Chapter 23

A point charge \(q_{1}=+2.40 \mu C\) is held stationary at the origin. A second point charge \(q_{2}=-4.30 \mu C\) moves from the point \(x=0.150 \mathrm{m}, y=0\) to the point \(x=0.250 \mathrm{m}, y=0.250 \mathrm{m} .\) How much work is done by the electric force on \(q_{2} ?\)

A point charge \(q_{1}\) is held stationary at the origin. A second charge \(q_{2}\) is placed at point \(a,\) and the electric potential energy of the pair of charges is \(+5.4 \times 10^{-8} \mathrm{J}\) . When the second charge is moved to point \(b\) , the electric force on the charge does \(-1.9 \times 10^{-8} \mathrm{J}\) of work. What is the electric potential energy of the pair of charges when the second charge is at point \(b ?\)

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 \(2.00 \times 10^{-13} \mathrm{m}\) with a proton at each vertex? Assume the protons started from very far away.

How far from a \(-7.20-\mu \mathrm{C}\) point charge must \(\mathrm{a}+2.30-\mu \mathrm{C}\) point charge be placed for the electric potential energy \(U\) of the pair of charges to be \(-0.400 \mathrm{J}\) ? (Take \(U\) to be zero when the charges have infinite separation.)

A point charge \(Q=+4.60 \mu \mathrm{C}\) is held fixed at the origin. A second point charge \(q=+1.20 \mu \mathrm{C}\) with mass of \(2.80 \times 10^{-4} \mathrm{kg}\) is placed on the \(x\) -axis, 0.250 \(\mathrm{m}\) 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 scparation.) (b) The scond point charge is released from rest. What is its speed when its distance from the origin is (i) \(0.500 \mathrm{m} ;\) (ii) \(5.00 \mathrm{m} ;\) (iii) 50.0 \(\mathrm{m} ?\)

Three equal \(1.20-\mu \mathrm{C}\) point charges are placed at the corners of an equilateral triangle whose sides are 0.500 \(\mathrm{m}\) 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)

Three point charges, which initially are infinitely far apart, are placed at the corners of an equilateral triangle with sides \(d\) . Two of the point charges are identical and have charge \(q\) . If zerv net work is required to place the three charges at the comers of the triangle, what must the value of the third charge be?

Two protons are aimed directly toward each other by a cyclotron accelerator with speeds of 1000 \(\mathrm{km} / \mathrm{s}\) , measured relative to the earth. Find the maximum electrical force that these protons will exert on each other.

Identical point charges \(q=+5.00 \mu \mathrm{C}\) are placed at opposite corners of a square. The length of each side of the square is 0.200 \(\mathrm{m}\) . A point charge \(q_{0}=-2.00 \mu \mathrm{C}\) is placed at one of the empty comers. How much work is done on \(q_{0}\) by the electric force when \(q_{0}\) is moved to the other empty comer?

A small particle has charge \(-5.00 \mu \mathrm{C}\) and mass \(2.00 \times 10^{-4} \mathrm{kg}\) . It moves from point \(A\) , where the electric potential is \(V_{A}=+200 \mathrm{V},\) to point \(B\) , where the electric potential is \(V_{B}=+800 \mathrm{V}\) . The electric force is the only force acting on the particle. The particle has speed 5.00 \(\mathrm{m} / \mathrm{s}\) at point \(A .\) What is its speed at point \(B ?\) Is it moving faster or slower at \(B\) than at \(A ?\) Explain.

A particle with a charge of \(+4.20 \mathrm{nC}\) is in a uniform electric field \(\overrightarrow{\boldsymbol{E}}\) directed to the left. It is released from rest and moves to the left; after it has moved \(6.00 \mathrm{cm},\) its kinetic energy is found to be \(+1.50 \times 10^{-6} \mathrm{J}\) . \(\mathrm{J}\) (a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the end point? (c) What is the magnitude of \(\overrightarrow{\boldsymbol{E}} ?\)

A charge of 28.0 \(\mathrm{nC}\) is placed in a uniform electric field that is directed vertically upward and has a magnitude of \(4.00 \times 10^{4} \mathrm{V} / \mathrm{m}\) . What work is done by the electric force when the charge moves (a) 0.450 \(\mathrm{m}\) to the right; (b) 0.670 \(\mathrm{m}\) upward; (c) 2.60 \(\mathrm{m}\) at an angle of \(45.0^{\circ}\) downward from the horizontal?

Two stationary point charges \(+3.00 \mathrm{nC}\) and \(+2.00 \mathrm{nC}\) are separated by a distance of 50.0 \(\mathrm{cm}\) . 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 \(\mathrm{cm}\) from the \(+3.00-\mathrm{charge}\) ?

A point charge has a charge of \(2.50 \times 10^{-11} \mathrm{C}\) . At what distance from the point charge is the electric potential (a) 90.0 \(\mathrm{V}\) and (b) 30.0 \(\mathrm{V}\) ? Take the potential to be zero at an infinite distance from the charge.

Two charges of equal magnitude \(Q\) are beld 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 zero at these points?), and (ii) the electric field is zero (is the potential zero at these points?). and (b) Repeat part (a) for two charges having opposite signs.

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 potenntial \(V_{0}\) at the origin? (c) Show that the potential at any point on the \(x\) -axis is $$ V=\frac{1}{4 \pi \epsilon_{0}} \frac{2 q}{\sqrt{a^{2}+x^{2}}} $$ (d) Graph the potential on the \(x\) -axis as a function of \(x\) over the range from \(x=-4 a\) to \(x=+4 a\) . (e) What is the potential when \(x \gg a ?\) Explain why this result is obtained.

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 coondinate \(x\) . Take \(V\) to be zero at an infinite distance from the charges. (c) Graph \(V\) at points on the \(x\) -axis as a function of \(x\) over the range from \(x=-4 a\) to \(x=+4 a\) . (d) What is the answer to part (b) if the two charges are interchanged so that \(+q\) is at \(y=+a\) and \(-q\) is at \(y=-a ?\)

A positive charge \(q\) is fixed at the point \(x=0, y=0,\) and a negative charge \(-2 q\) 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 infinite distance from the charges. (c) At which positions on the \(x\) -axis is \(V=0 ?(\text { d) Graph }\) \(V\) at points on the \(x\) -axis as a function of \(x\) in the range from \(x=-2 a\) to \(x=+2 a\) (e) What does the answer to part \((b)\) become when \(x \gg a ?\) Explain why this result is obtained.

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 emits electrons from its surface. A potential difference of a few hundred volts is maintained between the cathode and the other electrode, known as the anode, with the anode at the higher potential. Suppose that in a particular vacuum tube the potential of the anode is 295 \(\mathrm{V}\) higher than that of the cathode. An electron leaves the surface of the cathode with zero initial speed. Find its speed when it strikes the anode.

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are 4.98 \(\mathrm{V}\) and \(12.0 \mathrm{V} / \mathrm{m},\) 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 field directed toward or away from the point charge?

A uniform electric field has magnitude \(E\) and is directed in the negative \(x\) -direction. The potential difference between point \(a\) (at \(x=0.60 \mathrm{m}\) ) and point \(b\) (at \(x=0.90 \mathrm{m} )\) is 240 \(\mathrm{V}\) . (a) Which point, \(a\) or \(b\) , is at the higher potential? (b) Calculate the value of E. (c) A negative point charge \(q=-0.200 \mu \mathrm{C}\) is moved from \(b\) to a. Calculate the work done on the point charge by the electric field.

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 \(+2 Q\) separated by a distance \(d\) , and (b) charges \(-Q\) and \(+2 Q\) separated by a distance \(d\) . (c) Are both \(V\) and \(E\) zero at the same places? Explain.

(a) An electron is to be accelerated from \(3.00 \times 10^{6} \mathrm{m} / \mathrm{s}\) to \(8.00 \times 10^{6} \mathrm{m} / \mathrm{s}\) . 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 \(8.00 \times 10^{6} \mathrm{m} / \mathrm{s}\) to a halt?

A total electric charge of 3.50 \(\mathrm{nC}\) is distributed uniformly over the surface of a metal sphere with a radius of \(24.0 \mathrm{cm} .\) 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) \(48.0 \mathrm{cm} ;(\mathrm{b}) 24.0 \mathrm{cm}\) (c) \(12.0 \mathrm{cm} .\)

A uniformly charged thin ring has radius 15.0 \(\mathrm{cm}\) and total charge \(+24.0 \mathrm{nC}\) . An electron is placed on the ring's axis a dis0tance 30.0 \(\mathrm{cm}\) 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 electron. (b) Find the speed of the electron when it reaches the center of the ring.

An infinitely long line of charge has linear charge density \(5.00 \times 10^{-12} \mathrm{C} / \mathrm{m} .\) A proton (mass \(1.67 \times 10^{-27} \mathrm{kg}\) . charge \(+1.60 \times 10^{-19} \mathrm{C} )\) is 18.0 \(\mathrm{cm}\) from the line and moving directly toward the line at \(1.50 \times 10^{3} \mathrm{m} / \mathrm{s}\) (a) Calculate the proton's initial kinetic energy. (b) How close does the proton get to the line of charge? (Hints See Example \(23.10 . )\)

A very long wire carries a uniform linear charge density \(\lambda\) . Using a voltmeter to measure potential difference, you find that when one probe of the meter is placed 2.50 \(\mathrm{cm}\) from the wire and the other probe is 1.00 \(\mathrm{cm}\) farther from the wire, the meter reads 575 \(\mathrm{V}\) . (a) What is \(\lambda 2\) (b) If you now place one probe at 3.50 \(\mathrm{cm}\) from the wire and the other probe 1.00 \(\mathrm{cm}\) farther away, will the voltmeter read 575 \(\mathrm{V} ?\) If not, will it read more or less than 575 \(\mathrm{V}\) ? Why? (c) If you place both probes 3.50 \(\mathrm{cm}\) from the wire but 17.0 \(\mathrm{cm}\) from each other, what will the voltmeter read?

A very long insulating cylinder of charge of radius 2.50 \(\mathrm{cm}\) carries a uniform linear density of 15.0 \(\mathrm{nC} / \mathrm{m}\) . 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 \(\mathrm{V} ?\)

A ring of diameter 8.00 \(\mathrm{cm}\) is fixed in place and carries a charge of \(+5.00 \mu \mathrm{C}\) unifurnly spread over its circumference. (a) How much work does it take to move a tiny \(+3.00-\mu \mathrm{C}\) charged ball of mass 1.50 \(\mathrm{g}\) from very far away to the center of the ring? (b) Is it necessary to take a path along the axis of the ring? Why? (c) If the ball is slightly displaced from the center of the ring, what will it do and what is the maxi- mum speed it will reach?

Two very large, parallel metal plates carry charge densities of the same magnitude but opposite signs (Fig, 23.32\()\) . 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 function of \(x\) . Include all regions from the left of the plates to the right of the plates.

Two large, parallel conducting plates carrying opposite charges of equal magnitude are separated by 2.20 \(\mathrm{cm}\) . (a) If the nsurface charge density for each plate has magnitude 47.0 \(\mathrm{nC} / \mathrm{m}^{2}\) , what is the magnitude of \(\overrightarrow{\boldsymbol{E}}\) in the region between the plates? (b) What is the potential difference between the two plates? (c) If the separation between the plates is doubled while the surface charge density is kept constant at the value in part (a), what happens to the magnitude of the electric field and to the potential difference?

Two large, parallel, metal plates carry opposite charges of equal magnitude. They are separated by \(45.0 \mathrm{mm},\) and the potential difference between them is 360 \(\mathrm{V}\) . (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 force this field exerts on a particle with charge \(+2.40 \mathrm{nC} ?\) (c) Use the results of part (b) to compute the work done by the field on the particle as it moves from the higher-potential plate to the lower. (d) Compare the result of part (c) to the change of potential energy of the same charge, computed from the electric potential.

(a) How much excess charge must be placed on a copper sphere 25.0 \(\mathrm{cm}\) in diameter so that the potential of its center, relative to infinity, is 1.50 \(\mathrm{kV}\) ? (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 \(\mathrm{rub}\) an inflated balloon on the carpet and it acquires a potential that is 1560 \(\mathrm{V}\) lower than its potential before it became charged. If the charge is uniformly distributed over the surface of the balloon and if the radius of the balloon is \(15 \mathrm{cm},\) what is the net charge on the balloon? (c) In light of its \(1200-\mathrm{V}\) potential difference relative to you, do you think this balloon is dangerous? Explain.

The electric field at the surface of a charged, solid, copper 0sphere with radius 0.200 \(\mathrm{m}\) is 3800 \(\mathrm{N} / \mathrm{C}\) , 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 \(\mathrm{V}\) is established between large, parallel, metal plates. Let the potential of one plate be 480 \(\mathrm{V}\) and the other be 0 \(\mathrm{V}\) . The plates are separated by \(d=1.70 \mathrm{cm} .\) (a) Sketch the equipotential surfaces that correspond to \(0,120,\) \(240,360,\) and 480 \(\mathrm{V}\) . (b) In your sketch, show the electric field lines. Does your sketch confirm that the field lines and equipotential surfaces are mutually perpendicular?

In a certain region of space, the electric potential is \(V(x, y, z)=A x y-B x^{2}+C y,\) 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?

A very long cylinder of radius 2.00 \(\mathrm{cm}\) carries a uniform charge density of 1.50 \(\mathrm{nC} / \mathrm{m}\) . (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 potentials of \(10.0 \mathrm{V}, 20.0 \mathrm{V},\) and 30.0 \(\mathrm{V}\) . (c) Are the equipotential surfaces equally spaced? If not, do they get closer together or farther apart as \(r\) increases?

A particle with charge \(+7.60 \mathrm{nC}\) 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 \(\mathrm{cm}\) , the additional force has done \(6.50 \times 10^{-5} \mathrm{J}\) of work and the particle has \(4.35 \times 10^{-5} \mathrm{J}\) of kinetic energy. (a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the end point? (c) What is the magnitude of the electric field?

In the Rohr 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 expression for the electron's kinetic energy, and show that its magnitude is just half that of the electric potential energy. (c) Obtain an expression for the total energy, and evaluate it using \(r=5.29 \times 10^{-11} \mathrm{m} .\) Give your numerical result in joules and in electron volts.

(a) Calculate the potential energy of a system of two small spheres, one carrying a charge of 2.00\(\mu \mathrm{C}\) and the other a charge of \(-3.50 \mu \mathrm{C},\) with their centers separated by a distance of 0.250 \(\mathrm{m}\) . Assume zero potential energy when the charges are infinitely separated. (b) Suppose that one of the spheres is held in place and the other sphere, which has a mass of 1.50 \(\mathrm{g}\) , is shot away from it. What minimum initial speed would the moving sphere need in order to escape completely from the attraction of the fixed sphere? (To escape, the moving sphere would have to reach a velocity of zero when it was infinitely distant from the fixed sphere.)

The \(\mathbf{H}_{2}^{+}\) Ion. The \(\mathbf{H}_{2}^{+}\) ion is composed of two protons, each of charge \(+e=1.60 \times 10^{-19} \mathrm{C},\) and an electron of charge \(-e\) and mass \(9.11 \times 10^{-31} \mathrm{kg} .\) The separation between the pro- tons is \(1.07 \times 10^{-10} \mathrm{m}\) . The protons and the electron may be treated as point charges. (a) Suppose the electron is located at the point midway between the two protons. What is the potential energy of the interaction between the electron and the two protons? (Do not include the potential energy due to the interaction between the two protons.) (b) Suppose the electron in part (a) has a velocity of magnitude \(1.50 \times 10^{6} \mathrm{m} / \mathrm{s}\) in a direction along the perpendicular bisector of the line connecting the two protons. How far from the point midway between the two protons can the electron move? Because the masses of the protons are much greater than the clectron mass, the motions of the protons are very slow and can be ignored. (Note: A realistic description of the electron motion requires the use of quantum mechanics, not Newtonian mechanics.)

Alpha particles \(\left(\text { mass }=6.7 \times 10^{-27} \mathrm{kg}, \text { charge }=+2 e\right)\) 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 \(5.6 \times\) \(10^{-15} \mathrm{m}\) , what minimum speed do the alpha particles need when they are far away to reach the surface of the gold nucleus? (lanore relativistic effects.) (b) Give good physical reasons why we can ignore the effects of the orbital electrons when the alpha particle is (i) outside the electron orbits and (ii) inside the electron orbits.

A thin insulating rod is bent into a semicircular are 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 are 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 through- out 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. (Hint: After you have assembled a charge \(q\) in a sphere of radius \(r\) , how much energy would it take to add a spherical shell of thickness \(d r\) having charge \(d q ?\) Then integrate to get the total energy.)

Two plastic spheres, each carrying charge uniformly distributed throughout its interior, are initially placed in contact and then released. One sphere is 60.0 \(\mathrm{cm}\) in diameter, has mass 50.0 \(\mathrm{g}\) and contains \(-10.0 \mu \mathrm{C}\) of charge. The other sphere is 40.0 \(\mathrm{cm}\) in diameter, has mass 150.0 \(\mathrm{g}\) , and contains \(-30.0 \mu \mathrm{C}\) of charge. Find the maximum acceleration and the maximum speed achieved by each sphere (relative to the fixed point of their initial location in space), assuming that no other forces are acting on them. (Hint: The uniformly distributed charges behave as though they were concentrated at the centers of the two spheres.)

(a) If a spherical raindrop of radius 0.650 \(\mathrm{mm}\) carries a charge of \(-1.20 \mathrm{pC}\) uniformly distributed over its volume, what is the potential at its surface? (Take the potential to be zero at an infinite distance from the raindrop.) (b) Two identical raindrops, each with radius and charge specified in part (a), collide and merge into one larger raindrop. What is the radius of this larger drop, and what is the potential at its surface, if its charge is uniformly distributed over its volume?

Two metal spheres of different sizes are charged such that the electric potential is the same at the surface of each. Sphere \(A\) has a radius three times that of sphere \(B\) . Let \(Q_{A}\) and \(Q_{B}\) be the charges on the two spheres, and let \(E_{A}\) and \(E_{B}\) be the electric-field magnitudes at the surfaces of the two spheres. What are (a) the ratio \(Q_{B} / Q_{A}\) and (b) the ratio \(E_{B} / E_{A} ?\)

An alpha particle with kinetic energy 11.0 MeV makes a head-on collision with a lead nucleus at rest. What is the distance of closest approach of the two particles? (Assume that the lead nucleus remains stationary and that it may be treated as a point charge. The atomic number of lead is \(82 .\) The alpha particle is a helium nucleus, with atomic number \(2 .\) )

A metal sphere with radius \(R_{1}\) has a charge \(Q_{1}\) . Take the electric potential to be zero at an infinite distance from the sphere. (a) What are the electric field and electric potential at the surface of the sphere? This sphere is now connected by a long, thin conducting wire to another sphere of radius \(R_{2}\) that is several meters from the first sphere. Before the connection is made, this second sphere is uncharged. After electrostatic equilibrium has been reached, what are (b) the total charge on each sphere; (c) the electric potential at the surface of each sphere (d) the electric field at the surface of each sphere? Assume that the amount of charge on the wire is much less than the charge on each sphere.