?A spherical conducting shell has a charge of \(-14\ \mu \mathrm{C}\) on its outer

A surface has the area vector \(\vec{A}=(2\hat{i}+3\hat{j}) \mathrm{\ m}^2\). What is the flux of a uniform electric field through the area if the field is (a) \(\vec{E}=4 \hat{\mathrm{i}} \mathrm{\ N}/\mathrm{C}\) and (b) \(\vec{E}=4\hat{\mathrm{k}} \mathrm{\ N}/\mathrm{C}\)? Text Transcription: vec A = (2 hat i + 3 hat j) m^2 vec E = 4 hat i N/C vec E = 4 hat k N/C

Figure 23-22 shows, in cross section, three solid cylinders, each of length L and uniform charge Q. Concentric with each cylinder is a cylindrical Gaussian surface, with all three surfaces having the same radius. Rank the Gaussian surfaces according to the electric field at any point on the surface, greatest first.

Figure 23-23 shows, in cross section, a central metal ball, two spherical metal shells, and three spherical Gaussian surfaces of radii R, 2R, and 3R, all with the same center. The uniform charges on the three objects are: ball, Q; smaller shell, 3Q; larger shell, 5Q. Rank the Gaussian surfaces according to the magnitude of the electric field at any point on the surface, greatest first.

Figure 23-24 shows, in cross section, two Gaussian spheres and two Gaussian cubes that are centered on a positively charged particle. (a) Rank the net flux through the four Gaussian surfaces, greatest first. (b) Rank the magnitudes of the electric fields on the surfaces, greatest first, and indicate whether the magnitudes are uniform or variable along each surface.

In Fig. 23-25, an electron is released between two infinite nonconducting sheets that are horizontal and have uniform surface charge densities \(\sigma_{(+)}\text{and }\sigma_{\left((-\right)}\), as indicated. The electron is subjected to the following three situations involving surface charge densities and sheet separations. Rank the magnitudes of the electron’s acceleration, greatest first. Text Transcription: sigma_(+) and sigma_(-)

Three infinite nonconducting sheets, with uniform positive surface charge densities \(\sigma, 2 \sigma, \text { and } 3 \sigma\), are arranged to be parallel like the two sheets in Fig. 23-19a. What is their order, from left to right, if the electric field \(\vec{E}\) produced by the arrangement has magnitude E = 0 in one region and \(E=2 \sigma / \varepsilon_{0}\) in another region? Text Transcription: sigma, 2sigma, and 3sigma vec E E = 2sigma/varepsilon_0

Figure 23-26 shows four situations in which four very long rods extend into and out of the page (we see only their cross sections). The value below each cross section gives that particular rod’s uniform charge density in microcoulombs per meter. The rods are separated by either d or 2d as drawn, and a central point is shown midway between the inner rods. Rank the situations according to the magnitude of the net electric field at that central point, greatest first.

Figure 23-27 shows four solid spheres, each with charge Q uniformly distributed through its volume. (a) Rank the spheres according to their volume charge density, greatest first. The figure also shows a point P for each sphere, all at the same distance from the center of the sphere. (b) Rank the spheres according to the magnitude of the electric field they produce at point P, greatest first.

Figure 23-28 shows a section of three long charged cylinders centered on the same axis. Central cylinder A has a uniform charge \(q_{A}=+3 q_{0}\). What uniform charges \(q_{B}\) and \(q_{C}\) should be on cylinders B and C so that (if possible) the net electric field is zero at (a) point 1, (b) point 2, and (c) point 3? Text Transcription q_A=+3q_0 q_B q_C

Figure 23-29 shows four Gaussian surfaces consisting of identical cylindrical midsections but different end caps. The surfaces are in a uniform electric field \(\vec{E}\) that is directed parallel to the central axis of each cylindrical midsection. The end caps have these shapes: \(S_{1}\), convex hemispheres; \(S_{2}\), concave hemispheres; \(S_{3}\), cones; \(S_{4}\), flat disks. Rank the surfaces according to (a) the net electric flux through them and (b) the electric flux through the top end caps, greatest first. Text Transcription vec E S_1 S_2 S_3 S_4

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 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 m? (b) What net charge is enclosed within the cylinder? Text Transcription: vec E=(x+2) hat i N/C

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

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

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

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 \(3.0 \times 10^{6} \mathrm{N} / \mathrm{C}\) or greater, so that electrical breakdown and thus sparking could occur. (2) The energy of a spark was 150 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 \(\rho\). (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.) Text Transcription: 3.0 times 10^6 N/C rho vec E rho=1.1 times 10^-3 C/m^3

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) r < a, (b) a < r < b, and (c) r > b. (d) Discuss the criterion you would use to determine how the charges are distributed on the inner and outer surfaces of the shells. Text Transcription: q_a q_b

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

A proton at speed \(v=3.00 \times 10^{5} \mathrm{m} / \mathrm{s}\) orbits at radius r = 1.00 cm outside a charged sphere. Find the sphere’s charge. Text Transcription: v=3.00 times 10^5 m/s

Equation 23-11 \(\left(E=\sigma / \varepsilon_{0}\right)\) gives the electric field at points near a charged conducting surface. Apply this equation to a conducting sphere of radius r and charge q, and show that the electric field outside the sphere is the same as the field of a charged particle located at the center of the sphere. Text Transcription: (E=sigma/varepsilon_0)

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 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? Text Transcription: -750 N cdot m^2/C

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? Text Transcription: 10^3 N cdot m^2/C

Figure 23-59 shows, in cross section, three infinitely large nonconducting sheets on which charge is uniformly spread. The surface charge densities are \(\sigma_1=+2.00\ \mu\mathrm{C}/\mathrm{m}^2,\ \sigma_2=+4.00\ \mu\mathrm{C}/\mathrm{m}^2,\quad\text{ and }\sigma_3=-5.00\ \mu \mathrm{C}/ \mathrm{m}^2\), and distance L = 1.50 cm. In unit vector notation, what is the net electric field at point P? Text Transcription: sigma_1=+2.00 mu C/m^2, sigma_2=+4.00 mu C/m^2, and sigma_3=-5.00 mu C/m^2

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

A nonconducting solid sphere has a uniform volume charge density \(\rho\). Let \(\vec{r}\) be the vector from the center of the sphere to a general point P within the sphere. (a) Show that the electric field at P is given by \(\overrightarrow{\vec{E}}=\rho \vec{r} / 3 \varepsilon_{0}\). (Note that the result is independent of the radius of the sphere.) (b) A spherical cavity is hollowed out of the sphere, as shown in Fig. 23-60. Using superposition concepts, show that the electric field at all points within the cavity is uniform and equal to \(\vec{E}=\rho \vec{a} / 3 \varepsilon_{0}\), where \(\vec{a}\) is the position vector from the center of the sphere to the center of the cavity. Text Transcription: rho vec r overrightarrow vec E=rho vec r/3varepsilon_0 vec E=rho vec a/3varepsilon_0 vec a

A uniform charge density of \(500\ \mathrm{nC}/ \mathrm{m}^3\) is distributed throughout a spherical volume of radius 6.00 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 cm and (b) 14.0 cm? Text Transcription: 500 nC/m^3

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 cm, and the length of the shell 16 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? Text Transcription: 25 mu m 2.9 times 10^4 N/C

Charge is distributed uniformly throughout the volume of an infinitely long solid cylinder of radius R. (a) Show that, at a distance r < R from the cylinder axis, \(E=\frac{\rho r}{2 \varepsilon_{0}}\), where \(\rho\) is the volume charge density. (b) Write an expression for E when r > R. Text Transcription: E=rho r/2 varepsilon_0 rho

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? Text Transcription: -14 mu C -10 mu C

Water in an irrigation ditch of width w = 3.22 m and depth d = 1.04 m flows with a speed of 0.207 m/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 wd, entirely in the water, perpendicular to the flow; (b) a surface with area 3wd/2, of which wd is in the water, perpendicular to the flow; (c) a surface of area wd/2, entirely in the water, perpendicular to the flow; (d) a surface of area wd, half in the water and half out, perpendicular to the flow; (e) a surface of area wd, entirely in the water, with its normal \(34.0^{\circ}\) from the direction of flow. Text Transcription: (1000 kg/m^3) 34.0^circ

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

A small charged ball lies within the hollow of a metallic spherical shell of radius R. For three situations, the net charges on the ball and shell, respectively, are (1) +4q, 0; (2) ?6q, +10q; (3) +16q, ?12q. Rank the situations according to the charge on (a) the inner surface of the shell and (b) the outer surface, most positive first.

Rank the situations of Question 9 according to the magnitude of the electric field (a) halfway through the shell and (b) at a point 2R from the center of the shell, greatest first.

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?

The electric field at point P just outside the outer surface of a hollow spherical conductor of inner radius 10 cm and outer radius 20 cm has magnitude 450 N/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 N/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?

A Gaussian surface in the form of a hemisphere of radius R = 5.68 cm lies in a uniform electric field of magnitude E = 2.50 N/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?

What net charge is enclosed by the Gaussian cube of Problem 2?

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

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 14 of the maximum magnitude of that field?