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electricity and magnetism

University Physics with Modern Physics 12th Edition Hugh D. Young, Roger A. Freedman, Lewis Ford - Solutions

In a rectangular coordinate system a positive point charge q = 6.00 X 10-9 C is placed at the point x = + 0.150 m, y = 0, and an identical point charge is placed at x = -0.150 m, y = 0. Find the x- and y-components, the magnitude, and the direction of the electric field at the following points: (a)
A point charge q1 = -4.00 nC is at the point x = 0.600 m, y = 0.800 m, and a second point charge q2 = +6.00 nC is at the point x = 0.600 m, y = 0. Calculate the magnitude and direction of the net electric field at the origin due to these two point charges.
Repeat Exercise 21.49 for the case where the point charge at x = +0.150 m, y = 0 is positive and the other is negative, each with magnitude 6.00 X 10-9 C.
A very long, straight wire has charge per unit length 1.50 X 10-10 C/m. At what distance from the wire is the electric field magnitude equal to 2.50 N/C?
Positive electric charge is distributed along the y-axis with charge per unit length λ (a) Consider the case where charge is distributed only between the points y = a and y = -a. For points on the + x-axis, graph the x-component of the electric field as a function of x for values of x
A straight, nonconducting plastic wire 8.50 cm long carries a charge density of + 175 nC/m distributed uniformly along its length. It is lying on a horizontal tabletop. (a) Find the magnitude and direction of the electric field this wire produces at a point 6.00 cm directly above its midpoint. (b)
A ring-shaped conductor with radius a = 2.50 cm has a total positive charge Q = + 0.125 nC uniformly distributed around it, as shown in Fig. 21.24. The center of the ring is at the origin of coordinates 0. (a) What is the electric field (magnitude and direction) at point P, which is on the x-axis
A charge of -6.50 nC is spread uniformly over the surface of one face of a nonconducting disk of radius 1.25 cm. (a) Find the magnitude and direction of the electric field this disk produces at a point P on the axis of the disk a distance of 2.00 cm from its center. (b) Suppose that the charge were
Two horizontal, infinite, plane sheets of charge are separated by a distance d. The lower sheet has negative charge with uniform surface charge density - σ < 0. The upper sheet has positive charge with uniform surface charge density σ > 0. What is the electric field (magnitude, and
Infinite sheet A carries a positive uniform charge density σ, and sheet B, which is to the right of A and parallel to it, carries a uniform negative charge density -2σ. (a) Sketch the electric field lines for this pair of sheets. Include the region between the sheets as well as the
Suppose the charge shown in Fig. 21.29a is fixed in position. A small, positively charged particle is then placed at some point in the figure and released. Will the trajectory of the particle follow an electric field line? Why or why not? Suppose instead that the particle is placed at some point in
Sketch the electric field lines for a disk of radius R with a positive uniform surface charge density σ. Use what you know about the electric field very close to the disk and very far from the disk to make your sketch.
(a) Sketch the electric field lines for an infinite line of charge. You may find it helpful to show the field lines in a plane containing the line of charge in one sketch and the field lines in a plane perpendicular to the line of charge in a second sketch. (b) Explain how your sketches show (i)
Figure 21.41 shows some of the electric field lines due to three point charges arranged along the vertical axis. All three charges have the same magnitude.(a) What are the signs of the three charges? Explain your reasoning.(b) At what point(s) is the magnitude of the electric field the smallest?
Point charges q1 = -4.5 nC and q2 = +4.5 nC are separated by 3.1 mm, forming an electric dipole. (a) Find the electric dipole moment (magnitude and direction). (b) The charges are in a uniform electric field whose direction makes an angle of 36.9° with the line connecting the charges. What is the
The ammonia molecule (NH3) has a dipole moment of 5.0 X 10-30 C • m. Ammonia molecules in the gas phase are placed in a uniform electric field E with magnitude 1.6 X 106 N/c. (a) What is the change in electric potential energy when the dipole moment of a molecule changes its orientation with
The ammonia molecule (NH3) has a dipole moment of 5.0 X 10-30 C • m. Ammonia molecules in the gas phase are placed in a uniform electric field E with magnitude 1.6 X 106 N/c. (a) What is the change in electric potential energy when the dipole moment of a molecule changes its orientation with
The dipole moment of the water molecule (H20) is 6.17 X 10-30 C. m. Consider a water molecule located at the origin whose dipole moment p points in the + x-direction. A chlorine ion (C1-), of charge -1.60 X 10-19 C, is located at x = 3.00 X 109 m. Find the magnitude and direction of the electric
Surface Tension The surface of a polar liquid, such as water, can be viewed as a series of dipoles strung together in the stable arrangement in which the dipole moment vectors are parallel to the surface and all point in the same direction. Suppose now that something presses inward on the surface,
Consider the electric dipole of Example 21.15. (a) Derive an expression for the magnitude of the electric field produced by the dipole at a point on the x-axis in Fig. 21.34. What is the direction of this electric field? (b) How does the electric field at points on the x-axis depend on x when x is
Torque on a Dipole. An electric dipole with dipole moment p is in a uniform electric field E. (a) Find the orientations of the dipole for which the torque on the dipole is zero. (b) Which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small displacement away
A dipole consisting of charges ± e, 220 nm apart, is placed between two very large (essentially infinite) sheets carrying equal but opposite charge densities of 125 µC/m2• (a) What is the maximum potential energy this dipole can have due to the sheets, and how should it be oriented relative to
Three charges are at the corners of an isosceles Triangle as shown in Fig. 21.43. The ±5.00-µC charges form a dipole.(a) Find the force (magnitude and direction) the €“l0.00-µC charge exerts on the(b) For an axis perpendicular to the line connecting the
A charge q1 = + 5.00 nC is placed at the origin of an x y coordinate system, and a charge q2 = - 2.00 nC is placed on the positive x-axis at x = 4.00 cm. (a) If a third charge q3 = +6.00 nC is now placed at the point x = 4.00 cm, y = 3.00 cm, find the x and y-components of the total force exerted
Two positive point charges Q are held fixed on the x-axis at x = a and x = -a. A third positive point charge q, with mass m, is placed on the x-axis away from the origin at a coordinate x such that |x| «a. The charge q, which is free to move along the x-axis, is then released. (a) Find the
Two identical spheres with mass m are hung from silk threads of length L, as shown in Fig. 21.44. Each sphere has the same charge, so q1 = q2 = q. The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. Show that if the angle
Two small spheres with mass m = 15.0 g are hung by silk threads of length L = 1.20 m from a common point (Fig. 21.44). When the spheres are given equal quantities of negative charge, so that q1 = q2 = q, each thread hangs at θ = 25.0° from the vertical. (a) Draw a diagram showing the forces
Two identical spheres are each attached to silk threads of length L = 0.500 m and hung from a common point (Fig. 21.44). Each sphere has mass m = 8.00 g. The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. One sphere is
Sodium chloride (NaCl, ordinary table salt) is made up of positive sodium ions (Na+) and negative chloride ions (c1-). (a) If a point charge with the same charge and mass as all the Na + ions in 0.100 mol of NaC1 is 2.00 cm from a point charge with the same charge and mass as all the C 1- ions,
Two point charges q1 and q2 are held in place 4.50 cm apart. Another point charge Q = -1.75µC of mass 5.00 g is initially located 3.00 cm from each of these charges (Fig. 21.45) and released from rest. You observe that the initial acceleration of Q is 324 m/s2 upward, parallel to the line
Three identical point charges q are placed at each of three corners of a square of side L. Find the magnitude and direction of the net force on a point charge - 3q placed (a) At the center of the square and (b) At the vacant corner of the square. In each case, draw a free. Body diagram showing the
Three point charges are placed on the y-axis: a charge q at y = a, a charge - 2q at the origin, and a charge q at y = -a. Such an arrangement is called an electric quadrupole. (a) Find the magnitude and direction of the elec1ric field at points on the positive x-axis. (b) Use the binomial expansion
Strength of the Electric Force Imagine two 1.0-g bags of protons, one at the earth's North Pole and the other at the South Pole. (a) How many protons are in each bag? (b) Calculate the gravitational attraction and the electrical repulsion that each bag exerts on the other. (c) Are the forces in
Electric Force With in the Nucleus Typical dimensions of atomic nuclei are of the order of l0-15 m (1 fm). (a) If two protons in a nucleus are 2.0 fm apart, find the magnitude of the electric force each one exerts on the other. Express the answer in newtons and in pounds. Would this force be large
If Atoms Were Not Neutral . Because the charges on the electron and proton have the same absolute value, atoms are electrically neutral. Suppose this were not precisely true, and the absolute value of the charge of the electron were less than the charge of the proton by 0.00100%. (a) Estimate what
Two tiny balls of mass m carry equal but opposite charges of magnitude q. They are tied to the same ceiling hook by light strings of length L. When a horizontal uniform electric field E is turned on, the balls hang with an angle θ between the strings (Fig. 21.46). (a) Which ball (the right
Two small, copper spheres each have radius 1.00 mm. (a) How many atoms does each sphere contain? (b) Assume that each copper atom contains 29 protons and 29 electrons. We know that electrons and protons have charges of exactly the same magnitude, but let's explore the effect of small differences
Operation or an Inkjet Printer In an inkjet printer, letters are built up by squirting drops of ink at the paper from a rapidly moving nozzle. The ink drops, which have a mass of 1.4 x 10-8 g each, leave the nozzle and travel toward the paper at 20 m/s, passing through a charging unit that gives
A proton is projected into a uniform electric field that points vertically upward and has magnitude E. The initial velocity of the proton has a magnitude U0 and is directed at an angle a below the horizontal. (a) Find the maximum distance hmax that the proton descends vertically below its initial
A negative point charge ql = - 4.00 nC is on the x-axis at x = 0.60 m. A second point charge q2 is on the x-axis at x = -1.20 m. What must the sign and magnitude of q2 be for the net electric field at the origin to be? (a) 50.0 N/C in the + x-direction and (b) 50.0 N/C in the -x-direction?
Positive charge Q is distributed uniformly along the x-axis from x = 0 to x = a. A positive point charge q is located on the positive x-axis at x = a + r, a distance, to the right of the end of Q (Fig. 21.47). (a) Calculate the x- and y-components of the electric field produced by the charge
Positive charge Q is distributed uniformly along the positive y-axis between y = 0 and y = a. A negative point charge -q lies on the positive x axis, a distance x from the origin (Fig. 21.48). (a) Calculate the x and y-components of the electric field produced by the charge distribution Q at
A charged line like that shown in Fig. 21.25 extends from y = 2.50 cm to y = - 2.50 cm. The total charge distributed uniformly along the line is -9.00 nC. (a) Find the electric field (magnitude and direction) on the x-axis at x = 10.0 cm. (b) Is the magnitude of the electric field you calcula1ed
A Parallel Universe Imagine a parallel universe in which the electric force has the same properties as in our universe but there is no gravity. In this parallel universe, the sun carries charge Q, the earth carries charge - Q, and the electric attraction between them keeps the earth in orbit. The
A uniformly charged disk like the disk in Fig. 21.26 has radius 2.50 cm and carries a total charge of 4.0 X 10-12 C. (a) Find the electric field (magnitude and direction) on the x-axis at x = 20.0 cm. (b) Show that for x » R, Eq. (21.11) becomes E = Q/4πє0x2. Where Q is the total
(a) Let f(x) be an even function of x so that f(x) = f (-x). Show that ƒa-a ƒ (x) dx = 2 ƒ a0 ƒ (x) dx. (Hint: Write the integral from -a to a as the sum of the integral from -a to 0 and the integral from 0 to a. In the first integral, make the change of variable x' = -x.) (b) Let g(x) be an
Positive charge + Q is distributed uniformly along the + x-axis from x = 0 to x = a. Negative charge - Q is distributed uniformly along the -x-axis from x = 0 to x = -a. (a) A positive point charge q lies on the positive y-axis, a distance y from the origin. Find the force (magnitude and direction)
Positive charge Q is uniformly distributed around a semicircle of radius a (Fig. 21.49). Find the electric field (magnitude and direction) at the center of curvature P.
Negative charge -Q is distributed uniformly around a quarter-circle of radius a that lies in the first quadrant, with the center of curvature at the origin. Find the x- and y-components of the net electric field at the origin.
A small sphere with mass m carries a positive charge q and is attached to one end of a silk fiber of length L The other end of the fiber is attached to a large vertical insulating sheet that has a positive surface charge density σ. Show that when the sphere is in equilibrium, the fiber makes
Two 1.20-m nonconducting wires meet at a right angle. One segment carries + 2.50 µC of charge distributed uniformly along its length, and the other carries - 2.50 µC distributed uniformly along it, as shown in Fig. 21.50. (a) Find the magnitude and direction of the electric field these
Two very large parallel sheets are 5.00 cm apart. Sheet A carries a uniform surface charge density of -9.50 µC/m2, and sheet B, which is to the right of A, carries a uniform charge of -11.6 µC/m2. Assume the sheets are large enough to be treated as infinite. Find the magnitude and direction of
Repeat Problem 21.100 for the case where sheet B is positive.
Two very large horizontal sheets are 4.25 cm apart and carry equal but opposite uniform surface charge densities of magnitude σ. You want to use these sheets to hold stationary in the region between them an oil droplet of mass 324 µg that carries an excess of five electrons. Assuming that the
An infinite sheet with positive charge per unit area σ lies in the xy-plane. A second infinite sheet with negative charge per unit area - σ lies in the yz-plane. Find the net electric field at all points that do not lie in either of these planes. Express your answer in terms of the unit
A thin disk with a circular hole at its center, called an annulus, has inner radius R 1 and outer radius R2 (Fig. 21.51). The disk has a uniform positive surface charge density σ on its surface.(a) Determine the total electric charge on the annulus.(b) The annulus lies in the yz-plane, with
Three charges are placed as shown in Fig. 21.52. The magnitude of q1 is 2.00 µC, but its sign and the value of the charge q2 are not known. Charge q3 is +4.00 µC, and the net force F on q3 is entirely in the negative direction.(a) Considering the different possible signs of q1 and there
Two charges are placed as shown in Fig. 21.53. The magnitude of q1 is 3.00 µC, but its sign and the value of the charge q2 are not known. The direction of the net electric field E at point P is entirely in the negative y- direction.(a) Considering the different possible signs of q1 and q2,
Two thin rods of length L lie along the x-axis, one between x = a/2 and x = a/2 + L and the other between x = -a/2 and x = -a/2 - L. Each rod has positive charge Q distributed uniformly along its length.(a) Calculate the electric field produced by the second rod at points along the positive
A Hat sheet of paper of area 0.250 m2 is oriented so that the normal to the sheet is at an angle of 60° to 8 uniform electric field of magnitude 14 N/C. (a) Find the magnitude of the electric flux through the sheet. (b) Does the answer to part (a) depend on the shape of the sheet? Why or why
A flat sheet is in the shape of a rectangle with sides of lengths 0.400 m and 0.600 m. The sheet is immersed in 8 uniform electric field of magnitude 75.0 N/C that is directed at 20o from the plane of the sheet (Fig.). Find the magnitude of the electric flux through the sheet.
You measure an electric field of 1.25 X 106 N/C at a distance of 0.150 m from a point charge. (a) What is the electric flux through a sphere at that distance from the charge? (b) What is the magnitude of the charge?
A cube has sides of length L = 0.300 m. It is placed with one comer at the origin as shown in Fig. The electric field is not uniform but is given by E = (-5.00 N/C .m) xi + (3.00 N/C .m) zk.(a) Find the electric flux through each of the six cubes faces S1, S2, S3, S4, S5, and S6.(b) Find the total
A hemispherical surface with radius r in a region of uniform electric field E has its axis aligned parallel to the direction of the field. Calculate the flux through the surface.
The cube in Fig. 22.32 has sides of length L = 10.0 cm. The electric field is uniform, has magnitude E = 4.00 X 105 N/C, and is parallel to the xy -plane at an angle of 36.9o measured from the +x-axis toward the +y-axis. (a) What is the electric flux through each of the six cube faces S1, S2, S3,
It was shown in Example 21.11 (Section 21.5) that the electric field due to an infinite line of charge is perpendicular to the line and has magnitude E = λ/2πє0r. Consider an imaginary cylinder with radius r = 0.250 m and length I = 0.400 m that has an infinite line of positive
The three small spheres shown in Fig. carry charges q1 = 4.00 nC, q2 = -7.80 nC, and q3 = 2.40 nC. Find the net electric flux through each of the following closed surfaces shown in cross section in the figure:(a) S1;(b) S2;(c) S3;(d) S4;(e) S5.(f) Do your answers to parts (a)-(e) depend on how the
A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter 12.0 cm, giving it a charge of -15.0 µC. Find the electric field (a) Just inside the paint layer; (b) Just outside the paint layer; (c) 5.00 cm outside the surface of the paint layer
A point charge q1 = 4.00 nC is located on the x-axis at x = 2.00 m, and a second point charge q2 = -6.00 nC is on the y-axis at y = 1.00 m. What is the total electric flux due to these two point charges through a spherical surface centered at the origin and with radius? (a) 0.500m, (b) 1.50m, (c)
In a certain region of space, the electric field E is uniform. (a) Use Gauss's law to prove that this region of space must be electrically neutral; that is, the volume charge density p must be zero. (b) Is the converse true? That is, in a region of space where there is no charge, must E be uniform?
(a) In a certain region of space, the volume charge density p has a uniform positive value. Can E be uniform in this region? Explain. (b) Suppose that in this region of uniform positive p there is a “bubble” within which p = 0. Can E be uniform within this bubble? Explain.
A 9.60-µC point charge is at the center of a cube with sides of length 0.500 m. (a) What is the electric flux through one of the six faces of the cube? (b) How would your answer to part (a) change if the sides were 0.250 m long? Explain.
Electric Fields in an Atom The nuclei of large atoms, such as uranium, with 92 protons, can be modeled as spherically symmetric spheres of charge. The radius of the uranium nucleus is approximately 7.4 X 10-15 m. (a) What is the electric field this nucleus produces just outside its surface? (b)
A point charge of +5.00 µC is located on the x-axis at x = 4.00 m, next to a spherical surface of radius 3.00 m centered at the origin. (a) Calculate the magnitude of the electric field at x = 3.00 m. (b) Calculate the magnitude of the electric field at x = -3.00 m. (c) According to Gauss's law,
A solid metal sphere with radius 0.450 m carries a net charge of 0.250 nC. Find the magnitude of the electric field (a) At a point 0.100 m outside the surface of the sphere and (b) At a point inside the sphere, 0.100 m below the surface.
On a humid day, an electric field of 2.00 X 104 N/c is enough to produce sparks about an inch long. Suppose that in your physics class, a van de Graaff generator (see Fig. 22.27) with a sphere radius of 15.0 cm is producing sparks 6 inches long. (a) Use Gauss's law to calculate the amount of charge
Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of 3.63 X 1016 N • m2/C at the planet's surface. Calculate:(a) The total electric charge on the planet;(b) The electric field at the planet's
How many excess electrons must be added to an isolated spherical conductor 32.0 cm in diameter to produce an electric field of 1150 N/C just outside the surface?
The electric field 0.400 m from a very long uniform line of charge is 840 N/C. How much charge is contained in a 2.00-cm section of the line?
A very long uniform line of charge has charge per unit length 4.80 µC/m and lies along the x-axis. A second long uniform line of charge has charge per unit length - 2.40 µC/m and is parallel to the x-axis at y = 0.400 m. What is the net electric field (magnitude and direction) at the following
(a) At a distance of 0.200 cm from the center of a charged conducting sphere with radius 0.100 cm, the electric field is 480 N/C. What is the electric field 0.600 cm from the center of the sphere? (b) At a distance of 0.200 cm from the axis of a very long charged conducting cylinder with radius
A hollow, conducting sphere with an outer radius of 0.250 m and an inner radius of 0.200 m has a uniform surface charge density of +6.37 X 10-6 C/m2 • A charge of -0.500 µC is now introduced into the cavity inside the sphere. (a) What is the new charge density on the outside of the sphere? (b)
A point charge of -2.00 µC is located in the center of a spherical cavity of radius 6.50 cm inside an insulating charged solid. The charge density in the solid is p = 7.35 X 10-4 C/m3. Calculate the electric field inside the solid at a distance of 9.50 cm from the center of the cavity.
The electric field at a distance of 0.145 m from the surface of a solid insulating sphere with radius 0.355 m is 1750 N/C. (a) Assuming the sphere's charge is uniformly distributed, what is the charge density inside it? (b) Calculate the electric field inside the sphere at a distance of 0.200 m
A conductor with an inner cavity, like that shown in Fig. 22.23c, carries a total charge of + 5.00 nC. The charge within the cavity, insulated from the conductor, is -6.00 nC. How much charge is on (a) The inner surface of the conductor and (b) The outer surface of the conductor?
Apply Gauss's law to the Gaussian surfaces S2, S3, and S4 in Fig. 22.21b to calculate the electric field between and outside the plates.
A square insulating sheet 80.0 cm on a side is held horizontally. The sheet has 7.50 nC of charge spread uniformly over its area (a) Calculate the electric field at a point 0.100 mm above the center of the sheet. (b) Estimate the electric field at a point 100 m above the center of the sheet. (c)
An infinitely long cylindrical conductor has radius R and uniform surface charge density σ.(a) In terms of σ and R, what is the charge per unit length λ for the cylinder?(b) In terms of σ, what is the magnitude of the electric field produced by the charged cylinder at a distance r > R from
Two very large, nonconducting plastic sheets, each 10.0 cm thick, carry uniform charge densities σ1, σ2, σ3, and σ4 on their surfaces. As shown in. 22.34.These surface charge densities have the values σ1= -6.00µC/m2, σ2 = +5.00µC/m2, σ3 =
A negative charge - Q is placed inside the cavity of a hollow metal solid. The outside of the solid is grounded by connecting a conducting wire between it and the earth. (a) Is there any excess charge induced on the inner surface of the piece of metal? If so, find its sign and magnitude. (b) Is
A cube has sides of length L. It is placed with one comer at the origin as shown in Fig. 22.32. The electric field is uniform and given by E = -Bi + Cj - Dk, where B, C, and D are positive constants. (a) Find the electric flux through each of the six cubes faces S1, S2, S3, S4, S5, and S6. (b) Find
The electric field E in Hg. 22.35 is everywhere parallel to the x-axis, so the components Ey and Ez are zero. The x-component of the field Ex depends on x but not on y and Z. At points in the yz-plane (where x = 0), Ex= 125 N/C.(a) What is the electric flux through surface I in Fig. ?(b) What is
A flat, square surface with sides of length L is described by the equations(a) Draw this square and show the x-, y-, and z-axes.(b) Find the electric flux through the square due to a positive point charge q located at the origin (x = 0, y = 0, z = 0). (Hint: Think of the square as part of a cube
The electric field E1 at one face of a parallelepiped is uniform over the entire face and is directed out of the face. At the opposite face, the electric field E2 is also uniform over the entire face and is directed into that face (Fig.). The two faces in question are inclined at 30.0° from the
A long line carrying a uniform linear charge density + 50.0 µC/m runs parallel to and 10.0 cm from the surface of a large, flat plastic sheet that has a uniform surface charge density of -100 µC/m2 on one side. Find the location of all points where an a particle would feel no force due to this
The Coaxial Cable A long coaxial cable consists of an inner cylindrical conductor with radius a and an outer coaxial cylinder with inner radius b and outer radius c. The outer cylinder is mounted on insulating supports and has DO net charge. The inner cylinder has a uniform positive charge per unit
A very long conducting tube (hollow cylinder) has inner radius a and outer radius b. It carries charge per unit length +a, where a is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length +a. (a) Calculate the electric
Repeat Problem 22.38, but now let the conducting tube have charge per unit length -a. As in Problem 22.38, the line of charge has charge per unit length +a.
A very long, solid cylinder with radius R has positive charge uniformly distributed throughout it, with charge per unit volume p. (a) Derive the expression for the electric field inside the volume at a distance r from the axis of the cylinder in terms of the charge density p. (b) What is the
A small sphere with a mass of 0.002 g and carrying a charge of 5.00 X 10-8 C hangs from a thread near a very large, charged conducting sheet, as shown in Fig. The charge density on the sheet is 2.50 X 10-9 C/m2€¢ Find the angle of the thread.

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