A solid sphere of radius R carries a nonuniform volume charge density r = r0er/R, where

Can electric field lines ever cross? Why or why not?

The electric flux through a closed surface is zero. Must the electric field be zero on that surface? If not, give an example.

If the flux of the gravitational field through a closed surface is zero, what can you conclude about the region interior to the surface?

Under what conditions can the electric flux through a surface be written as EA, where A is the surface area?

Eight field lines emerge from a closed surface surrounding an isolated point charge. Would the number of field lines change if a second identical charge were brought to a point just outside the surface? If not, would anything change? Explain.

If a charged particle were released from rest on a curved field line, would its subsequent motion follow the field line? Explain.

In Gausss law, A E S # dA S = q/P0, does the field E S necessarily arise only from charges within the closed surface?

In a certain region the electric field points to the right and its magnitude increases as you move to the right, as shown in Fig. 21.30. Does the region contain net positive charge, net negative charge, or zero net charge?

A point charge is located a fixed distance outside of a uniformly charged sphere. If the sphere shrinks in size without losing any charge, what happens to the force on the point charge?

The field of an infinite charged line decreases as 1/r. Why isnt this a violation of the inverse-square law?

Why cant you use Gausss law to determine the field of a uniformly charged cube? Why couldnt you use a cubical Gaussian surface?

Youre sitting inside an uncharged, hollow spherical shell. Suddenly someone dumps a billion coulombs of charge on the shell, distributed uniformly. What happens to the electric field at your location?

Does Gausss law apply to a spherical Gaussian surface not centered on a point charge, as shown in Fig. 21.31? Would this be a useful surface to use in calculating the electric field?

An insulating sphere carries charge spread uniformly throughout its volume. A conducting sphere has the same radius and net charge, but of course the charge is spread over its surface only (assume uniformly in this case). Compare the electric fields outside these two charge distributions.

Why must the electric field be zero inside a conductor in electrostatic equilibrium?

The electric field of a flat sheet of charge is s/2P0. Yet the field of a flat conducting sheeteven a thin one, like a piece of aluminum foilis s/P0. Explain this apparent discrepancy

In Fig. 21.32, the magnitude of the middle charge is 3 C. Whats the net charge shown?

Charges +2q and -q are near each other. Sketch some field lines for this charge distribution, using eight lines for a charge of magnitude q.

The net charge shown in Fig. 21.33 is +Q. Identify each of the charges A, B, and C shown.

A flat surface with area 2.0 m2 is in a uniform 850-N/C electric field. Find the electric flux through the surface when its (a) at right angles to the field, (b) at 45 to the field, and (c) parallel to the field.

Whats the electric field strength in a region where the flux through a 1.0 cm * 1.0 cm flat surface is 65 N # m2 /C, if the field is uniform and the surface is at right angles to the field?

A flat surface with area 0.14 m2 lies in the x@y plane, in a uniform electric field E S = 5.1ni + 2.1nj + 3.5kn kN/C. Find the flux through the surface.

The electric field on the surface of a 10-cm-diameter sphere is perpendicular to the sphere and has magnitude 47 kN/C. Whats the electric flux through the sphere?

In the figure with GOT IT? 21.2, take E = 1.75 kN/C and s = 125 cm. Find the flux through faces B and C of cubes (a) and (b).

In Fig. 21.8, take the half-cylinders radius and length to be 3.4 cm and 15 cm, respectively. If the electric field has magnitude 6.8 kN/C, find the flux through the half-cylinder. Hint: You dont need to do an integral! Why not?

A sock comes out of the dryer with a trillion 110122 excess electrons. Whats the electric flux through a surface surrounding the sock?

Whats the electric flux through the closed surfaces marked (a), (b), (c), and (d) in Fig. 21.34?

A 6.8@C charge and a -4.7@C charge are inside an uncharged sphere. Whats the electric flux through the sphere?

A 2.6@C charge is at the center of a cube 7.5 cm on each side. Whats the electric flux through one face of the cube? (Hint: Think about symmetry, and dont do an integral.)

The electric field at the surface of a 5.0-cm-radius uniformly charged sphere is 90 kN/C. Whats the field strength 10 cm from the surface?

A solid sphere 25 cm in radius carries 14 C, distributed uniformly throughout its volume. Find the electric field strength (a) 15 cm, (b) 25 cm, and (c) 50 cm from its center

A 15-nC point charge is at the center of a thin spherical shell of radius 10 cm, carrying -22 nC of charge distributed uniformly over its surface. Find the magnitude and direction of the electric field (a) 2.2 cm, (b) 5.6 cm, and (c) 14 cm from the point charge.

The electric field strength outside a charge distribution and 18 cm from its center has magnitude 55 kN/C. At 23 cm the field strength is 43 kN/C. Does the distribution have spherical or line symmetry?

An electron close to a large, flat sheet of charge is repelled from the sheet with a 1.8-pN force. Find the surface charge density on the sheet

Find the field produced by a uniformly charged sheet carrying 87 pC/m2

What surface charge density on an infinite sheet will produce a 1.4-kN/C electric field?

A rod 50 cm long and 1.0 cm in radius carries a 2.0@C charge distributed uniformly over its length. Find the approximate magnitude of the electric field (a) 4.0 mm from the rod surface, not near either end, and (b) 23 m from the rod.

Whats the approximate field strength 1 cm above a sheet of paper carrying uniform surface charge density s = 45 nC/m2 ?

The disk in Fig. 21.22 has area 0.14 m2 and is uniformly charged to 5.0 C. Find the approximate field strength (a) 1 mm from the disk, not near the edge, and (b) 2.5 m from the disk.

What is the electric field strength just outside the surface of a conducting sphere carrying surface charge density 1.4 C/m2 ?

A net charge of 5.0 C is applied on one side of a solid metal sphere 2.0 cm in diameter. Once electrostatic equilibrium is reached, and assuming no other conductors or charges nearby, what are (a) the volume charge density inside the sphere and (b) the surface charge density on the sphere?

A positive point charge q lies at the center of a spherical conducting shell carrying net charge 3 2 q. Sketch the field lines both inside and outside the shell, using eight field lines to represent a charge of magnitude q.

A total charge of 18C is applied to a thin, square metal plate 75 cm on a side. Find the electric field strength near the plates surface.

Whats the flux through the hemispherical open surface of radius R in a uniform field of magnitude E shown in Fig. 21.35? (Hint: Dont do a messy integral!)

An electric field is given by E S = E01y/a2kn, where E0 and a are constants. Find the flux through the square in the x@y plane bounded by the points 10, 02, 10, a2, 1a, a2, 1a, 02

The electric field in a certain region is given by E S = axni, where a = 40 N/C # m and x is in meters. Find the volume charge density in the region. (Hint: Apply Gausss law to a cube 1 m on a side.)

A study shows that mammalian red blood cells (RBCs) carry electric charge resulting from 4.4 million (for rabbit cells) to 15 million (for human cells) excess electrons spread over their surfaces. Approximating rabbit and human RBCs as spheres with radii 30 m and 36 m, respectively, find the electric field strengths at the cells surfaces.

Positive charge is spread uniformly over the surface of a spherical balloon 70 cm in radius, resulting in an electric field of 26 kN/C at the balloons surface. Find the field strength (a) 50 cm from the balloons center and (b) 190 cm from the center. (c) Whats the net charge on the balloon?

A solid sphere 2.0 cm in radius carries a uniform volume charge density. The electric field 1.0 cm from the spheres center has magnitude 39 kN/C. (a) At what other distance does the field have this magnitude? (b) Whats the net charge on the sphere?

A point charge of -2Q is at the center of a spherical shell of radius R carrying charge Q spread uniformly over its surface. Find the electric field at (a) r = 1 2R and (b) r = 2R. (c) How would your answers change if the charge on the shell were doubled?

A friend is working on a biology experiment and needs to create an electric field of magnitude 430 N/C at 10 cm from the central portion of a large nonconducting square plate 4.5 m on each side. She needs to know how much charge to put on the plate. What do you tell her?

A spherical shell of radius 15 cm carries 4.8 C distributed uniformly over its surface. At the center of the shell is a point charge. If the electric field at the spheres surface is 750 kN/C and points outward, what are (a) the point charge and (b) the field just inside the shell?

A spherical shell 30 cm in diameter carries 85 C distributed uniformly over its surface. A 1.0@C point charge is located at the shells center. Find the electric field strength (a) 5.0 cm from the center and (b) 45 cm from the center. (c) How would your answers change if the charge on the shell were doubled?

A thick, spherical shell of inner radius a and outer radius b carries a uniform volume charge density r. Find an expression for the electric field strength in the region a 6 r 6 b, and show that your result is consistent with Equation 21.5 when a = 0.

A long, thin wire carrying 5.6 nC/m runs down the center of a long, thin-walled, pipe with radius 1.0 cm carrying -4.2 nC/m spread uniformly over its surface. Find the electric field (a) 0.50 cm from the wire and (b) 1.5 cm from the wire.

An infinitely long rod of radius R carries a uniform volume charge density r. Show that the electric field strengths outside and inside the rod are given, respectively, by E = rR2 /2P0r and E = rr/2P0, where r is the distance from the rod axis. (Although an infinite rod is an impossibility, your answer is a good approximation for the field of a finite rod whose length is much greater than its diameter.)

A long, solid rod 4.5 cm in radius carries a uniform volume charge density. If the electric field strength at the surface of the rod (not near either end) is 16 kN/C, whats the volume charge density?

If you painted positive charge on the floor, what surface charge density would be necessary to suspend a 15C, 5.0-g particle above the floor?

A charged slab extends infinitely in two dimensions and has thickness d in the third dimension, as shown in Fig. 21.36. The slab carries a uniform volume charge density r. Find expressions for the electric field (a) inside and (b) outside the slab, as functions of the distance x from the center plane. (Although the infinite slab is impossible, your answer is a good approximation to the field of a finite slab whose width is much greater than its thickness.)

A solid sphere 10 cm in radius carries a 40@C charge distributed uniformly throughout its volume. Its surrounded by a concentric shell 20 cm in radius, also uniformly charged with 40 C. Find the electric field (a) 5.0 cm, (b) 15 cm, and (c) 30 cm from the center.

A nonconducting square plate 75 cm on a side carries a uniform surface charge density. The electric field strength 1 cm from the plate, not near an edge, is 45 kN/C. Whats the approximate field strength 15 m from the plate?

A 250-nC point charge is placed at the center of an uncharged spherical conducting shell 20 cm in radius. Find (a) the surface charge density on the outer surface of the shell and (b) the electric field strength at the shells outer surface

An irregular conductor containing an irregular, empty cavity carries a net charge Q. (a) Show that the electric field inside the cavity must be zero. (b) If you put a point charge inside the cavity, what value must it have in order to make the charge density on the outer surface of the conductor everywhere zero?

You measure the electric field strength at points directly above the center of a square plate carrying charge spread uniformly over its surface. The data are tabulated in the next column, with x the perpendicular distance from the center of the plate. Use the data to determine (a) the total charge on the plate and (b) the plates size. Hint: Youll need to consider separately data taken lose to the plate and also far away. For the latter, plot E versus a quantity that should yield a straight line. x (cm) 0.01 0.02 1.2 6.0 12.0 24.0 E (N/C) 5870 5860 4840 1960 754 221 x (cm) 48.0 72.0 96.0 120 240 E (N/C) 57.6 26.7 16.1 8.45 2.34

A point charge -q is at the center of a spherical shell carrying charge +2q. That shell, in turn, is concentric with a larger shell carrying -3 2 q. Draw a cross section of this structure, and sketch the electric field lines using the convention that eight lines correspond to a charge of magnitude q

A point charge q is at the center of a spherical shell of radius R carrying charge 2q spread uniformly over its surface. Write expressions for the electric field strength at (a) 1 2 R and (b) 2R.

The volume charge density inside a solid sphere of radius a is r = r0r/a, where r0 is a constant. Find (a) the total charge and (b) the electric field strength within the sphere, as a function of distance r from the center.

Figure 21.37 shows a rectangular box with sides 2a and length L surrounding a line carrying uniform line charge density l. The line passes directly through the center of the box faces. Integrate the field of the line charge over strips of width dx as shown to find the electric flux through one face of the box. Multiply by 4 to get the total flux, and show that your result is consistent with Gausss law.

The charge density within a charged sphere of radius R is given by r = r0 - ar2 , where r0 and a are constants and r is the distance from the center. Find an expression for a such that the electric field outside the sphere is zero

Calculate the electric fields in Example 21.2 directly, using the superposition principle and integration. Consider the shell to be composed of charge elements that are coaxial rings, whose axes pass through the field point, which is a distance r from the center. (Hint: Consult Example 20.6. Youll have to evaluate the cases r 6 R and r 7 R separately.)

A solid sphere of radius R carries a nonuniform volume charge density r = r0er/R, where r0 is a constant and r is the distance from the center. Find an expression for the electric field strength at the spheres surface.

Problem 76 of Chapter 13 explored what happened to a person falling into a hole extending all the way through Earths center and out the other side, assuming that g1r2 = g01r/RE2 for points inside Earth 1r 6 RE2. Prove this assumption, treating Earth as a uniform sphere and using the gravitational version of Gausss law: Ag ! # dA S = -4pGMenclosed.

An infinitely long solid cylinder of radius R carries a nonuniform charge density given by r = r01r/R2, where r0 is a constant and r is the distance from the cylinders axis. Find an expression for the magnitude of the electric field as a function of position r within the cylinder.

A solid sphere of radius R carries a uniform volume charge density r. A hole of radius R/2 occupies a region from the center to the edge of the sphere, as shown in Fig. 21.38. Show that the electric field everywhere in the hole points horizontally and has magnitude rR/6P0. Hint: Treat the hole as a superposition of two charged spheres with opposite charges

Repeat Problem 59 for the case where the charge density in the slab is given by r = r0  x/d , where r0 is a constant.

Coaxial cables are widely used with audio-visual technology, electronic instrumentation, and radio broadcasting, because they minimize interference with or from signals traveling on the cable. Coaxial cables consist of a wire inner conductor surrounded by a thin cylindrical conducting shield, usually of braided copper (Fig. 21.39). Flexible insulation separates the conductors. A straight length of coaxial cable can be approximated as an infinitely long wire surrounded by a cylindrical shell. Normally the two conductors carry charges of equal magnitude but opposite sign. (Charge actually varies with time and position as signals travel down the cable, but for these problems consider the charge to be fixed and spread uniformly.) For a coaxial cable in electrostatic equilibrium carrying equal but opposite charges on its two conductors, theres a nonzero electric field a. only in the space between the wire and shield. b. in the space between wire and shield, and outside the shield. c. inside the metal conducting wire and shield, as well as between the wires and outside the shield. d. only outside the shield.

Coaxial cables are widely used with audio-visual technology, electronic instrumentation, and radio broadcasting, because they minimize interference with or from signals traveling on the cable. Coaxial cables consist of a wire inner conductor surrounded by a thin cylindrical conducting shield, usually of braided copper (Fig. 21.39). Flexible insulation separates the conductors. A straight length of coaxial cable can be approximated as an infinitely long wire surrounded by a cylindrical shell. Normally the two conductors carry charges of equal magnitude but opposite sign. (Charge actually varies with time and position as signals travel down the cable, but for these problems consider the charge to be fixed and spread uniformly.) A coaxial cable carries equal but opposite charges on its two conductors. In electrostatic equilibrium, charge on the shield a. lies entirely on its outer surface. b. is divided evenly between inner and outer surfaces. c. lies entirely on its inner surface. d. distributes itself differently depending on the magnitude of the charge.

Coaxial cables are widely used with audio-visual technology, electronic instrumentation, and radio broadcasting, because they minimize interference with or from signals traveling on the cable. Coaxial cables consist of a wire inner conductor surrounded by a thin cylindrical conducting shield, usually of braided copper (Fig. 21.39). Flexible insulation separates the conductors. A straight length of coaxial cable can be approximated as an infinitely long wire surrounded by a cylindrical shell. Normally the two conductors carry charges of equal magnitude but opposite sign. (Charge actually varies with time and position as signals travel down the cable, but for these problems consider the charge to be fixed and spread uniformly.) How does the electric field between the conductors in a coaxial cable in electrostatic equilibrium depend on the radial distance r from the cables axis? a. its constant b. as 1/r c. as 1/r2 d. as 1/r3

Coaxial cables are widely used with audio-visual technology, electronic instrumentation, and radio broadcasting, because they minimize interference with or from signals traveling on the cable. Coaxial cables consist of a wire inner conductor surrounded by a thin cylindrical conducting shield, usually of braided copper (Fig. 21.39). Flexible insulation separates the conductors. A straight length of coaxial cable can be approximated as an infinitely long wire surrounded by a cylindrical shell. Normally the two conductors carry charges of equal magnitude but opposite sign. (Charge actually varies with time and position as signals travel down the cable, but for these problems consider the charge to be fixed and spread uniformly.) A coaxial cable in electrostatic equilibrium carries charge -Q on its inner conductor and +Q on its shield. If the charge on the shield only is doubled, a. the magnitude of the electric field between the conductors will double. b. the magnitude of the electric field outside the shield will double. c. the magnitude of the electric field at the outer surface of the shield will become twice the magnitude of the field at the shields inner surface. d. the magnitude of the electric field at the outer surface of the shield will equal the magnitude of the field at the shields inner surface