A spherical shell has inner radius Rin and outer radius

What is Gausss law?

What good is symmetry?

What is electric flux?

How is Gausss law used?

What can we learn about conductors?

A uniformly charged rod has a finite length L. The rod is symmetric under rotations about the axis and under reflection in any plane containing the axis. It is not symmetric under translations or under reflections in a plane perpendicular to the axis unless that plane bisects the rod. Which field shape or shapes match the symmetry of the rod?

Two 100 cm2 parallel electrodes are spaced 2.0 cm apart. One is charged to +5.0 nC, the other to -5.0 nC. A 1.0 cm * 1.0 cm surface between the electrodes is tilted to where its normal makes a 45 angle with the electric field. What is the electric flux through this surface?

This box containsa. A positive charge. b. A negative charge. c. No charge. d. A net positive charge. e. A net negative charge. f. No net charge.

A charge distribution with cylindrical symmetry has created the electric field E u = E01r2 /r0 2 2nr, where E0 and r0 are constants and where unit vector nr lies in the xy-plane. Calculate the electric flux through a closed cylinder of length L and radius R that is centered along the z-axis.

The total electric flux through this box is a. 0 N m2 /C b. 1 N m2 /C c. 2 N m2 /C d. 4 N m2 /C e. 6 N m2 /C f. 8 N m2 /C

In Chapter 23 we asserted, without proof, that the electric field outside a sphere of total charge Q is the same as the field of a point charge Q at the center. Use Gausss law to prove this result.

These are two-dimensional cross sections through three-dimensional closed spheres and a cube. Rank in order, from largest to smallest, the electric fluxes a to e through surfaces a to e.

What is the electric field inside a uniformly charged sphere?

In Chapter 23, we used superposition to find the electric field of a long, charged wire with linear charge density l 1C/m2. It was not an easy derivation. Find the electric field using Gausss law.

Which Gaussian surface would allow you to use Gausss law to calculate the electric field outside a uniformly charged cube? a. A sphere whose center coincides with the center of the charged cube b. A cube whose center coincides with the center of the charged cube and that has parallel faces c. Either a or b d. Neither a nor b

Use Gausss law to find the electric field of an infinite plane of charge with surface charge density h 1C/m2 2.

A 2.0-cm-diameter brass sphere has been given a charge of 2.0 nC. What is the electric field strength at the surface?

An infinite slab of charge of thickness 2a is centered in the xyplane. The charge density is r = r011 - 0z0 /a2. Find the electric field strengths inside and outside this slab of charge

Suppose you have the uniformly charged cube in FIGURE Q24.1. Can you use symmetry alone to deduce the shape of the cubes electric field? If so, sketch and describe the field shape. If not, why not?

FIGURE Q24.2 shows cross sections of three-dimensional closed surfaces. They have a flat top and bottom surface above and below the plane of the page. However, the electric field is everywhere parallel to the page, so there is no flux through the top or bottom surface. The electric field is uniform over each face of the surface. For each, does the surface enclose a net positive charge, a net negative charge, or no net charge? Explain.

The square and circle in FIGURE Q24.3 are in the same uniform field. The diameter of the circle equals the edge length of the square. Is square larger than, smaller than, or equal to circle? Explain

In FIGURE Q24.4, where the field is uniform, is the magnitude of 1 larger than, smaller than, or equal to the magnitude of 2? Explain

What is the electric flux through each of the surfaces in FIGURE Q24.5? Give each answer as a multiple of q/P0.

What is the electric flux through each of the surfaces A to E in FIGURE Q24.6? Give each answer as a multiple of q/P0.

The charged balloon in FIGURE Q24.7 expands as it is blown up, increasing in size from the initial to final diameters shown. Do the electric field strengths at points 1, 2, and 3 increase, decrease, or stay the same? Explain your reasoning for each

The two spheres in FIGURE Q24.8 on the next page surround equal charges. Three students are discussing the situation. Student 1: The fluxes through spheres A and B are equal because they enclose equal charges. Student 2: But the electric field on sphere B is weaker than the electric field on sphere A. The flux depends on the electric field strength, so the flux through A is larger than the flux through B.

The sphere and ellipsoid in FIGURE Q24.9 surround equal charges. Four students are discussing the situation. Student 1: The fluxes through A and B are equal because the average radius is the same. Student 2: I agree that the fluxes are equal, but thats because they enclose equal charges. Student 3: The electric field is not perpendicular to the surface for B, and that makes the flux through B less than the flux through A. Student 4: I dont think that Gausss law even applies to a situation like B, so we cant compare the fluxes through A and B.

A small, metal sphere hangs by an insulating thread within the larger, hollow conducting sphere of FIGURE Q24.10. A conducting wire extends from the small sphere through, but not touching, a small hole in the hollow sphere. A charged rod is used to transfer positive charge to the protruding wire. After the charged rod has touched the wire and been removed, are the following surfaces positive, negative, or not charged? Explain. a. The small sphere. b. The inner surface of the hollow sphere. c. The outer surface of the hollow sphere

FIGURE EX24.1 shows two cross sections of two infinitely long coaxial cylinders. The inner cylinder has a positive charge, the outer cylinder has an equal negative charge. Draw this figure on your paper, then draw electric field vectors showing the shape of the electric field.

FIGURE EX24.2 shows a cross section of two concentric spheres. The inner sphere has a negative charge. The outer sphere has a positive charge larger in magnitude than the charge on the inner sphere. Draw this figure on your paper, then draw electric field vectors showing the shape of the electric field.

FIGURE EX24.3 shows a cross section of two infinite parallel planes of charge. Draw this figure on your paper, then draw electric field vectors showing the shape of the electric field.

The electric field is constant over each face of the cube shown in FIGURE EX24.4. Does the box contain positive charge, negative charge, or no charge? Explain.

The electric field is constant over each face of the cube shown in FIGURE EX24.5. Does the box contain positive charge, negative charge, or no charge? Explain.

The cube in FIGURE EX24.6 contains negative charge. The electric field is constant over each face of the cube. Does the missing electric field vector on the front face point in or out? What strength must this field exceed?

The cube in FIGURE EX24.7 contains negative charge. The electric field is constant over each face of the cube. Does the missing electric field vector on the front face point in or out? What strength must this field exceed?

The cube in FIGURE EX24.8 contains no net charge. The electric field is constant over each face of the cube. Does the missing electric field vector on the front face point in or out? What is the field strength?

What is the electric flux through the surface shown in FIGURE EX24.9?

What is the electric flux through the surface shown in FIGURE EX24.10?

The electric flux through the surface shown in FIGURE EX24.11 is 25 N m2 /C. What is the electric field strength?

A 2.0 cm * 3.0 cm rectangle lies in the xy-plane. What is the magnitude of the electric flux through the rectangle if a. E u = 1100ni - 200k n2 N/C? b. E u = 1100ni - 200nj2 N/C?

A 2.0 cm * 3.0 cm rectangle lies in the xz-plane. What is the magnitude of the electric flux through the rectangle if a. E u = 1100ni - 200k n2 N/C? b. E u = 1100ni - 200nj2 N/C?

A 3.0-cm-diameter circle lies in the xz-plane in a region where the electric field is E u = 11500ni + 1500nj - 1500k n2 N/C. What is the electric flux through the circle?

A 1.0 cm * 1.0 cm * 1.0 cm box with its edges aligned with the xyz-axes is in the electric field E u = 1350x + 1502ni N/C, where x is in meters. What is the net electric flux through the box?

What is the net electric flux through the two cylinders shown in FIGURE EX24.16? Give your answer in terms of R and E.

FIGURE EX24.17 shows three charges. Draw these charges on your paper four times. Then draw two-dimensional cross sections of three-dimensional closed surfaces through which the electric flux is (a) 2q/P0, (b) q/P0, (c) 0, and (d) 5q/P0.

FIGURE EX24.18 shows three charges. Draw these charges on your paper four times. Then draw two-dimensional cross sections of three-dimensional closed surfaces through which the electric flux is (a) -q/P0, (b) q/P0, (c) 3q/P0, and (d) 4q/P0.

What is the net electric flux through the torus (i.e., doughnut shape) of FIGURE EX24.20?

What is the net electric flux through the cylinder of FIGURE EX24.21?

The net electric flux through an octahedron is -1000 N m2 /C. How much charge is enclosed within the octahedron?

55.3 million excess electrons are inside a closed surface. What is the net electric flux through the surface?

A spark occurs at the tip of a metal needle if the electric field strength exceeds 3.0 * 106 N/C, the field strength at which air breaks down. What is the minimum surface charge density for producing a spark?

The electric field strength just above one face of a copper penny is 2000 N/C. What is the surface charge density on this face of the penny?

The conducting box in FIGURE EX24.26 has been given an excess negative charge. The surface density of excess electrons at the center of the top surface is 5.0 * 1010 electrons/m2 . What are the electric field strengths E1 to E3 at points 1 to 3?

FIGURE EX24.27 shows a hollow cavity within a neutral conductor. A point charge Q is inside the cavity. What is the net electric flux through the closed surface that surrounds the conductor?

A thin, horizontal, 10-cm-diameter copper plate is charged to 3.5 nC. If the electrons are uniformly distributed on the surface, what are the strength and direction of the electric field a. 0.1 mm above the center of the top surface of the plate? b. at the plates center of mass? c. 0.1 mm below the center of the bottom surface of the plate?

Find the electric fluxes 1 to 5 through surfaces 1 to 5 in FIGURE P24.29.

FIGURE P24.30 shows four sides of a 3.0 cm * 3.0 cm * 3.0 cm cube. a. What are the electric fluxes 1 to 4 through sides 1 to 4? b. What is the net flux through these four sides?

A tetrahedron has an equilateral triangle base with 20-cm-long edges and three equilateral triangle sides. The base is parallel to the ground, and a vertical uniform electric field of strength 200 N/C passes upward through the tetrahedron. a. What is the electric flux through the base? b. What is the electric flux through each of the three sides?

Charges q1 = -4Q and q2 = +2Q are located at x = -a and x = +a, respectively. What is the net electric flux through a sphere of radius 2a centered (a) at the origin and (b) at x = 2a?

A 10 nC charge is at the center of a 2.0 m * 2.0 m * 2.0 m cube. What is the electric flux through the top surface of the cube?

A spherically symmetric charge distribution produces the electric field E u = 15000r2 2nr N/C, where r is in m. a. What is the electric field strength at r = 20 cm? b. What is the electric flux through a 40-cm-diameter spherical surface that is concentric with the charge distribution? c. How much charge is inside this 40-cm-diameter spherical surface?

A neutral conductor contains a hollow cavity in which there is a +100 nC point charge. A charged rod then transfers -50 nC to the conductor. Afterward, what is the charge (a) on the inner wall of the cavity, and (b) on the exterior surface of the conductor?

A hollow metal sphere has inner radius a and outer radius b. The hollow sphere has charge +2Q. A point charge +Q sits at the center of the hollow sphere. a. Determine the electric fields in the three regions r a, a 6 r 6 b, and r b. b. How much charge is on the inside surface of the hollow sphere? On the exterior surface?

A 20-cm-radius ball is uniformly charged to 80 nC. a. What is the balls volume charge density 1C/m3 2? b. How much charge is enclosed by spheres of radii 5, 10, and 20 cm? c. What is the electric field strength at points 5, 10, and 20 cm from the center?

FIGURE P24.38 shows a solid metal sphere at the center of a hollow metal sphere. What is the total charge on (a) the exterior of the inner sphere, (b) the inside surface of the hollow sphere, and (c) the exterior surface of the hollow sphere?

The earth has a vertical electric field at the surface, pointing down, that averages 100 N/C. This field is maintained by various atmospheric processes, including lightning. What is the excess charge on the surface of the earth?

Figure 24.32b showed a conducting box inside a parallel-plate capacitor. The electric field inside the box is E u = 0 u . Suppose the surface charge on the exterior of the box could be frozen. Draw a picture of the electric field inside the box after the box, with its frozen charge, is removed from the capacitor. Hint: Superposition.

A hollow metal sphere has 6 cm and 10 cm inner and outer radii, respectively. The surface charge density on the inside surface is -100 nC/m2 . The surface charge density on the exterior surface is +100 nC/m2 . What are the strength and direction of the electric field at points 4, 8, and 12 cm from the center?

A positive point charge q sits at the center of a hollow spherical shell. The shell, with radius R and negligible thickness, has net charge -2q. Find an expression for the electric field strength (a) inside the sphere, r 6 R, and (b) outside the sphere, r 7 R. In what direction does the electric field point in each case?

Find the electric field inside and outside a hollow plastic ball of radius R that has charge Q uniformly distributed on its outer surface

A uniformly charged ball of radius a and charge -Q is at the center of a hollow metal shell with inner radius b and outer radius c. The hollow sphere has net charge +2Q. Determine the electric field strength in the four regions r a, a 6 r 6 b, b r c, and r 7 c

The three parallel planes of charge shown in FIGURE P24.45 have surface charge densities - 1 2 h, h, and - 1 2 h. Find the electric fields E u 1 to E u 4 in regions 1 to 4.

An infinite slab of charge of thickness 2z0 lies in the xyplane between z = -z0 and z = +z0. The volume charge density r 1C/m3 2 is a constant. a. Use Gausss law to find an expression for the electric field strength inside the slab 1-z0 z z02. b. Find an expression for the electric field strength above the slab 1z z02. c. Draw a graph of E from z = 0 to z = 3z0.

FIGURE P24.47 shows an infinitely wide conductor parallel to and distance d from an infinitely wide plane of charge with surface charge density h. What are the electric fields E u 1 to E u 4 in regions 1 to 4?

FIGURE P24.48 shows two very large slabs of metal that are parallel and distance l apart. The top and bottom of each slab has surface area A. The thickness of each slab is so small in comparison to its lateral dimensions that the surface area around the sides is negligible. Metal 1 has total charge Q1 = Q and metal 2 has total charge Q2 = 2Q. Assume Q is positive. In terms of Q and A, determine a. The electric field strengths E1 to E5 in regions 1 to 5. b. The surface charge densities ha to hd on the four surfaces a to d.

A long, thin straight wire with linear charge density l runs down the center of a thin, hollow metal cylinder of radius R. The cylinder has a net linear charge density 2l. Assume l is positive. Find expressions for the electric field strength (a) inside the cylinder, r 6 R, and (b) outside the cylinder, r 7 R. In what direction does the electric field point in each of the cases?

A very long, uniformly charged cylinder has radius R and linear charge density l. Find the cylinders electric field strength (a) outside the cylinder, r R, and (b) inside the cylinder, r R. (c) Show that your answers to parts a and b match at the boundary, r = R

The electric field must be zero inside a conductor in electrostatic equilibrium, but not inside an insulator. It turns out that we can still apply Gausss law to a Gaussian surface that is entirely within an insulator by replacing the right-hand side of Gausss law, Qin /P0, with Qin /P, where P is the permittivity of the material. (Technically, P0 is called the vacuum permittivity.) Suppose that a 50 nC point charge is surrounded by a thin, 32-cm-diameter spherical rubber shell and that the electric field strength inside the rubber shell is 2500 N/C. What is the permittivity of rubber?

The electric field must be zero inside a conductor in electrostatic equilibrium, but not inside an insulator. It turns out that we can still apply Gausss law to a Gaussian surface that is entirely within an insulator by replacing the right-hand side of Gausss law, Qin /P0, with Qin /P, where P is the permittivity of the material. (Technically, P0 is called the vacuum permittivity.) Suppose a long, straight wire with linear charge density 250 nC/m is covered with insulation whose permittivity is 2.5P0. What is the electric field strength at a point inside the insulation that is 1.5 mm from the axis of the wire?

A long cylinder with radius b and volume charge density r has a spherical hole with radius a 6 b centered on the axis of the cylinder. What is the electric field strength inside the hole at radial distance r 6 a in a plane that is perpendicular to the cylinder through the center of the hole? Hint: Can you create this charge distribution as a superposition of charge distributions for which you can use Gausss law to find the electric field?

A spherical shell has inner radius Rin and outer radius Rout . The shell contains total charge Q, uniformly distributed. The interior of the shell is empty of charge and matter. a. Find the electric field strength outside the shell, r Rout . b. Find the electric field strength in the interior of the shell, r Rin. c. Find the electric field strength within the shell, Rin r Rout . d. Show that your solutions match at both the inner and outer boundaries.

An early model of the atom, proposed by Rutherford after his discovery of the atomic nucleus, had a positive point charge +Ze (the nucleus) at the center of a sphere of radius R with uniformly distributed negative charge -Ze. Z is the atomic number, the number of protons in the nucleus and the number of electrons in the negative sphere. a. Show that the electric field strength inside this atom is Ein = Ze 4pP0 1 1 r2 - r R3 2 b. What is E at the surface of the atom? Is this the expected value? Explain. c. A uranium atom has Z = 92 and R = 0.10 nm. What is the electric field strength at r = 1 2 R?

Newtons law of gravity and Coulombs law are both inversesquare laws. Consequently, there should be a Gausss law for gravity. a. The electric field was defined as E u = F u on q /q, and we used this to find the electric field of a point charge. Using analogous reasoning, what is the gravitational field g u of a point mass?Write your answer using the unit vector nr, but be careful with signs; the gravitational force between two like masses is attractive, not repulsive. b. What is Gausss law for gravity, the gravitational equivalent of Equation 24.18? Use G for the gravitational flux, g u for the gravitational field, and Min for the enclosed mass. c. A spherical planet is discovered with mass M, radius R, and a mass density that varies with radius as r = r011 - r/2R2, where r0 is the density at the center. Determine r0 in terms of M and R. Hint: Divide the planet into infinitesimal shells of thickness dr, then sum (i.e., integrate) their masses. d. Find an expression for the gravitational field strength inside the planet at distance r 6 R.

All examples of Gausss law have used highly symmetric surfaces where the flux integral is either zero or EA. Yet weve claimed that the net e = Qin/P0 is independent of the surface. This is worth checking. FIGURE CP24.57 shows a cube of edge length L centered on a long thin wire with linear charge density l. The flux through one face of the cube is not simply EA because, in this case, the electric field varies in both strength and direction. But you can calculate the flux by actually doing the flux integral.a. Consider the face parallel to the yz-plane. Define area dA u as a strip of width dy and height L with the vector pointing in the x-direction. One such strip is located at position y. Use the known electric field of a wire to calculate the electric flux d through this little area. Your expression should be written in terms of y, which is a variable, and various constants. It should not explicitly contain any angles. b. Now integrate d to find the total flux through this face. c. Finally, show that the net flux through the cube is e = Qin/P0.

An infinite cylinder of radius R has a linear charge density l. The volume charge density 1C/m3 2 within the cylinder 1r R2 is r1r2 = rr0/R, where r0 is a constant to be determined. a. Draw a graph of r versus x for an x-axis that crosses the cylinder perpendicular to the cylinder axis. Let x range from -2R to 2R. b. The charge within a small volume dV is dq = r dV. The integral of r dV over a cylinder of length L is the total charge Q = lL within the cylinder. Use this fact to show that r0 = 3l/2pR2 . Hint: Let dV be a cylindrical shell of length L, radius r, and thickness dr. What is the volume of such a shell? c. Use Gausss law to find an expression for the electric field strength E inside the cylinder, r R, in terms of l and R. d. Does your expression have the expected value at the surface, r = R? Explain

A sphere of radius R has total charge Q. The volume charge density (C/m3 ) within the sphere is r(r) = C/r2 , where C is a constant to be determined. a. The charge within a small volume dV is dq = r dV. The integral of r dV over the entire volume of the sphere is the total charge Q. Use this fact to determine the constant C in terms of Q and R. Hint: Let dV be a spherical shell of radius r and thickness dr. What is the volume of such a shell? b. Use Gausss law to find an expression for the electric field strength E inside the sphere, r R, in terms of Q and R. c. Does your expression have the expected value at the surface, r = R? Explain.

A sphere of radius R has total charge Q. The volume charge density (C/m3 ) within the sphere is r = r011 - r R2 This charge density decreases linearly from r0 at the center to zero at the edge of the sphere. a. Show that r0 = 3Q/pR3 . b. Show that the electric field inside the sphere points radially outward with magnitude E = Qr 4pP0R3 14 - 3 r R2 c. Show that your result of part b has the expected value at r = R

A spherical ball of charge has radius R and total charge Q. The electric field strength inside the ball 1r R2 is E1r2 = r4 Emax /R4 . a. What is Emax in terms of Q and R? b. Find an expression for the volume charge density r1r2 inside the ball as a function of r. c. Verify that your charge density gives the total charge Q when integrated over the volume of the ball.