Answer: Starting from Eq. 24-30, derive an expression for

In Fig. 24-21, eight particles -4q -2q +q form a square, with distance d ~----~'-----d-'-- between adjacent particles. What is the electric potential at point P at the center of the square if the electric potential is zero at infinity?

Figure 24-22 shows three sets of cross sections of equipotential surfaces; all three cover the same +5q p -5q size region of space. (a) Rank the -q -2q +4q arrangements according to the Fig. 24-21 Question l. magnitude of the electric field present in the region, greatest first. (b) In which is the electric field directed down the page?

Figure 24-23 shows four pairs of charged particles. For each pair, let V = 0 at infinity and consider ~,et at points on the x axis. For which pairs is there a point at which Vnet = 0 (a) between the particles and (b) to the right of the particles? (c) At such a point is Enet due to the particles equal to zero? (d) For each pair, are there off-axis points (other than at infinity) where Vnet = O?

Figure 24-24 gives the electric j! potential V as a function of x. (a) Rank the five regions according to the magnitude of the x component of the electric field within them, greatest first. What is the direction @ x -4q " x -2q 5 -- x of the field along the x axis in Fig. 24-24 Question 4. (b) region 2 and (c) region 4?

Figure 24-25 shows three paths along which we can move the positively charged sphere A closer to positively charged sphere B, which is held fixed in place. (a) Would sphere A be moved to a higher or lower electric potential? Is the work Fig.24-25 Question 5. done (b) by our force and (c) by the electric field due to B positive, negative, or zero? (d) Rank the paths according to the work our force does, greatest first.

Figure 24-26 shows four arrangements of charged particles, all the same distance from the origin. Rank the situations according to the net electric potential at the origin, most positive first. Take the potential to be zero at infinity.

Figure 24-27 shows a system of three charged particles. If you move the particle of charge +q from point A to point D, are the following quantities positive, negative, or zero: (a) the change in the electric potential energy of the three-particle system, (b) the work done by the net electrostatic force on the particle you moved (that is, the net force due to the other two particles), and (c) the work done by your force? (d) What are the answers to (a) through (c) if, instead, the particle is moved from E to C?

In the situation of Question 7, is the work done by your force positive, negative, or zero if the particle is moved (a) from A to E, (b) from A to C, and (c) from E to D? (d) Rank those moves according to the magnitUde of the work done by your force, greatest first.

Figure 24-23 shows four pairs of charged particles with identical separations. (a) Rank the pairs according to their electric potential energy (that is, the energy of the two-particle system), greatest (most positive) first. (b) For each pair, if the separation between the particles is increased, does the potential energy of the pair increase or decrease?

(a) In Fig. 24-28a, what is the potential at point P due to charge Q at distance R from P? Set V = o at infinity. (b) In Fig. 24-28b, the same charge Q has been spread uniformly over a circular arc of radius R and central angle 40. What is the potential at point P, the center of curvature of the arc? (c) In Fig. 24-28c, the same charge Q has been spread uniformly over a circle of radius R. What is the potential at point P, the center of the circle? (d) Rank the three situations according to the magnitude of the electric field that is set up at P, greatest first.

A nonconducting sphere has radius R = 2.31 cm and uniformly distributed charge q = +3.50 fC. Take the electric potential at the sphere's center to be Vo = O. What is Vat radial distance (a) r = 1.45 cm and (b) r = R. (Hint: See Section 23-9.)

As a space shuttle moves through the dilute ionized gas of Earth's ionosphere, the shuttle's potential is typically changed by -1.0 V during one revolution. Assuming the shuttle is a sphere of radius 10 m, estimate the amount of charge it collects.

What are (a) the charge and (b) the charge density on the surface of a conducting sphere of radius 0.15 m whose potential is 200 V (with V = 0 at infinity)?

Consider a point charge q = 1.0 j.LC, point A at distance d1 = 2.0 m from q, and point B at distance d2 = 1.0 ill. (a) If A and B are diametrically opposite each other, as in Fig. 24-31a, what is the electric potential difference VA - VB? (b) What is that electric potential difference if A and B are located as in Fig. 24-31b?

A spherical drop of water carrying a charge of 30 pC has a potential of 500 Vat its surface (with V = 0 at infinity). (a) What is the radius of the drop? (b) If two such drops of the same charge and radius combine to form a single spherical drop, what is the potential at the surface of the new drop?

Figure 24-32 shows a rectangular array of charged particles fixed in place, with distance a = 39.0 cm and the charges shown as integer multiples of qj = 3.40 pC and q2 = 6.00 pc. With V = 0 at infinity, what is the net electric potential at the rectangle'S center? (Hint: Thoughtful examination can reduce Fig. 24-32 Problem 16. the calculation.)

In Fig. 24-33, what is the net electric potential at point P due to the four particles if V = 0 at infinity, q = 5.00 fC, and d = 4.00 cm?

Two charged particles are shown in Fig. 24-34a. Particle 1, with charge qj, is fixed in place at distance d. Particle 2, with charge q2, can be moved along the x axis. Figure 24-34b gives the net electric potential Vat the origin due to the two particles as a function of the x coordinate of particle 2. The scale of the x axis is set by Xs = 16.0 cm. The plot has an asymptote of V = 5.76 X 10-7 Vas x ~ 00. What is q2 in terms of e?

In Fig. 24-35, particles with the charges ql = +5e and q2 = -15e are fixed in place with a separation of d = 24.0 cm. With electric potential defined to be V = 0 at infinity, what are the finite (a) positive and (b) negative values of x at which the net electric potential on the x axis is zero?

Tho particles, of charges qj and q2, are separated by distance d in Fig. 24-35. The net electric field due to the particles is zero at x = d/4. With V = 0 at infinity, locate (in terms of d) any point on the x axis (other than at infinity) at which the electric potential due to the two particles is zero.

The ammonia molecule NH3 has a permanent electric dipole moment equal to 1.47 D, where 1 D = 1 debye unit = 3.34 X 10-30 C m. Calculate the electric potential due to an ammonia molecule at a point 52.0 nm away along the axis of the dipole. (Set V = 0 at infinity.)

In Fig. 24-36a, a particle of elementary charge +e is initially at coordinate z = 20 nm on the dipole axis (here a z axis) through an electric dipole, on the positive side of the dipole. (The origin of z is at the center of the dipole.) The particle is then moved along a circular path around the dipole center until it is at coordinate z = -20 nm, on the negative side of the dipole axis. Figure 24-36b gives the work Wa done by the force moving the particle versus the angle () that locates the particle relative to the positive direction of the z axis. The scale of the vertical axis is set by Was = 4.0 X 10-30 J. What is the magnitude of the dipole moment?

(a) Figure 24-37a shows a nonconducting rod of length L = 6.00 cm and uniform linear charge density A = +3.68 pC/m. Assume that the electric potential is defined to be V = 0 at infinity. What is V at point P at distance d = 8.00 cm along the rod's perpendicular bisector? (b) Figure 24-37 b shows an identical rod except that one half is now negatively charged. Both halves have a linear charge density of magnitude 3.68 pC/m. With V = 0 at infinity, what is Vat P?

In Fig. 24-38, a plastic rod having a uniformly distributed charge Q = -25.6 pC has been bent into a circular arc of radius R = 3.71 cm and central angle cp = 120. With V = o at infinity, what is the electric potential at P, the center of curvature of the rod?

A plastic rod has been bent into a circle of radius R = 8.20 cm. It has a charge Ql = +4.20 pC uniformly distributed along onequarter of its circumference and a charge Q2 = -6Ql uniformly distributed along the rest of the circumference (Fig. 24-39). With V = 0 at infinity, what is the electric potential / / / / / / / tCP-E~~, ." ... P R ' \ : \ \ \ \ \ \ \ Fig. 24-38 Problem 24. at (a) the center C of the circle and (b) point P, on the central axis of the circle at distance D = 6.71 cm from the center?

Figure 24-40 shows a thin rod with a uniform charge density of 2.00 f-LC/m. Evaluate the electric potential at point P if d = D = Ll4.00.

In Fig. 24-41, three thin plastic V d ~* ____ ~===R=o=d==~r-x ~L~---II rods form quarter-circles with a com- Fig. 24-40 Problem 26. mon center of curvature at the origin. The uniform charges on the rods are Ql = +30 nC, Q2 = + 3.0Qb and Q3 = -8.0Ql' What is the net electric potential at the origin due to the rods?

Figure 24-42 shows a thin plastic rod of length L = 12.0 cm and uniform positive charge Q = 56.1 fC lying on anx axis. With V = 0 at infinity, find the electric potential at point Pi on the axis, at distance d = 2.50 cm from one end of the rod.

In Fig. 24-43, what is the net electric potential at the origin due to the circular arc of charge Ql = +7.21 pC and the two particles of charges Q2 = 4.00Ql and Q3 = -2.00Ql? The arc's center of curvature is at the origin and its radius is R = 2.00 m; the angle indicated is () = 20.0.

The smiling face of Fig. 24-44 consists of three items: 1. a thin rod of charge -3.0 f-LC that forms a full circle of radius 6.0cm; 2. a second thin rod of charge 2.0 f-LC that forms a circular arc of radius 4.0 cm, subtending an angle of 90 about the center of the full circle; 3. an electric dipole with a dipole moment that is perpendicular to a radial line and has magnitude 1.28 X 10-21 C . m. What is the net electric potential at the center?

A plastic disk of radius R = 64.0 cm is charged on one side with a uniform surface charge density (T = 7.73 fC/m2, and then three quadrants of the disk are removed. The remaining quadrant is shown in Fig. 24-45. With V = 0 at infinity, what is the PT D 1 potential due to the remaining Fig. 24-45 Problem 31. quadrant at point P, which is on the central axis of the original disk at distance D = 25.9 cm from the original center?

A nonuniform linear charge distribution given by A = bx, where b is a constant, is located along an x axis from x = 0 to x = 0.20 m. If b = 20 nC/m2 and V = 0 at infinity, what is the electric potential at (a) the origin and (b) the point y = 0.l5 m on the y axis?

The thin plastic rod shown in Fig. 24-42 has length L = 12.0 cm and a nonuniform linear charge density A = ex, where e = 28.9 pC/mZ. With V = 0 at infinity, find the electric potential at point PIon the axis, at distance d = 3.00 cm from one end.

Two large parallel metal plates are 1.5 cm apart and have charges of equal magnitudes but opposite signs on their facing surfaces. Take the potential of the negative plate to be zero. If the potential halfway between the plates is then +5.0 V, what is the electric field in the region between the plates?

The electric potential at points in an xy plane is given by V = (2.0 V/mZ)xZ - (3.0 V/mZ)yz. In unit-vector notation, what is the electric field at the point (3.0 m, 2.0 m)?

The electric potential V in the space between two fiat parallel plates 1 and 2 is given (in volts) by V = 1500xz, where x (in meters) is the perpendicular distance from plate 1. At x = 1.3 cm, (a) what is the magnitude of the electric field and (b) is the field directed toward or away from plate 1 ?

What is the magnitude of the electric field at the point (3.001 - 2.00J + 4.00k) m if the electric potential is given by V = 2.00xyzZ, where V is in volts and x,y, and z are in meters?

Figure 24-42 shows a thin plastic rod of length L = 13.5 cm and uniform charge 43.6 fC. (a) In terms of distance d, find an expression for the electric potential at point Pj. (b) Next, substitute variable x for d and find an expression for the magnitude of the component Ey of the electric field at PI' (c) What is the direction of Ey relative to the positive direction of the x axis? (d) What is the value of Ey at Pj for x = d = 6.20 cm? (e) From the symmetry in Fig. 24-42, determine Ey at PI

An electron is placed in an xy plane where the electric potential depends on x and y as shown in Fig. 24-46 (the potential does not depend on z). The scale of the vertical axis is set by Vs = 500 V. In unit-vector notation, what is the electric force on the electron?

The thin plastic rod of length L = 10.0 cm in Fig. 24-42 has a nonuniform linear charge density ;\ = cx, where c = 49.9 pC/mz. (a) With V = 0 at infinity, find the electric potential at point P2 on the y axis at y = D = 3.56 cm. (b) Find the electric field component Ey at Pz. (c) Why cannot the field component Ey at Pz be found using the result of (a)?

A particle of charge +7.5/LC is released from rest at the point x = 60 cm on an x axis. The particle begins to move due to the presence of a charge Q that remains fixed at the origin. What is the kinetic energy of the particle at the instant it has moved 40 cm if (a) Q = +20 /LC and (b) Q = -20/LC?

(a) What is the electric potential energy of two electrons separated by 2.00 nm? (b) If the separation increases, does the potential energy increase or decrease?

How much work is required to set up the arrangement of Fig. 24- 47 if q = 2.30 pC, a = 64.0 cm, and the p31'ticles are initially infinitely far apart and at rest?

In Fig. 24-48, seven charged particles are fixed in place to form a square with an edge length of 4.0 cm. How much work must we do -q to bring a particle of charge +6e initially at rest from an infinite distance to the center of the square?

A particle of charge q is fixed at point P, and a second particle of mass In and the same charge q is initially held a distance 1'1 from P. The second particle is then released. Determine its speed when it is a distance I'z from P. Let q = 3.1 /LC, In = 20 mg, 1'1 = 0.90 mm, and 1'2 = 2.5 mm.

A charge of -9.0 nC is uniformly distributed around a thin plastic ring lying in a yz plane with the ring center at the origin. A -6.0 pC point charge is located on the x axis at x = 3.0 m. For a ring radius of 1.5 m, how much work must an external force do on the point charge to move it to the origin?

What is the escape speed for an electron initially at rest on the surface of a sphere with a radius of 1.0 cm and a uniformly distributed charge of 1.6 X 10- 15 C? That is, what initial speed must the electron have in order to reach an infinite distance from the sphere and have zero kinetic energy when it gets there?

A thin, spherical, conducting shell of radius R is mounted on an isolating support and charged to a potential of -125 V. An electron is then fired directly toward the center of the shell, from point P at distance I' from the center of the shell (I' P R). What initial speed Vo is needed for the electron to just reach the shell before reversing direction?

Two electrons are fixed 2.0 cm apart. Another electron is shot from infinity and stops midway between the two. What is its initial speed?

In Fig. 24-49, how much work must we do to bring a particle, of charge Q = + 16e and initially at rest, along the dashed line from infinity to the indicated point near two fixed particles of charges qj = +4e and q2 = -q/2? Distance d = 1.40 cm, ()j = 43, and ()2 = 60.

In the rectangle of Fig. 24- 50, the sides have lengths 5.0 cm and 15 cm, ql = -5.0 ftC, and q2 = +2.0 ftc. With V = 0 at infinity, what is the electric potential at (a) corner A and (b) corner B? (c) How much work is Fig.24-50 Problem51. required to move a charge q3 = +3.0 ftC from B to A along a diagonal of the rectangle? (d) Does this work increase or decrease the electric potential energy of the threecharge system? Is more, less, or the same work required if q3 is moved along a path that is (e) inside the rectangle but not on a diagonal and (f) outside the rectangle?

Figure 24-51a shows an electron moving along an electric dipole axis toward the negative side of the dipole. The dipole is fixed in place. The electron was initially very far from the dipole, with kinetic energy 100 e V. Figure 24-51b gives the kinetic energy K of the electron versus its distance r from the dipole center. The scale of the horizontal axis is set by rs = 0.10 m. What is the magnitude of the dipole moment?

Two tiny metal spheres A and B, mass mA = 5.00 g and mB = 10.0 g, have equal positive charge q = 5.00 ftc. The spheres are connected by a massless nonconducting string of length d = 1.00 m, which is much greater than the radii of the spheres. (a) What is the electric potential energy of the system? (b) Suppose you cut the string. At that instant, what is the acceleration of each sphere? (c) A long time after you cut the string, what is the speed of each sphere?

A positron (charge +e, mass equal to the electron mass) is moving at 1.0 X 107 mls in the positive direction of an x axis when, at x = 0, it encounters an electric field directed along the x axis. The electric potential V associated with the field is given in Fig. 24-52. The scale of the vertical axis is set by Vs = 500.0 V. (a) Does the positron emerge from the field at x = 0 (which means its motion is reversed) or at x = 0.50 m (which means its motion is not reversed)? (b) What is its speed when it emerges?

An electron is projected with an initial speed of 3.2 X 105 mls directly toward a proton that is fixed in place. If the electron is initially a great distance from the proton, at what distance from the proton is the speed of the electron instantaneously equal to twice the initial value?

Figure 24-53a shows three particles on an x axis. Particle 1 (with a charge of + 5.0 ftC) and particle 2 (with a charge of + 3.0 ftC) are fixed in place with separation d = 4.0 cm. Particle 3 can be moved along the x axis to the right of particle 2. Figure 24- 53b gives the electric potential energy U of the three-particle system as a function of the x coordinate of particle 3. The scale of the vertical axis is set by Us = 5.01. What is the charge of particle 3?

Identical 50 ftC charges are fixed on an x axis at x = 3.0 m. A particle of charge q = -15 ftC is then released from rest at a point on the positive part of the y axis. Due to the symmetry of the situation, the particle moves along the y axis and has kinetic energy 1.2 J as it passes through the point x = 0, y = 4.0 m. (a) What is the kinetic energy of the particle as it passes through the origin? (b) At what negative value of y will the particle momentarily stop?

Proton in a well. Figure 24-54 shows electric potential V along an x axis. The scale of the vertical axis is set by Vs = 10.0 V. A proton is to be released at x = 3.5 cm with initial kinetic energy 4.00 eV. (a) If it is initially moving in the negative direction of the axis, does it reach a turning point (if so, what is the x coordinate of that point) or does it escape from the plotted region (if so, what is its speed at x = O)? (b) If it is initially moving in the positive direction of the axis, does it reach a turning point (if so, what is the x coordinate of that point) or does it escape from the plotted region (if so, what is its speed at x = 6.0 cm)? What are the ( c) magnitude F and (d) direction (positive or negative direction of the x axis) of the electric force on the proton if the proton moves just to the left of x = 3.0 cm? What are (e) F and (f) the direction if the proton moves just to the right of x = 5.0 cm?

In Fig. 24-55, a charged particle (either an electron or a proton) is moving rightward between two parallel charged plates separated by distance d = 2.00 mm. The plate potentials are VI = -70.0 V and V2 = - 50.0 V. The particle is slowing from an initial speed of 90.0 kmls at the left plate. (a) Is the particle an electron or a proton? (b) What is its speed just as it reaches plate 2?

In infinite Fig. 24-56a, distance we move to a point an electron at dis- --- ~ tance R = 8.00 cm from a tiny charged Vj ball. The move requires work W = 2.16 X 10-13 J by us. (a) What is the charge Q on the ball? In Fig. 24-56b, the ball has been Fig. 24-55 Problem 59. sliced up and the slices spread out so that an equal amount of charge is at the hour positions on a circular clock face of radius R = 8.00 cm. Now the electron is brought from an infinite distance to the center of the circle. (b) With that addition of the electron to the system of 12 charged particles, what is the change in the electric potential energy of the system?

Suppose N electrons can be placed in either of two configurations. In configuration 1, they are all placed on the circumference of a narrow ring of radius R and are uniformly distributed so that the distance between adjacent electrons is the same everywhere. In configuration 2, N - 1 electrons are uniformly distributed on the ring and one electron is placed in the center of the ring. (a) What is the smallest value of N for which the second configuration is less energetic than the first? (b) For that value of N, consider anyone circumference electron-call it eo. How many other circumference electrons are closer to eo than the central electron is?

Sphere 1 with radius Rj has positive charge q. Sphere 2 with radius 2.00R j is far from sphere 1 and initially uncharged. After the separated spheres are connected with a wire thin enough to retain only negligible charge, (a) is potential VI of sphere 1 greater than, less than, or equal to potential Vz of sphere 2? What fraction of q ends up on (b) sphere 1 and (c) sphere 2? (d) What is the ratio uj/uz of the surface charge densities of the spheres?

Two metal spheres, each of radius 3.0 cm, have a center-to-center separation of 2.0 m. Sphere 1 has charge + 1.0 X 10-8 C; sphere 2 has charge -3.0 X 10-8 C. Assume that the separation is large enough for us to say that the charge on each sphere is uniformly distributed (the spheres do not affect each other). With V = 0 at infinity, calculate (a) the potential at the point halfway between the centers and the potential on the surface of (b) sphere 1 and (c) sphere 2.

A hollow metal sphere has a potential of +400 V with respect to ground (defined to be at V = 0) and a charge of 5.0 X 10-9 C. Find the electric potential at the center of the sphere.

What is the excess charge on a conducting sphere of radius r = 0.15 m if the potential of the sphere is 1500 V and V = 0 at infinity?

Tho isolated, concentric, conducting spherical shells have radii R j = 0.500 m and Rz = 1.00 m, uniform charges ql = +2.00 pC and qz = + 1.00 /-LC, and negligible thicknesses. What is the magnitude of the electric field E at radial distance (a) r = 4.00 m, (b) r = 0.700 m, and (c) r = 0.200 m? With V = 0 at infinity, what is Vat (d) r = 4.00 m, (e) r = 1.00 m, (f) r = 0.700 m, (g) r = 0.500 m, (h) r = 0.200 m, and (i) r = O? (j) Sketch E(r) and V(r).

A metal sphere of radius 15 cm has a net charge of 3.0 X 10-8 C. (a) What is the electric field at the sphere's surface? (b) If V = 0 at infinity, what is the electric potential at the sphere's surface? (c) At what distance from the sphere's surface has the electric potential decreased by 500 V?

Here are the charges and coordinates of two point charges located in an xy plane: qj = +3.00 X 10-6 C, X = +3.50 cm, y = +0.500 cm and qz = -4.00 X 10-6 C, X = -2.00 cm, y = + 1.50 cm. How much work must be done to locate these charges at their given positions, starting from infinite separation?

A long, solid, conducting cylinder has a radius of 2.0 cm. The electric field at the surface of the cylinder is 160 N/C, directed radially outward. Let A, B, and C be points that are 1.0 cm, 2.0 cm, and 5.0 cm, respectively, from the central axis of the cylinder. What are (a) the magnitude of the electric field at C and the electric potential differences (b) VB - Vcand (c) VA - VB?

The chocolate crumb mystery. This story begins with Problem 60 in Chapter 23. (a) From the answer to part (a) of that problem, find an expression for the electric potential as a function of the radial distance r from the center of the pipe. (The electric potential is zero on the grounded pipe wall.) (b) For the typical volume charge density p = -1.1 X 10-3 C/m3, what is the difference in the electric potential between the pipe's center and its inside wall? (The story continues with Problem 60 in Chapter 25.)

Starting from Eq. 24-30, derive an expression for the electric field due to a dipole at a point on the dipole axis.

The magnitude E of an electric field depends on the radial distance r according to E = Alr4, where A is a constant with the unit volt-cubic meter. As a mUltiple of A, what is the magnitude of the electric potential difference between r = 2.00 m and r = 3.00 m?

(a) If an isolated conducting sphere 10 cm in radius has a net charge of 4.0 /-LC and if V = 0 at infinity, what is the potential on the surface of the sphere? (b) Can this situation actually occur, given that the air around the sphere undergoes electrical breakdown when the field exceeds 3.0 MV/m?

Three particles, charge q1 = + 10 /-LC, qz = -20 /-LC, and q3 = +30/-LC, are positioned at the vertices of an isosceles triangle as shown in Fig. 24-57. If a = 10 cm and b = 6.0 cm, how much work must an external agent do to exchange the positions of (a) ql and q3 and, instead, (b) q1 and qz?

An electric field of approximately 100 Vim is often observed near the surface of Earth. If this were the field over the entire surface, what would be the electric potential of a point on the surface? (Set V = 0 at infinity.)

A Gaussian sphere of radius 4.00 cm is centered on a ball that has a radius of 1.00 cm and a uniform charge distribution. The total (net) electric flux through the surface of the Gaussian sphere is +5.60 X 104 N m2/C. What is the electric potential 12.0 cm from the center of the ball?

In a Millikan oil-drop experiment (Section 22-8), a uniform electric field of 1.92 X 105 N/C is maintained in the region between two plates separated by 1.50 cm. Find the potential difference between the plates

Figure 24-58 shows three circular, nonconducting arcs of radius R = 8.50 cm. The charges on the arcs are ql = 4.52 pC, q2 = -2.00qj, q3 = +3.00ql' With V = 0 at infinity, what is the net electric potential of the arcs at the common center of curvature?

An electron is released from rest on the axis of an electric dipole that has charge e and charge separation d = 20 pm and that is fixed in place. The release point is on the positive side of the dipole, at distance 7.0d from the dipole center. What is the electron's speed when it reaches a point 5.0d from the dipole center?

Figure 24-59 shows a ring of outer radius R = 13.0 cm, inner radius r = 0.200R, and uniform surface charge density if = 6.20 pC/m2. With V = 0 at infinity, find the electric potential at point P on the central axis of the ring, at distance z = 2.00R from the center of the ring.

Electron in a well. Figure 24-60 shows electric potential V along an x axis. The scale of the vertical axis is set by Vs = 8.0 V. An electron is to be released at x = 4.5 cm with initial kinetic energy 3.00 e V. (a) If it is initially moving in the negative direction of the axis, does it reach a turning point (if so, what is the x coordinate of that point) or does it escape from the plotted region (if so, what is its speed at x = O)? (b) If it is initially moving in the positive direction of the axis, does it reach a turning point (if so, what is the x coordinate of that point) or does it escape from the plotted region (if so, what is its speed at x = 7.0 cm)? What are the (c) magnitude F and (d) direction (positive or negative direction of the x axis) of the electric force on the electron if the electron moves just to the left of x = 4.0 cm? What are (e) F and (f) the direction if it moves just to the right ofx = 5.0 cm?

(a) If Earth had a uniform surface charge density of 1.0 electron/m2 (a very artificial assumption), what would its potential be? (Set V = 0 at infinity.) What would be the (b) magnitude and (c) direction (radially inward or outward) of the electric field due to Earth just outside its surface?

In Fig. 24-61, point P is at distance dl = 4.00 m from particle 1 ql (ql = -2e) and distance d2 = 2.00 m from particle 2 (q2 = +2e), with both particles fixed in place. (a) With V = 0 ----~p I I : d2 I Q q2 at infinity, what is Vat P? If we bring a Fig. 24-61 Problem 83. particle of charge q3 = +2e from infinity to P, (b) how much work do we do and (c) what is the potential energy of the three-particle sytem?

A solid conducting sphere of radius 3.0 cm has a charge of 30 nC distributed uniformly over its surface. Let A be a point 1.0 cm from the center of the sphere, S be a point on the surface of the sphere, and B be a point 5.0 cm from the center of the sphere. What are the electric potential differences (a) Vs- VB and (b) VA - VB?

In Fig. 24-62, we move a particle of charge + 2e in from infinity to the x axis. How much work do we do? Distance D is 4.00 m.

Figure 24-63 shows a hemisphere with a charge of 4.00 j.LC distributed uniformly through its volume. The hemisphere lies on an xy plane the way half a grapefruit Fig. 24-63 Problem 86. might lie face down on a kitchen table. Point P is located on the plane, along a radial line from the hemisphere'S center of curvature, at radial distance 15 cm. What is the electric potential at point P due to the hemisphere?

Three +0.12 C charges form an equilateral triangle 1.7 m on a side. Using energy supplied at the rate of 0.83 kW, how many days would be required to move one of the charges to the midpoint of the line joining the other two charges?

Two charges q = +2.0 f.LC are fixed a distance d = 2.0 cm apart (Fig. 24-64). (a) With V = 0 at infinc d/2 d/2~~ q ity, what is the electric potential at point C? (b) You bring a third charge q = +2.0 f.LC from infinity to q C. How much work must you do? (c) What is the potential energy U of Fig. 24-64 Problem 88. the three-charge configuration when the third charge is in place?

Initially two electrons are fixed in place with a separation of 2.00 f.Lm. How much work must we do to bring a third electron in from infinity to complete an equilateral triangle?

A particle of positive charge Q is fixed at point P. A second particle of mass m and negative charge -q moves at constant speed in a circle of radius 1'10 centered at P. Derive an expression for the work W that must be done by an external agent on the second particle to increase the radius of the circle of motion to 1'2'

Two charged, parallel, flat conducting surfaces are spaced d = 1.00 cm apart and produce a potential difference Ll V = 625 V between them. An electron is projected from one surface directly toward the second. What is the initial speed of the electron if it stops just at the second surface?

In Fig. 24-65, point P is at the r---- I d~------dt center of the rectangle. With V = 0 I at infinity, ql = 5.00 fC, q2 = 2.00 I p I fC, q3 = 3.00 fC, and d = 2.54 cm, 'L --1' what is the net electric potential at d~ ______ d P due to the six charged particles?

A uniform charge of Fig.24-65 Problem 92. + 16.0 f.LC is on a thin circular ring lying in an xy plane and centered on the origin. The ring's radius is 3.00 cm. If point A is at the origin and point B is on the z axis at z = 4.00 cm, what is VB VA?

Consider a point charge q = 1.50 X 10-8 C, and take V = 0 at infinity. (a) What are the shape and dimensions of an equipotential surface having a potential of 30.0 V due to q alone? (b) Are surfaces whose potentials differ by a constant amount (1.0 V, say) evenly spaced?

A thick spherical shell of charge Q and uniform volume charge density p is bounded by radii rl and 1'2 > 1'1' With V = 0 at infinity, find the electric potential Vas a function of distance I' from the center of the distribution, considering regions (a) I' > 1'2, (b) 1'2> I' > 1'10 and (c) I' < 1'1' (d) Do these solutions agree with each other at r = 1'2 and I' = 1'1? (Hint: See Section 23-9.)

A charge q is distributed uniformly throughout a spherical volume of radius R. Let V = 0 at infinity. What are (a) Vat radial distance I' < Rand (b) the potential difference between points at I' = R and the point at I' = O?

Figure 24-35 shows two charged particles on an axis. Sketch the electric field lines and the equipotential surfaces in the plane of the page for (a) ql = +q, q2 = +2q and (b) ql = +q, q2 = -3q

What is the electric potential energy of the charge configuraPROBLEMS 655 tion of Fig. 24-8a? Use the numerical values provided in the associated sample problem.

(a) Using Eq. 24-32, show that the electric potential at a point on the central axis of a thin ring (of charge q and radius R) and at distance z from the ring is 1 V= 47TBO (b) From this result, derive an expression for the electric field magnitude E at points on the ring's axis; compare your result with the calculation of E in Section 22-6.

An alpha particle (which has two protons) is sent directly toward a target nucleus containing 92 protons. The alpha particle has an initial kinetic energy of 0.48 pI What is the least center-to-center distance the alpha particle will be from the target nucleus, assuming the nucleus does not move?

In the quark model of fundamental particles, a proton is composed of three quarks: two "up" quarks, each having charge +2eI3, and one "down" quark, having charge -eI3. Suppose that the three quarks are equidistant from one another. Take that separation distance to be 1.32 X 10-15 m and calculate the electric potential energy of the system of (a) only the two up quarks and (b) all three quarks.

(a) A proton of kinetic energy 4.80 MeV travels head-on toward a lead nucleus. Assuming that the proton does not penetrate the nucleus and that the only force between proton and nucleus is the Coulomb force, calculate the smallest center-to-center separation dp between proton and nucleus when the proton momentarily stops. If the proton were replaced with an alpha particle (which contains two protons) of the same initial kinetic energy, the alpha particle would stop at center-to-center separation dQ (b) What is daldp?

In Fig. 24-66, two particles of 1 2 P charges ql and q2 are fixed to an x --0-----0 x axis. If a third particle, of charge r- d-+-1.5d-1 +6.0 f.LC, is brought from an infi- Fig.24-66 Problem 103. nite distance to point P, the threeparticle system has the same electric potential energy as the original two-particle system. What is the charge ratio q/q2?

A charge of 1.50 X 10-8 C lies on an isolated metal sphere of radius 16.0 cm. With V = 0 at infinity, what is the electric potential at points on the sphere's surface?

A solid copper sphere whose radius is 1.0 cm has a very thin surface coating of nickel. Some of the nickel atoms are radioactive, each atom emitting an electron as it decays. Half of these electrons enter the copper sphere, each depositing 100 ke V of energy there. The other half of the electrons escape, each carrying away a charge -e. The nickel coating has an activity of 3.70 X 108 radioactive decays per second. The sphere is hung from a long, nonconducting string and isolated from its surroundings. (a) How long will it take for the potential of the sphere to increase by 1000 V? (b) How long will it take for the temperature of the sphere to increase by 5.0 K due to the energy deposited by the electrons? The heat capacity of the sphere is 14 11K.