long conducting cylindrical shell (inner radius \(=5.0 \mathrm{~cm},\) outer radius \(=10 \mathrm{~cm}\) ). The net charge on the shell is zero. (a) What is the magnitude of the electric field \(15 \mathrm{~cm}\) from the axis of the shell? What is the surface charge density on the (b) inner and (c) outer surface of the shell?

Figure 23-42 is a section of a conducting rod of radius \(\quad R_{1}=1.30 \mathrm{~mm}\) and length \(L=11.00 \mathrm{~m}\) inside a thin-walled coaxial conducting cylindrical shell of radius \(R_{2}=10.0 R_{1}\) and the (same) length \(L .\) The net charge on the rod is \(Q_{1}=+3.40 \times 10^{-12} \mathrm{C} ;\) that on the shell is \(Q_{2}=-2.00 Q_{1} .\) What are the (a) magnitude \(E\) and (b) direction (radially inward or outward) of the electric field at radial distance \(r=2.00 R_{2} ?\) What are \((\mathrm{c}) E\) and \((\mathrm{d})\) the direction at \(r=5.00 R_{1} ?\) What is the charge on the (e) interior and (f) exterior surface of the shell?

Two long, charged, thin-walled, concentric cylindrical shells have radii of 3.0 and \(6.0 \mathrm{~cm} .\) The charge per unit length is \(5.0 \times 10^{-6} \mathrm{C} / \mathrm{m}\) on the inner shell and \(-7.0 \times 10^{-6} \mathrm{C} / \mathrm{m}\) on the outer shell. What are the (a) magnitude \(E\) and (b) direction (radially inward or outward) of the electric field at radial distance \(r=4.0 \mathrm{~cm} ?\) What are (c) \(E\) and (d) the direction at \(r=8.0 \mathrm{~cm} ?\)

A long, nonconducting, solid cylinder of radius \(4.0 \mathrm{~cm}\) has a nonuniform volume charge density \(\rho\) that is a function of radial distance \(r\) from the cylinder axis: \(\rho=A r^{2} .\) For \(A=2.5 \mu \mathrm{C} / \mathrm{m}^{5},\) what is the magnitude of the electric field at (a) \(r=3.0 \mathrm{~cm}\) and (b) \(r=5.0 \mathrm{~cm} ?\)

A square metal plate of edge length \(8.0 \mathrm{~cm}\) and negligible thickness has a total charge of \(6.0 \times 10^{-6} \mathrm{C}\). (a) Estimate the magnitude \(E\) of the electric field just off the center of the plate (at, say, a distance of \(0.50 \mathrm{~mm}\) from the center by assuming that the charge is spread uniformly over the two faces of the plate. (b) Estimate \(\bar{E}\) at a distance of \(30 \mathrm{~m}\) (large relative to the plate size) by assuming that the plate is a charged particle.

An electron is shot directly toward the center of a large metal plate that has surface charge density \(-2.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2} .\) If the initial kinetic energy of the electron is \(1.60 \times 10^{-17} \mathrm{~J}\) and if the electron is to stop (due to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be?

Two large metal plates of area \(1.0 \mathrm{~m}^{2}\) face each other, \(5.0 \mathrm{~cm}\) apart, with equal charge magnitudes \(|q|\) but opposite signs. The field magnitude \(E\) between them (neglect fringing) is \(55 \mathrm{~N} / \mathrm{C}\). Find \(|q|\).

Two charged concentric spherical shells have radii \(10.0 \mathrm{~cm}\) and \(15.0 \mathrm{~cm} .\) The charge on the inner shell is \(4.00 \times 10^{-8} \mathrm{C},\) and that on the outer shell is \(2.00 \times 10^{-8} \mathrm{C}\). Find the electric field (a) at \(r=12.0 \mathrm{~cm}\) and (b) at \(r=20.0 \mathrm{~cm}\)

An unknown charge sits on a conducting solid sphere of radius \(10 \mathrm{~cm}\). If the electric field \(15 \mathrm{~cm}\) from the center of the sphere has the magnitude \(3.0 \times 10^{3} \mathrm{~N} / \mathrm{C}\) and is directed radially inward, what is the net charge on the sphere?

The volume charge density of a solid nonconducting sphere of radius \(R=5.60 \mathrm{~cm}\) varies with radial distance \(r\) as given by \(\rho=\left(14.1 \mathrm{pC} / \mathrm{m}^{3}\right) r / R .\) (a) What is the sphere's total charge? What is the field magnitude \(E\) at (b) \(r=0,(\mathrm{c}) r=R / 2.00,\) and (d) \(r=R ?\) (e) Graph \(E\) versus \(r\)

A charge distribution that is spherically symmetric but not uniform radially produces an electric field of magnitude \(E=K r^{4}\), directed radially outward from the center of the sphere. Here \(r\) is the radial distance from that center, and \(K\) is a constant. What is the volume density \(\rho\) of the charge distribution?

The electric field in a particular space is \(\vec{E}=(x+2) \hat{\mathrm{i}} \mathrm{N} / \mathrm{C}\) with \(x\) in meters. Consider a cylindrical Gaussian surface of radius \(20 \mathrm{~cm}\) that is coaxial with the \(x\) axis. One end of the cylinder is at \(x=0 .\) (a) What is the magnitude of the electric flux through the other end of the cylinder at \(x=2.0 \mathrm{~m} ?\) (b) What net charge is enclosed within the cylinder?

A thin-walled metal spherical shell has radius \(25.0 \mathrm{~cm}\) and charge \(2.00 \times 10^{-7} \mathrm{C}\). Find \(E\) for a point (a) inside the shell, (b) just outside it, and (c) \(3.00 \mathrm{~m}\) from the center.

A uniform surface charge of density \(8.0 \mathrm{nC} / \mathrm{m}^{2}\) is distributed over the entire \(x y\) plane. What is the electric flux through a spherical Gaussian surface centered on the origin and having a radius of \(5.0 \mathrm{~cm} ?\)

Charge of uniform volume density \(\rho=1.2 \mathrm{nC} / \mathrm{m}^{3}\) fills an infinite slab between \(x=-5.0 \mathrm{~cm}\) and \(x=+5.0 \mathrm{~cm} .\) What is the magnitude of the electric field at any point with the coordinate (a) \(x=4.0 \mathrm{~cm}\) and (b) \(x=6.0 \mathrm{~cm} ?\)

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 \(3.0 \times 10^{6} \mathrm{~N} / \mathrm{C}\) or greater, so that electrical breakdown and thus sparking could occur. ( 2 ) The energy of a spark was \(150 \mathrm{~mJ}\) 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 \(R=5.0 \mathrm{~cm}\). Assume that the powder and its charge were spread uniformly through the pipe with a volume charge density \(\rho\). (a) Using Gauss' law, find an expression for the magnitude of the electric field \(\vec{E}\) 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 \(\vec{E}\) directed radially inward or outward? (d) For \(\rho=1.1 \times 10^{-3} \mathrm{C} / \mathrm{m}^{3}\) (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 thin-walled metal spherical shell of radius \(a\) has a charge \(q_{a}\). Concentric with it is a thin-walled metal spherical shell of radius \(b>a\) and charge \(q_{b} .\) Find the electric field at points a distance \(r\) from the common center, where (a) \(rb\). (d) Discuss the criterion you would use to determine how the charges are distributed on the inner and outer surfaces of the shells.

A particle of charge \(q=1.0 \times 10^{-7} \mathrm{C}\) is at the center of a spherical cavity of radius \(3.0 \mathrm{~cm}\) in a chunk of metal. Find the electric field (a) \(1.5 \mathrm{~cm}\) from the cavity center and (b) anyplace in the metal.

Charge \(Q\) is uniformly distributed in a sphere of radius \(R\). (a) What fraction of the charge is contained within the radius \(r=R / 2.00 ?\) (b) What is the ratio of the electric field magnitude at \(r=R / 2.00\) to that on the surface of the sphere?

A charged particle causes an electric flux of \(-750 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}\) to pass through a spherical Gaussian surface of \(10.0 \mathrm{~cm}\) radius centered on the charge. (a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the charge of the particle?

ssim The electric field at point \(P\) just outside the outer surface of a hollow spherical conductor of inner radius \(10 \mathrm{~cm}\) and outer radius \(20 \mathrm{~cm}\) has magnitude \(450 \mathrm{~N} / \mathrm{C}\) and is directed outward. When a particle of unknown charge \(Q\) is introduced into the center of the sphere, the electric field at \(P\) is still directed outward but is now \(180 \mathrm{~N} / \mathrm{C}\). (a) What was the net charge enclosed by the outer surface before \(Q\) was introduced? (b) What is charge \(Q\) ? After \(Q\) is introduced, what is the charge on the (c) inner and (d) outer surface of the conductor?

The net electric flux through each face of a die (singular of dice) has a magnitude in units of \(10^{3} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}\) that is exactly equal to the number of spots \(N\) on the face ( 1 through 6 ). The flux is inward for \(N\) odd and outward for \(N\) even. What is the net charge inside the die?

Charge of uniform volume density \(\rho=3.2 \mu \mathrm{C} / \mathrm{m}^{3}\) fills a nonconducting solid sphere of radius \(5.0 \mathrm{~cm}\). What is the magnitude of the electric field (a) \(3.5 \mathrm{~cm}\) and (b) \(8.0 \mathrm{~cm}\) from the sphere's center?

A Gaussian surface in the form of a hemisphere of radius \(R=5.68 \mathrm{~cm}\) lies in a uniform electric field of magnitude \(E=2.50 \mathrm{~N} / \mathrm{C}\). 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 uniform charge density of \(500 \mathrm{nC} / \mathrm{m}^{3}\) is distributed throughout a spherical volume of radius \(6.00 \mathrm{~cm}\). Consider a cubical Gaussian surface with its center at the center of the sphere. What is the electric flux through this cubical surface if its edge length is (a) \(4.00 \mathrm{~cm}\) and \((\mathrm{b}) 14.0 \mathrm{~cm} ?\)

Figure \(23-61\) shows a Geiger counter, a device used to detect ionizing radiation, which causes ionization of atoms. A thin, positively charged central wire is surrounded by a concentric, circular, conducting cylindrical shell with an equal negative charge, creating a strong radial electric field. The shell contains a low-pressure inert gas. A particle of radiation entering the device through the shell wall ionizes a few of the gas atoms. The resulting free electrons (e) are drawn to the positive wire. However, the electric field is so intense that, between collisions with gas atoms, the free electrons gain energy sufficient to ionize these atoms also. More free electrons are thereby created, and the process is repeated until the electrons reach the wire. The resulting "avalanche" of electrons is collected by the wire, generating a signal that is used to record the passage of the original particle of radiation. Suppose that the radius of the central wire is \(25 \mu \mathrm{m},\) the inner radius of the shell \(1.4 \mathrm{~cm},\) and the length of the shell \(16 \mathrm{~cm} .\) If the electric field at the shell's inner wall is \(2.9 \times 10^{4} \mathrm{~N} / \mathrm{C},\) what is the total positive charge on the central wire?

Charge is distributed uniformly throughout the volume of an infinitely long solid cylinder of radius \(R\). (a) Show that, at a distance \(rR\)

A spherical conducting shell has a charge of \(-14 \mu \mathrm{C}\) on its outer surface and a charged particle in its hollow. If the net charge on the shell is \(-10 \mu \mathrm{C},\) what is the charge (a) on the inner surface of the shell and (b) of the particle?

A charge of \(6.00 \mathrm{pC}\) is spread uniformly throughout the volume of a sphere of radius \(r=4.00 \mathrm{~cm}\). What is the magnitude of the electric field at a radial distance of (a) \(6.00 \mathrm{~cm}\) and (b) \(3.00 \mathrm{~cm} ?\)

Water in an irrigation ditch of width \(w=3.22 \mathrm{~m}\) and depth \(d=1.04 \mathrm{~m}\) flows with a speed of \(0.207 \mathrm{~m} / \mathrm{s}\). The mass flux of the flowing water through an imaginary surface is the product of the water's density \(\left(1000 \mathrm{~kg} / \mathrm{m}^{3}\right)\) and its volume flux through that surface. Find the mass flux through the following imaginary surfaces: (a) a surface of area \(w d\), entirely in the water, perpendicular to the flow; (b) a surface with area \(3 w d / 2,\) of which \(w d\) is in the water, perpendicular to the flow; (c) a surface of area \(w d / 2,\) entirely in the water, perpendicular to the flow; (d) a surface of area \(w d\), half in the water and half out, perpendicular to the flow; (e) a surface of area \(w d\), entirely in the water, with its normal \(34.0^{\circ}\) from the direction of flow.

Charge of uniform surface density \(8.00 \mathrm{nC} / \mathrm{m}^{2}\) is distributed over an entire \(x y\) plane; charge of uniform surface density \(3.00 \mathrm{nC} / \mathrm{m}^{2}\) is distributed over the parallel plane defined by \(z=2.00 \mathrm{~m}\). Determine the magnitude of the electric field at any point having a \(z\) coordinate of (a) \(1.00 \mathrm{~m}\) and (b) \(3.00 \mathrm{~m}\).

A spherical ball of charged particles has a uniform charge density. In terms of the ball's radius \(R,\) at what radial distances (a) inside and (b) outside the ball is the magnitude of the ball's electric field equal to \(\frac{1}{4}\) of the maximum magnitude of that field?