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

A Sphere in a Sphere A solid conducting sphere carrying charge q has radius a. It is inside a concentric hollow conducting sphere with inner radius b and outer radius c. The hollow sphere has no net charge. (a) Derive expressions for the electric-field magnitude in terms of the distance r from the
A solid conducting sphere with radius R that carries positive charge Q is concentric with a very thin insulating shell of radius 2R that also carries charge Q. The charge Q is distributed uniformly over the insulating shell. (a) Find the electric field (magnitude and direction) in each of the
A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell is - 3Q, and it is insulated from its surroundings (Fig.).(a) Derive expressions for the electric-field magnitude in terms of the distance r from
Concentric Spherical Shells A small conducting spherical shell with inner radius a and outer radius b is concentric with a larger conducting spherical shell with inner radius c and outer radius d (Fig. ). The inner shell has total charge +2q, and the outer shell has charge +4q.(a) Calculate the
Repeat Problem 22.45, hut now let the outer shell have charge -2q. As in Problem 22.45, the inner shell has charge +2q.
Repeat Problem 22.45, but now let the outer shell have charge -4q. As in Problem 22.45, the inner shell has charge + 2q.
A solid conducting sphere with radius R carries a positive total charge Q. The sphere is surrounded by an insulating shell with inner radius R and outer radius 2R. The insulating shell has a uniform charge density p. (a) Find the value of p so that the net charge of the entire system is zero. (b)
Negative charge -Q is distributed uniformly over the surface of a thin spherical insulating shell with radius R. Calculate the force (magnitude and direction) that the shell exerts on a positive point charge q located (a) a distance r > R from the center of the shell (outside the shell) and (b) a
(a) How many excess electrons must be distributed uniformly within the volume of an isolated plastic sphere 30.0 cm in diameter to produce an electric field of 1150 N/C just outside the surface of the sphere? (b) What is the electric field at a point 10.0 cm outside the surface of the sphere?
A single isolated, large conducting plate (Fig.) has a charge per unit area σ on its surface. Because the plate is a conductor, the electric field at its surface is perpendicular to the surface and has magnitude E = σ /є0. (a) In Example 22.7 (Section 22.4) it was shown that the
Thomson's Model of the Atom In the early years of the 20th century, a leading model of the structure of the atom was that of the English physicist J. J. Thomson (the discoverer of the electron). In Thomson's model, an atom consisted of a sphere of positively charged material in which was embedded
Thomson's Model of the Atom Continued. Using Thomson's (outdated) model of the atom described in Problem 22.52, consider an atom consisting of two electrons, each of charge -e, embedded in a sphere of charge + 2e and radius R. In equilibrium, each electron is a distance d from the center of the
A Uniformly Charged Slab A slab of insulating material has thickness 2d and is oriented so that its faces are parallel to the yz-plane and given by the planes x = d and x = -d. The y- and z-dimensions of the slab are very large compared to d and may be treated as essentially infinite. The slab has
A Nonuniformly Charged Slab Repeat Problem 22.54, but now let the charge density of the slab be given by p(x) = P0(x/d)2, where Po is a positive constant.
Can Electric Forces Alone Give Stable Equilibrium? In Chapter 21, several examples were given of calculating the force exerted on a point charge by other point charges in its surroundings. (a) Consider a positive point charge +q. Give an example of how you would place two other point charges of
A nonuniform, but spherically symmetric, distribution of charge has a charge density p (r) given as follows:Where P0 = 3Q/πR3 is a positive constant.(a) Show that the total charge contained in the charge distribution is Q.(b) Show that the electric field in the region r ≥ R is identical to
A nonuniform, but spherically symmetric, distribution of charge has a charge density p (r) given as follows:Where P0 is a positive constant(a) Find the total charge contained in the charge distribution.(b) Obtain an expression for the electric field in the region r≥ R.(c) Obtain an expression
Gauss's Law for Gravitation The gravitational force between two point masses separated by a distance r is proportional to l/r2, just like the electric force between two point charges. Because of this similarity between gravitational and electric interactions, there is also a Gauss's law for
Applying Gauss's Law for Gravitation Using Gauss's law for gravitation (derived in part (b) of Problem 22.59), show that the following statements are true: (a) For any spherically symmetric mass distribution with total mass M, the acceleration due to gravity outside the distribution is the same as
(a) An insulating sphere with radius a has a uniform charge density p. The sphere is not centered at the origin but at r = b. Show that the electric field inside the sphere is given by E = p (r - b)/3є0.(b) An insulating sphere of radius R has a spherical hole of radius a located within its
A very long, solid insulating cylinder with radius R has a cylindrical hole with radius a bored along its entire length. The axis of the hole is a distance b from the axis of the cylinder, where a
Positive charge Q is distributed uniformly over each of two spherical volumes with radius R. One sphere of charge is centered at the origin and the other at x = 2R (Fig. 22.44). Find the magnitude and direction of the net electric field due to these two distributions of charge at the following
Repeat Problem 22.63, but now let the left-hand sphere have positive charge Q and let the right-hand sphere have negative charge -Q.
Electric Field inside a Hydrogen Atom A hydrogen atom is made up of a proton of charge + Q = 1.60 X 10-19 C and an electron of charge - Q = -1.60 X 10-19 C. The proton may be regarded as a point charge at r = 0, the center of the atom. The motion of the electron causes its charge to be
A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density p (r) is given byHere a is a positive constant having units of C/m3 €¢(a) Determine a in terms of Q and R.(b) Using Gauss's law, derive an expression for the magnitude
A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density p (r} is given byHere a is a positive constant having units of C/m3.(a) Determine a in terms of Q and R.(b) Using Gauss's law, derive an expression for the magnitude of the
A point charge q1 = + 2.40 µC is held stationary at the origin. A second point charge q2 = -4.30 µC moves from the point x = 0.150 m, Y = 0 to the point x = 0.250 m, Y = 0.250 m. How much work is done by the electric force on q2?
A point charge q1 is held stationary at the origin. A second charge q2 is placed at point a, and the electric potential energy of the pair of charges is + 5.4 X 10-8 J. When the second charge is moved to point b, the electric force on the charge does -1.9 X 10-8 J of work. What is the electric
Energy of the Nucleus How much work is needed to assemble an atomic nucleus containing three protons (such as Be) if we model it as an equilateral triangle of side 2.00 X 10-15 m with a proton at each vertex? Assume the protons started from very far away.
(a) How much work would it take to push two protons very slowly from a separation of 2.00 X 10-10 m (a typical atomic distance) to 3.00 X 10-15 m (a typical nuclear distance)? (b) If the protons are both released from rest at the closer distance in part (a), how fast are they moving when they reach
A small metal sphere, carrying a net charge of q1 = 2.80 µC, is held in a stationary position by insulating supports. A second small metal sphere, with a net charge of q2 = -7.80 µC and mass 1.50 g, is projected toward q1. When the two spheres are 0.800 m apart, q2 is moving toward q1
How far from a -7.20-µC point charge must a +2.30-µC point charge be placed for the electric potential energy U of the pair of charges to be -0.400 J? (Take U to be zero when the charges have infinite separation.)
A point charge Q = +4.60 µC is held fixed at the origin. A second point charge q = + 1.20 µC with mass of 2.80 X 10-4 kg is placed on the x-rods, 0.250 m from the origin. (a) What is the electric potential energy U of the pair of charges? (Take U to be zero when the charges have infinite
Three equal 1.20-µC point charges are placed at the comers of an equilateral triangle whose sides are 0.500 m long. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)
A point charge q1 = 4.00 nC is placed at the origin, and a second point charge q2 = -3.00 nC is placed on the x-axis at x = + 20.0 cm. A third point charge q3 = 2.00 nC is to be placed on the x-axis between q1 and q2. (Take as zero the potential energy of the three charges when they are infinitely
Four electrons are located at the comers of a square 10.0 nm on a side, with an alpha particle at its midpoint. How much work is needed to move the alpha particle to the midpoint of one of the sides of the square?
Three point charges, which initially are infinitely far apart, are placed at the comers of an equilateral triangle with sides d. Two of the point charges are identical and have charge q. If zero net work is required to place the three charges at the comers of the triangle, what must the value of
Two protons are aimed directly toward each other by a cyclotron accelerator with speeds of 1000 km/s, measured relative to the earth. Find the maximum electrical force that these protons will exert on each other.
A uniform electric field is directed due east. Point B is 2.00 m west of point A, point C is 2.00 m east of point A and point D is 2.00 m south of A. For each point, B, C, and D, is the potential at that point larger, smaller or the same as at point A? Give the reasoning behind your answers.
Identical point charges q = +5.00 µC are placed at opposite comers of a square. The length of each side of the square is 0.200 m. A point charge q0 = - 2.00 µC is placed at one of the empty comers. How much work is done on q0 by the electric force when q0 is moved to the other empty comer?
A small particle has charge -5.00µC and mass 2.00 X 10-4 kg. It moves from point A, where the electric potential is VA = + 200 V, to point B, where the electric potential isVB = +500 V. The electric force is the only force acting on the particle. The particle has speed 5.00 m/s at point A. What is
A particle with a charge of +4.20 nC is in a uniform electric field E directed to the left. It is released from rest and moves to the left; after it has moved 6.00 cm, its kinetic energy is found to be + 1.50 X 10-6 J. (a) What work was done by the electric force? (b) What is the potential of the
A charge of 28.0 nC is placed in a uniform electric field that is directed vertically upward and has a magnitude of 4.00 X 104 V/m. What work is done by the electric force when the charge moves? (a) 0.450 m to the right; (b) 0.670 m upward; (c) 2.60 m at an angle of 45.0o downward from the
Two stationary point charges +3.00 nC and +2.00 nC are separated by a distance of 50.0 cm. An electron is released from rest at a point midway between the two charges and moves along the line connecting the two charges. What is the speed of the electron when it is 10.0 cm from the + 3.00-nC charge?
A point charge has a charge of 2.50 X 10-11 C. At what distance from the point charge is the electric potential (a) 90.0 V and (b) 30.0 V? Take the potential to be zero at an infinite distance from the charge.
Two charges of equal magnitude Q are held a distance d apart. Consider only points on the line passing through both charges. (a) If the two charges have the same sign, find the location of all points (if there are any) at which (i) the potential (relative to infinity) is zero (is the electric field
Two point charges ql = +2.40 nC and q2 = -6.50 nC are 0.100 m apart. Point A is midway between them; point B is 0.050 m from ql and 0.060 m from q2 (Fig). Take the electric potential to be zero at infinity. Find(a) The potential at point A;(b) The potential at point B;(c) The work done by the
Two positive point charges, each of magnitude q, are fixed on the y-axis at the points y = +a and y = -a. Take the potential to be zero at an infinite distance from the charges.(a) Show the positions of the charges in a diagram.(b) What is the potential V0 at the origin?(c) Show that the potential
A positive charge +q is located at the point x = 0, y = -a, and a negative charge -q is located at the point x = 0, y = +a. (a) Show the positions of the charges in a diagram. (b) Derive an expression for the potential V at points on the x-axis as a function of the coordinate x. Take V to be zero
Consider the arrangement of charges described in Exercise 23.23. (a) Derive an expression for the potential V at points on the y-axis as a function of the coordinate y. Take V to be zero at an infinite distance from the charges. (b) Graph V at points on the y-axis as a function of y over the
A positive charge q is fixed at the point x = 0, y = 0, and a negative charge - 2q is fixed at the point x = a, y = 0. (a) Show the positions of the charges in a diagram. (b) Derive an expression for the potential V at points on the x-axis as a function of the coordinate x. Take V to be zero at an
Consider the arrangement of point charges described in Exercise 23.25. (a) Derive an expression for the potential V at points on the y-axis as a function of the coordinate y. Take V to be zero at an infinite distance from the charges. (b) At which positions on the y-axis is V = 0? (c) Graph Vat
Before the advent of solid-state electronics, vacuum tubes were widely used in radios and other devices. A simple type of vacuum tube known as a diode consists essentially of two electrodes within a highly evacuated enclosure. One electrode, the cathode, is maintained at a high temperature and
At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are 4.98 V and 12.0 V/m, respectively. (Take the potential to be zero at infinity.) (a) What is the distance to the point charge? (b) What is the magnitude of the charge? (c) Is the electric
A uniform electric field has magnitude E and is directed in the negative x-direction. The potential difference between point a (at x = 0.60 m) and point b (at x = 0.90 m) is 240 V. (a) which point, a or b, is at the higher potential? (b) Calculate the value of E. (c) A negative point charge q =
For each of the following arrangements of two point charges, find all the points along the line passing through both charges for which the electric potential V is zero (take V = 0 infinitely far from the charges) and for which the electric field E is zero: (a) Charges +Q and +2Q separated by a
(a)An electron is to be accelerated from 3.00 X 106 m/s to 8.00 X 106 m/s. Through what potential difference must the electron pass to accomplish this? (b)Through what potential difference must the electron pass if it is to be slowed from 8.00 X 106 m/s to a halt?
A total electric charge of 3.50 nC is distributed uniformly over the surface of a metal sphere with a radius of 24.0 cm. If the potential is zero at a point at infinity, find the value of the potential at the following distances from the center of the sphere: (a) 48.0 cm; (b) 24.0 cm; (c) 12.0 cm.
A uniformly charged thin ring has radius 15.0 cm and total charge +24.0 nC. An electron is placed on the ring's axis a distance 30.0 cm from the center of the ring and is constrained to stay on the axis of the ring. The electron is then released from rest. (a) Describe the subsequent motion of the
An infinitely long line of charge has linear charge density 5.00 X 10-12 C/m. A proton (mass 1.67 X 10-27 kg, charge + 1.60 X 10-19 C) is 18.0 cm from the line and moving directly toward the line at 1.50 X 103 m/s. (a) Calculate the proton's initial kinetic energy. (b) How close does the proton get
A very long wire carries a uniform linear charge density A. Using a voltmeter to measure potential difference, you find that when one probe of the meter is placed 2.50 cm from the wire and the other probe is 1.00 cm farther from the wire, the meter reads 575 V. (a) What is A? (b) If you now place
A very long insulating cylinder of charge of radius 2.50 cm carries a uniform linear density of 15.0 nC/m. If you put one probe of a voltmeter at the surface, how far from the surface must the other probe be placed so that the voltmeter reads 175 V?
A very long insulating cylindrical shell of radius 6.00 cm carries charge of linear density 8.50 µC/m spread uniformly over its outer surface. What would a voltmeter read if it were connected between (a) The surface of the cylinder and a point 4.00 cm above the surface, and (b) The surface and a
A ring of diameter 8.00 cm is fixed in place and carries a charge of +5.00 µC uniformly spread over its circumference. (a) How much work does it take to move a tiny + 3.00-µC charged ball of mass 1.50 g from very far away to the center of the ring? (b) Is it necessary to take a path along the
Two very large, parallel metal plates carry charge densities of the same magnitude but opposite signs (Fig). Assume they are close enough together to be treated as ideal infinite plates. Taking the potential to be zero at the left surface of the negative plate, sketch a graph of the potential as a
Two large, parallel conducting plates carrying opposite charges of equal magnitude are separated by 2.20 cm. (a) If the surface charge density for each plate has magnitude 47.0nC/m2, what is the magnitude of E in the region between the plates? (b) What is the potential difference between the two
Two large, parallel, metal plates carry opposite charges of equal magnitude. They are separated by 45.0 mm, and the potential difference between them is 360 V. (a) What is the magnitude of the electric field (assumed to be uniform) in the region between the plates? (b) What is the magnitude of the
(a) How much excess charge must be placed on a copper sphere 25.0 cm in diameter so that the potential of its center, relative to infinity, is 1.50 kV? (b) What is the potential of the sphere's surface relative to infinity?
(a) Show that V for a spherical shell of radius R, that has charge q distributed uniformly over its surface, is the same as V for a solid conductor with radius R and charge q. (b) You rub an inflated balloon on the carpet and it acquires a potential that is 1560 V lower than its potential before it
The electric field at the surface of a charged, solid, copper sphere with radius 0.200 m is 3800 N/c, directed toward the center of the sphere. What is the potential at the center of the sphere, if we take the potential to be zero infinitely far from the sphere?
A potential difference of 480 V is established between large, parallel, metal plates. Let the potential of one plate be 480 V and the other be 0 V. The plates are separated by d = 1.70 cm. (a) Sketch the equipotential surfaces that correspond to 0, 120, 240, 360, and 480 V. (b) In your sketch, show
A very large plastic sheet carries a uniform charge density of -6.00nC/m2 on one face. (a) As you move away from the sheet along a line perpendicular to it, does the potential increase or decrease? How do you know, without doing any calculations? Does your answer depend on where you choose the
In a certain region of space, the electric potential is V(x, y, z) = Axy – Bx2 + Cy, where A, B, and C are positive constants. (a) Calculate the x-, y-, and z-components of the electric field. (b) At which points is the electric field equal to zero?
The potential due to a point charge Q at the origin may be written as (a) Calculate E., EY' and E, using Eqs. (23.19). (b) Show that the result of part (a) agrees with Eq. (21.7) for the electric field of a point charge.
A metal sphere with radius ra is supported on an insulating stand at the center of a hollow, metal. Spherical shell with radius r b- There is charge +q on the inner sphere and charge -q on the outer spherical shell.(a) Calculate the potential V(r) for (i) r r b; Take V to be zero when r is
A metal sphere with radius ra = 1.20 cm is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius r b = 9.60 cm. Charge +q is put on the inner sphere and charge -q on the outer spherical shell. The magnitude of q is chosen to make the potential difference
A very long cylinder of radius 2.00 cm carries a uniform charge density of 1.50nC/m. (a) Describe the shape of the equipotential surfaces for this cylinder. (b) Taking the reference level for the zero of potential to be the surface of the cylinder, find the radius of equipotential surfaces having
Figure shows the potential of a charge distribution as a function of x. Sketch a graph of the electric field E. over the region shown.
A particle with charge +7.60nC is in a uniform electric field directed to the left. Another force, in addition to the electric force, acts on the particle so that when it is released from rest, it moves to the right After it has moved 8.00 cm, the additional force has done 6.50 x 10-5 J of work
In the Bohr model of the hydrogen atom, a single electron revolves around a single proton in a circle of radius r. Assume that the proton remains at rest (a) By equating the electric force to the electron mass times its acceleration, derive an expression for the electron's speed. (b) Obtain an
A vacuum tube diode (see Exercise 23.27) consists of concentric cylindrical electrodes, the negative cathode and the positive anode. Because of the accumulation of charge near the cathode, the electric potential between the electrodes is not a linear function of the position, even with planar
Two oppositely charged. Identical insulating spheres, each 50.0 cm in diameter and carrying a uniform charge of magnitude 175 µC are placed 1.00 m apart center to center (Fig).(a) If a voltmeter is connected between the nearest points (a and b) on their surfaces, what will it read?(b) Which
An Ionic Crystal Figure shows eight point charges arranged at the corners of a cube with sides of length d. The values of the charges are +q and -q, as shown. This is a model of one cell of a cubic ionic crystal. In sodium chloride (NaC1), for instance, the positive ions are Na + and the negative
(a) Calculate the potential energy of a system of two small spheres, one carrying a charge of 2.00 µC and the other a charge of - 3.50 µC, with their centers separated by a distance of 0.250 m. Assume zero potential energy when the charges are infinitely separated. (b) Suppose that one of the
The H2+ Ion. The H2+ ion is composed of two protons, each of charge +e = 1.60 X 10-19 C, and an electron of charge -e and mass 9.11 X 10-31 kg. The separation between the protons is 1.07 X 10-10 m. The protons and the electron may be treated as point charges. (a) Suppose the electron is located at
A small sphere with mass 1.50 g hangs by a thread between two parallel vertical plates 5.00 cm apart (Fig). The plates are insulating and have uniform surface charge densities +σ and - σ. The charge on the sphere is q = 8.90 X 10-6 C. What potential difference between the plates will
(a) Calculate the potential V (r} for (i) r b. Take V = 0 at r = b.(b) Show that the potential of the inner cylinder with respect to the outer is(c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitudeE(r) = Vab/in (b/a)
A Geiger counter detects radiation such as alpha particles by using the fact that the radiation ionizes the air along its path. A thin wire lies on the axis of a hollow metal cylinder and is insulated from it (Fig). A large potential difference is established between the wire and the outer
Deflection in a CRT Cathode ray tubes (CRTs) are often found in oscilloscopes and computer monitors. In Fig an electron with an initial speed of 6.50 X 106 m/s is projected along he axis midway between the deflection plates of a cathode-ray tube. The uniform electric field between the plates has a
Deflecting Plates of an Oscilloscope the vertical deflecting plates of a typical classroom oscilloscope are a pair of parallel square metal plates carrying equal but opposite charges. Typical dimensions are about 3.0 cm on a side, with a separation of about 5.0 mm. The plates are close enough that
Electrostatic precipitators use electric forces to remove pollutant particles from smoke, in particular in the smokestacks of coal-burning power plants. One form of precipitator consists of a vertical, hollow, metal cylinder with a thin wire, insulated from the cylinder, running along its axis
A disk with radius R has uniform surface charge density σ (a) By regarding the disk as a series of thin concentric rings, calculate the electric potential V at a point on the disk's axis a distance x from the center of the disk. Assume that the potential is zero at infinity (b) Calculate
(a) From the expression for E obtained in Problem 22.40, find the expressions for the electric potential V as a function of r, both inside and outside the cylinder. Let V = 0 at the surface of the cylinder. In each case, express your result in terms of the charge per unit length λ of the
Alpha particles (mass = 6.7 X 10-27 kg, charge = + 2e) are shot directly at a gold foil target. We can model the gold nucleus as a uniform sphere of charge and assume that the gold does not move. (a) If the radius of the gold nucleus is 5.6 X 10-15 m, what minimum speed do the alpha particles need
For the ring of charge described in Example 23.11 (Section 23.3), integrate the expression for E, found in Example 21.10 (Section 21.5) to find the potential at point P on the ring's axis. Assume that V = 0 at infinity. Compare your result to that obtained in Example 23.11 using Eq. (23.16).
A thin insulating rod is bent into a semicircular arc of radius a, and a total electric charge Q is distributed uniformly along the rod. Calculate the potential at the center of curvature of the arc if the potential is assumed to be zero at infinity.
Self-Energy of a Sphere of Charge A solid sphere of radius R contains a total charge Q distributed uniformly throughout its volume. Find the energy needed to assemble this charge by bringing infinitesimal charges from far away. This energy is called the “self-energy” of the charge distribution.
(a) From the expression for E obtained in Example 22.9 (Section 22.4), find the expression for the electric potential V as a function of r both inside and outside the uniformly charged sphere. Assume that V = 0 at infinity. (b) Graph V and E as functions of r from r = 0 to r = 3R.
A solid insulating sphere with radius R has charge Q uniformly distributed throughout its volume. (a) Use the results of Problem 23.72 to find the magnitude of the potential difference between the surface of the sphere and its center. (b) Which is at higher potential, the surface or the center, if
An insulating spherical shell with inner radius 25.0 cm and outer radius 60.0 cm carries a charge of + 150.0 µC uniformly distributed over its outer surface (see Exercise 23.43). Point a is at the center of the shell, point b is on the inner surface, and point c is on the outer surface. (a) What

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