(II) A long cylindrical shell of radius R q and length (Rq

If the electric flux through a closed surface is zero, is the electric field necessarily zero at all points on the surface? Explain. What about the converse: If E = 0 at all points on the surface is the flux through the surface zero?

Is the electric field E in Gausss law, <|>E dA = Qencl/eo> created only by the charge Qenci ?

A point charge is surrounded by a spherical gaussian surface of radius r. If the sphere is replaced by a cube of side r, will be larger, smaller, or the same? Explain.

What can you say about the flux through a closed surface that encloses an electric dipole?

The electric field E is zero at all points on a closed surface; is there necessarily no net charge within the surface? If a surface encloses zero net charge, is the electric field necessarily zero at all points on the surface?

Define gravitational flux in analogy to electric flux. Are there sources and sinks for the gravitational field as there are for the electric field? Discuss.

Would Gausss law be helpful in determining the electric field due to an electric dipole?

A spherical basketball (a nonconductor) is given a charge Q distributed uniformly over its surface. What can you say about the electric field inside the ball? A person now steps on the ball, collapsing it, and forcing most of the air out without altering the charge. What can you say about the field inside now?

In Example 22-6, it may seem that the electric field calculated is due only to the charge on the wire that is enclosed by the cylinder chosen as our gaussian surface. In fact, the entire charge along the whole length of the wire contributes to the field. Explain how the charge outside the cylindrical gaussian surface of Fig. 22-15 contributes to E at the gaussian surface. [Hint: Compare to what the field would be due to a short wire.l

Suppose the line of charge in Example 22-6 extended only a short way beyond the ends of the cylinder shown in Fig. 22-15. How would the result of Example 22-6 be altered?

A point charge Q is surrounded by a spherical surface of radius rQ, whose center is at C. Later, the charge is moved to the right a distance | r 0, but the sphere remains where it was, Fig. 22-23. How is the electric flux OE through the sphere changed? Is the electric field at the surface of the sphere changed? For each yes answer, describe the change.

A solid conductor carries a net positive charge Q. There is a hollow cavity within the conductor, at whose center is a negative point charge q q (Fig. 22-24). What is the charge on (a) the outer surface of the conductor and (b) the inner surface of the conductors cavity?

A point charge q is placed at the center of the cavity of a thin metal shell which is neutral. Will a charge Q placed outside the shell feel an electric force? Explain.

A small charged ball is inserted into a balloon. The balloon is then blown up slowly. Describe how the flux through the balloons surface changes as the balloon is blown up. Consider both the total flux and the flux per unit surface area of the balloon.

(I) A long thin wire, hundreds of meters long, carries a uniformly distributed charge of 7.2 juC per meter of length. Estimate the magnitude and direction of the electric field at points (a) 5.0 m and (b) 1.5 m perpendicular from the center of the wire.

(I) A metal globe has 1.50 mC of charge put on it at the north pole. Then 3.00 mC of charge is applied to the south pole. Draw the field lines for this system after it has come to equilibrium.

(II) A nonconducting sphere is made of two layers. The innermost section has a radius of 6.0 cm and a uniform charge density of -5.0 C /m3. The outer layer has a uniform charge density of +8.0 C/m3 and extends from an inner radius of 6.0 cm to an outer radius of 12.0 cm. Determine the electric field for (a) 0 < r < 6.0 cm, (b) 6.0 cm < r < 12.0 cm, and (c) 12.0 cm < r < 50.0 cm. (d) Plot the magnitude of the electric field for 0 < r < 50.0 cm. Is the field continuous at the edges of the layers?

(II) A solid metal sphere of radius 3.00 m carries a total charge of -5.50 fxC. What is the magnitude of the electric field at a distance from the spheres center of (a) 0.250 m, (b) 2.90 m, (c) 3.10 m, and (d) 8.00 m? How would the answers differ if the sphere were (e) a thin shell, or ( /) a solid nonconductor uniformly charged throughout?

(II) A 15.0-cm-diameter nonconducting sphere carries a total charge of 2.25 /aC distributed uniformly throughout its volume. Graph the electric field E as a function of the distance r from the center of the sphere from r = 0 to r = 30.0 cm.

(II) A flat square sheet of thin aluminum foil, 25 cm on a side, carries a uniformly distributed 275 nC charge. What, approximately, is the electric field (a) 1.0 cm above the center of the sheet and (b) 15 m above the center of the sheet?

(II) A spherical cavity of radius 4.50 cm is at the center of a metal sphere of radius 18.0 cm. A point charge Q = 5.50 /jlC rests at the very center of the cavity, whereas the metal conductor carries no net charge. Determine the electric field at a point (a) 3.00 cm from the center of the cavity, (b) 6.00 cm from the center of the cavity, (c) 30.0 cm from the center.

(II) A point charge Q rests at the center of an uncharged thin spherical conducting shell. What is the electric field E as a function of r (a) for r less than the radius of the shell, (b) inside the shell, and (c) beyond the shell? (d) Does the shell affer.t the field due to (1 alone? D o e s the r.harae O

(II) A solid metal cube has a spherical cavity at its center as shown in Fig. 22-29. At the center of the cavity there is a point charge Q = +8.00 /jlC. The metal cube carries a net charge q = -6.10 /xC (not including Q). Determine (a) the total charge on the surface of the spherical cavity and (b) the total charge on the outer surface of the cube.

(II) Two large, flat metal plates are separated by a distance that is very small compared to their height and width. The conductors are given equal but opposite uniform surface charge densities + cr. Ignore edge effects and use Gausss law to show (a) that for points far from the edges, the electric field between the plates is E = cr/e0 and (b) that outside the plates on either side the field is zero, (c) How would your results be altered if the two plates were nonconductors? (See Fig. 22-30).

(II) Suppose the two conducting plates in Problem 24 have the same sign and magnitude of charge. What then will be the electric field (a) between them and (b) outside them on either side? (c) What if the plates are nonconducting?

(II) The electric field between two square metal plates is 160 N/C. The plates are 1.0 m on a side and are separated by 3.0 cm, as in Fig. 22-30. What is the charge on each plate? Neglect edge effects.

(II) Two thin concentric spherical shells of radii and r2 (t*i < r2) contain uniform surface charge densities ct\ and cr2, respectively (see Fig. 22-31). Determine the electric field for (a) 0 < r < r \, (b) < r < r2, and (c) r > r2. (<d) Under what conditions will E = 0 for r > r2l (e) Under what conditions will E = 0 for r\ < r < r2l v Neglect the thickness of the shells.

(II) A spherical rubber balloon carries a total charge Q uniformly distributed on its surface. At t = 0 the nonconducting balloon has radius rQ and the balloon is then slowly blown up so that r increases linearly to 2r0 in a time t. Determine the electric field as a function of time (a) just outside the balloon surface and (b) at r = 3.2r0.

(II) Suppose the nonconducting sphere of Example 22-4 has a spherical cavity of radius centered at the spheres center (Fig. 22-32). Assuming the charge Q is distributed uniformly in the shell (between r = r\ and r = r0), determine the electric field as a function of r for (a) 0 < r < rx, (b) rx < r < r0, and (c) r > r0.

(II) Suppose in Fig. 22-32, Problem 29, there is also a charge n at the center nf the cavitv. Determine the electric

(II) Suppose the thick spherical shell of Problem 29 is a conductor. It carries a total net charge Q and at its center there is a point charge q. What total charge is found on (a) the inner surface of the shell and (b) the outer surface of the shell? Determine the electric field for (c) 0 < r < rl5 (id) rx < r < r0, and (e) r > r0.

(II) Suppose that at the center of the cavity inside the shell (charge Q) of Fig. 22-11 (and Example 22-3), there is a point charge q Q). Determine the electric field for (a) 0 < r < r0, and for (b) r > r0. What are your answers if (c) q = Q and (d) q = -Q l

(II) A long cylindrical shell of radius R q and length (Rq . ) possesses a uniform surface charge density (charge per unit area) cr (Fig. 22-33). Determine the electric field at points (a) outside the cylinder (R > Rq) and (b) inside the cylinder (0 < R < R0); assume the points are far from the ends and not , too far from the shell ( / ? ) . (c) Compare to the result for a longline of charge, Example 22-6. Neglect | the thickness of shell.

(II) A very long solid nonconducting cylinder of radius R0 and length (R0 ) possesses a uniform volume charge density pE (C/m3), Fig. 22-34. Determine the electric field at points (a) outside the cylinder (R > R0) and (b) inside the cylinder (R < R0). Do only for points far from the ends and for which R L

(II) A thin cylindrical shell of radius R\ is surrounded by a second concentric cylindrical shell of radius R2 (Fig. 22-35). The inner shell has a total charge + Q and the outer shell Q. Assuming the length of the shells is much greater than R\ or R2, determine the electric field as a function of R (the perpendicular distance from the common axis of the cylinders) for (a) 0 < R < R i , (b) Ri < R < R2, and (c) R > R2. (d) What is the kinetic energy of an electron if it moves between (and concentric with) the shells in a circular orbit of radius (Ri + R2)/21 Neglect thickness of shells.

(II) A thin cylindrical shell of radius Ri = 6.5 cm is surrounded by a second cylindrical shell of radius R2 = 9.0 cm, as in Fig. 22-35. Both cylinders are 5.0 m long and the inner one carries a total charge Q\ = 0.88 /jlC and the outer one Q2 = +1.56 fxC. For points far from the ends of the cylinders, determine the electric field at a radial distance r from the central axis of (a) 3.0 cm, (b) 7.0 cm, and (c) 12.0 cm.

(II) (a) If an electron (m = 9.1 X 10-31kg) escaped from the surface of the inner cylinder in Problem 36 (Fig. 22-35) with negligible speed, what would be its speed when it reached the outer cylinder? (b) If a proton (m = 1.67 X 10-27kg) revolves in a circular nrhit o f radius r = 7 0 cm about the

(II) A very long solid nonconducting cylinder of radius Rx is uniformly charged with a charge density pE. It is surrounded by a concentric cylindrical tube of inner radius R2 and outer radius R3 as shown in Fig. 22-36, and it too carries a uniform charge density pE. Determine the electric field as a function of the distance R from the center of the cylinders for (a) 0 < R < Rlt (b) Rx < R < R2, (c) R2 < R < R3, and (d) R > R3. (e) If pE = 15pC/m3 and Ri = \R 2 = \R 3 = 5.0 cm, plot as a function of R from R = 0 to R = 20.0 cm. Assume the cylinders are very long compared to R3.

(II) A nonconducting sphere of radius r0 is uniformly charged with volume charge density pE. It is surrounded by a concentric metal (conducting) spherical shell of inner radius rx and outer radius r2, which carries a net charge +Q. Determine the resulting electric field in the regions (a) 0 < r < r0, (b) r0 < r < rx, (c) rx < r < r2, and (d) r > r2 where the radial distance r is measured from the center of the nonconducting sphere.

(II) A very long solid nonconducting cylinder of radius Rx is uniformly charged with charge density pE. It is surrounded by a cylindrical metal (conducting) tube of inner radius R2 and outer radius R3, which has no net charge (cross-sectional view shown in Fig. 22-37). If the axes of the two cylinders are parallel, but displaced from each other by a distance d, determine the resulting electric field in the region R > R3, where the radial distance R is measured from the metal cylinders axis. Assume d < (R2 - i^).

(II) A flat ring (inner radius R0, outer radius 4R0) is uniformly charged. In terms of the total charge Q, determine the electric field on the axis at points (a) 0.25R0 and (b) 75Rq from the center of the ring. [Hint: The ring can be replaced with two oppositely charged superposed disks.]

(II) An uncharged solid conducting sphere of radius r0 contains two spherical cavities of radii rx and r2, respectively. Point charge Q\ is then placed within the cavity of radius rx and point charge Q2 is placed within the cavity of radius r2 (Fig. 22-38). Determine the resulting electric field (magnitude and direction) at locations outside the solid sphere (r > r0), . where r is the distance from its center.

(Ill) A very large (i.e., assume infinite) flat slab of nonconducting material has thickness d and a uniform volume charge density +pE. (a) Show that a uniform electric field exists outside of this slab. Determine its magnitude E and its direction (relative to the slabs surface), (b) As shown in Fig. 22-39, the slab is now aligned so that one of its surfaces lies on the line y = x. At time t = 0, a pointlike particle (mass ra, charge +q) is located at position r = + y0j and has velocity v = v0i. Show that the particle will collide with the slab if v0 ^ V V 2 q y 0pEd/e0 ra. Ignore gravity.

(Ill) Suppose the density of charge between rx and r0 of the hollow sphere of Problem 29 (Fig. 22-32) varies as Pe = Pori / r Determine the electric field as a function of r for (a) 0 < r < rx, (b) rx < r < r0, and (c) r > r0. (d) Plot E versus r from r = 0 to r = 2r0.

(Ill) Suppose two thin flat plates measure 1.0 m X 1.0 m and are separated by 5.0 mm. They are oppositely charged with + 15 fiC. (a) Estimate the total force exerted by one plate on the other (ignore edge effects), (b) How much work would be required to move the plates from 5.0 mm apart to 1.00 cm apart?

(Ill) A flat slab of nonconducting material (Fig. 22-40) carries a uniform charge per unit volume, pE. The slab has thickness d which is small compared to the v height and breadth of the slab. Determine the electric field as a function of x (a) inside the slab and (b) outside the slab (at distances much less than the slabs height or breadth). Take the origin at the center of the slab.

(Ill) A flat slab of nonconducting material has thickness 2d, which is small compared to its height and breadth. Define the x axis to be along the direction of the slabs thickness with the origin at the center of the slab (Fig. 22-41). If the slab carries a volume charge density pE(x) = p0 in the region d < x < 0 and pE(jc) = +p0 in the region 0 < x < +d, determine the electric field E as a function of x in the regions (a) outside the slab, (b) 0 < x < +d, and (c) d < x < 0. Let po be a positive constant.

(Ill) An extremely long, solid nonconducting cylinder has a radius R0. The charge density within the cylinder is a function of the distance R from the axis, given by pE (R) = p0(R/R0)2. What is the electric field everywhere inside and outside the cylinder (far away from the ends) in terms of p0 and R07

(Ill) Charge is distributed within a solid sphere of radius r0 in such a way that the charge density is a function of the radial position within the sphere of the form: pE(r) = po(r/r0). If the total charge within the sphere is Q (and positive), what is the electric field everywhere within the sphere in terms of Q, r0, and the radial position rl

A point charge Q is on the axis of a short cylinder at its center. The diameter of the cylinder is equal to its length i (Fig. 22-42). What is the total flux through the curved sides of the cylinder? [Hint. First calculate the flux through the ends.]

Write Gausss law for the gravitational field g (see Section 6-6).

The Earth is surrounded by an electric field, pointing inward at every point, of magnitude E 150 N/C near the surface, (a) What is the net charge on the Earth? (b) How many excess electrons per square meter on the Earths surface does this correspond to?

A cube of side i has one corner at the origin of coordinates, and extends along the positive x, y, and z axes. Suppose the electric field in this region is given by E = (ay + b)\. Determine the charge inside the cube.

A solid nonconducting sphere of radius rQ has a total charge Q which is distributed according to pE = br, where pE is the charge per unit volume, or charge density (C/m3), and b is a constant. Determine (a) b in terms of Q, (b) the electric field at points inside the sphere, and (c) the electric field at points outside the sphere.

A point charge of 9.20 nC is located at the origin and a second charge of 5.00 nC is located on the x axis at x = 2.75 cm. Calculate the electric flux through a sphere centered at the origin with radius 1.00 m. Repeat the calculation for a sphere of radius 2.00 m.

A point charge produces an electric flux of +235N-m2/C through a gaussian sphere of radius 15.0 cm centered on the charge, (a) What is the flux through a gaussian sphere with a radius 27.5 cm? (b) What is the magnitude and sign of the charge?

A point charge Q is placed a distance r0/2 above the surface of an imaginary spherical surface of radius r0 (Fig. 22-43). (a) What is the electric flux through the sphere? (b) What range of values does E have at the surface of the sphere? (c) Is E perpendicular to the sphere at all points? (d) Is Gausss law useful for obtaining E at the surface of the sphere?

Three large but thin charged sheets are parallel to each other as shown in Fig. 22-44. Sheet I has a total surface charge density of 6.5nC/m2, sheet II a charge of 2.0nC/m2, and sheet III a charge of 5.0nC/m2. Estimate the force per unit area on each sheet, in N/m2?

Neutral hydrogen can be modeled as a positive point charge +1.6 X 10-19C surrounded by a distribution of negative charge with volume density given by p^(r) = A e~ 2r^a where a0 = 0.53 X 10-10 m is called the Bohr radius, A is a constant such that the total amount of negative charge is 1.6 X 10-19 C, and e = 2.718 is the base of the natural log. (a) What is the net charge inside a sphere of radius aQ ? (b) What is the strength of the electric field at a distance a0 from the nucleus? [Hint: Do not confuse the exponential number e with the elementary charge e which uses the same symbol but has a completely different meaning and value (e = 1.6 X 1019C).]

A very large thin plane has uniform surface charge density a. Touching it on the right (Fig. 22-45) is a long wide slab of thickness d with uniform volume charge density pE. Determine the electric field (a) to the left of the plane, (b) to the right of the slab, and (c) everywhere inside the slab.

A sphere of radius r0 carries a volume charge density pE (Fig. 22-46). A spherical cavity of radius r0/2 is then scooped out and left empty, as shown, (a) What is the magnitude and direction of the electric field at point A? (b) What is the direction and magnitude of the electric field at point B? Points A and C are at the centers of the respective spheres.

Dry air will break down and generate a spark if the electric field exceeds about 3 X 106N/C. How much charge could he narked onto the surf a re of a oreen nea ^diameter

Three very large sheets are separated by equal distances of 15.0 cm (Fig. 22-47). The first and third sheets are very thin and nonconducting and have charge per unit area a of +5.00 f j i C / m 2 and -5.00 /iC/m2 respectively. The middle sheet is conducting but has no net charge. (a) What is the electric field inside the middle sheet? What is the electric field (b) between the left and middle sheets, and (c) between the middle and right sheets? ((d) What is the charge density on the surface of the middle sheet facing the left sheet, and (e) on the surface facing the right sheet?

In a cubical volume, 0.70 m on a side, the electric field is E = o( l + f ) i + og ) j where E0 = 0.125 N /C and a = 0.70 m. The cube has its sides parallel to the coordinate axes, Fig. 22-48. Determine the net charge within the cube.

A conducting spherical shell (Fig. 22-49) has inner radius = 10.0 cm, outer radius = 15.0 cm, and has a +3.0 fxC point charge at the . - center. A charge of -3.0 jjlC is put on the conductor, (a) Where on the conductor does the -3.0 fj,C end up? (b) What is the electric field both inside and outside the shell?

A hemisphere of radius R is placed in a charge-free region of space where a uniform electric field exists of magnitude E directed perpendicular to the hemispheres circular base (Fig. 22-50). (a) Using the definition of through an open surface, calculate (via explicit integration) the electric flux through the hemisphere. [Hint: In Fig. 22-50 you can see that, on the surface of a sphere, the infinitesimal area located between the angles 0 and 0 + dd is dA = (2ttR sin 6)(Rdd) = 2ttR2 sin 6 d0.\ (b) Choose an appropriate gaussian surface and use Gausss law to much more easily obtain the same result for the electric flux through the hemisphere.

(Ill) An electric field is given by (x+y)2 (x+yV E = Ex o e \ a J i + Ey0e \ a ) J where Exq = 50 N/C, Eyo = 25 N/C, and a = 1.0 m. Given a cube with sides parallel to the coordinate axes, with one corner at the origin (as in Fig. 22-48), and with sides of length 1.0 m, estimate the flux out of the cube using a spreadsheet or other numerical method. How much total charge is enclosed by the cube?