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 m, \(y =\) 0 to the point \(x =\) 0.250 m, \(y =\) 0.250 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 \(+5.4 \times 10^{-8} \)J. When the second charge is moved to point \(b\), the electric force on the charge does \(-1.9 \times 10^{-8}\) J of work. What is the electric potential energy of the pair of charges when the second charge is at point \(b\)?

(a) How much work would it take to push two protons very slowly from a separation of \(2.00 \times 10^{-10}\) m (a typical atomic distance) to \(3.00 \times 10^{-15}\) 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 their original separation?

A small metal sphere, carrying a net charge of \(q_1 = -\)2.80 \(\mu\)C, is held in a stationary position by insulating supports. A second small metal sphere, with a net charge of \(q_2 = -\)7.80 \(\mu\)C and mass 1.50 g, is projected toward \(q_1\). When the two spheres are 0.800 m apart, \(q_2\), is moving toward \(q_1\) with speed 22.0 m\(/\)s (\(\textbf{Fig. E23.5}\)). Assume that the two spheres can be treated as point charges. You can ignore the force of gravity. (a) What is the speed of \(q_2\) when the spheres are 0.400 m apart? (b) How close does \(q_2\) get to \(q_1\)?

Two protons, starting several meters apart, are aimed directly at each other with speeds of \(2.00 \times 10^5\) m\(/\)s, measured relative to the earth. Find the maximum electric force that these protons will exert on each other.

Three equal 1.20-\(\mu$$C\) point charges are placed at the corners of an equilateral triangle with sides 0.400 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.)

Four electrons are located at the corners 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 corners 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 corners of the triangle, what must the value of the third charge be?

An object with charge \(q = -6.00 \times 10^{-9}\) C is placed in a region of uniform electric field and is released from rest at point \(A\). After the charge has moved to point \(B\), 0.500 m to the right, it has kinetic energy \(3.00 \times 10^{-7}\) J. (a) If the electric potential at point \(A\) is \(+\)30.0 V, what is the electric potential at point \(B\)? (b) What are the magnitude and direction of the electric field?

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

A particle with charge \(+\)4.20 nC is in a uniform electric field \(\overrightarrow{E}\) directed to the left. The charge is released from rest and moves to the left; after it has moved 6.00 cm, its kinetic energy is \(+2.20 \times 10^{-6}\) J. What are (a) the work done by the electric force, (b) the potential of the starting point with respect to the end point, and (c) the magnitude of \(\overrightarrow{E}\) ?

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 \(\times 10^4\) 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 45.0\(^\circ\) downward from the horizontal?

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?

Point charges \(q_1 = +\)2.00 \(\mu\)C and \(q_2 = -\)2.00 \(\mu\)C are placed at adjacent corners of a square for which the length of each side is 3.00 cm. Point \(a\) is at the center of the square, and point \(b\) is at the empty corner closest to \(q_2\) . Take the electric potential to be zero at a distance far from both charges. (a) What is the electric potential at point a due to \(q_1\) and \(q_2\)? (b) What is the electric potential at point \(b\)? (c) A point charge \(q_3 = -\)5.00 \(\mu\)C moves from point \(a\) to point \(b\). How much work is done on \(q_3\) by the electric forces exerted by \(q_1\) and \(q_2\)? Is this work positive or negative?

Two point 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 zero at these points?), and (ii) the electric field is zero (is the potential zero at these points?). (b) Repeat part (a) for two point charges having opposite signs.

Two point charges \(q_1 = +\)2.40 nC and \(q_2 = -\)6.50 nC are 0.100 m apart. Point \(A\) is midway between them; point \(B\) is 0.080 m from \(q_1\) and 0.060 m from \(q_2\) (\(\textbf{Fig. E23.19}\)). 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 electric field on a charge of 2.50 nC that travels from point \(B\) to point \(A\).

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

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 \(V = 0\)? (d) Graph \(V\) at points on the \(x\)-axis as a function of \(x\) in the range from \(x = -2a\) to \(x = +2a\). (e) What does the answer to part (b) become when \(x \gg a\)? Explain why this result is obtained.

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are 4.98 V and 16.2 V\(/\)m, respectively. (Take \(V = 0\) 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 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 \(q = -\)0.200 \(\mu\)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 \(+2Q\) separated by a distance \(d\), and (b) charges \(-Q\) and \(+2Q\) separated by a distance \(d\). (c) Are both \(V\) and \(E\) zero at the same places? Explain.

A thin spherical shell with radius \(R_1 =\) 3.00 cm is concentric with a larger thin spherical shell with radius \(R_2 =\) 5.00 cm. Both shells are made of insulating material. The smaller shell has charge \(q_1 = +\)6.00 nC distributed uniformly over its surface, and the larger shell has charge \(q_2 = -\)9.00 nC distributed uniformly over its surface. Take the electric potential to be zero at an infinite distance from both shells. (a) What is the electric potential due to the two shells at the following distance from their common center: (i) \(r =\) 0; (ii) \(r =\) 4.00 cm; (iii) \(r =\) 6.00 cm? (b) What is the magnitude of the potential difference between the surfaces of the two shells? Which shell is at higher potential: the inner shell or the outer shell?

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 electron. (b) Find the speed of the electron when it reaches the center of the ring.

A solid conducting sphere has net positive charge and radius \(R =\) 0.400 m. At a point 1.20 m from the center of the sphere, the electric potential due to the charge on the sphere is 24.0 V. Assume that \(V = 0\) at an infinite distance from the sphere. What is the electric potential at the center of the sphere?

Charge \(Q =\) 5.00 mC is distributed uniformly over the volume of an insulating sphere that has radius \(R =\) 12.0 cm. A small sphere with charge \(q = +\)3.00 \(\mu\)C and mass 6.00 \(\times 10^{-5}\) kg is projected toward the center of the large sphere from an initial large distance. The large sphere is held at a fixed position and the small sphere can be treated as a point charge. What minimum speed must the small sphere have in order to come within 8.00 cm of the surface of the large sphere?

An infinitely long line of charge has linear charge density 5.00 \(\times 10^{-12}\) C\(/\)m. A proton (mass 1.67 \(\times 10^{-27}\) kg, charge \(+\)1.60 \(\times 10^{-19}\) C) is 18.0 cm from the line and moving directly toward the line at 3.50 \(\times 10^3\) m\(/\)s. (a) Calculate the proton's initial kinetic energy. (b) How close does the proton get to the line of charge?

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 ring of diameter 8.00 cm is fixed in place and carries a charge of \(+\)5.00 \(\mu\)C uniformly spread over its circumference. (a) How much work does it take to move a tiny \(+\)3.00-\(\mu\)C 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 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 maximum speed it will reach?

A very small sphere with positive charge \(q = +\)48.00 \(\mu\)C is released from rest at a point 1.50 cm from a very long line of uniform linear charge density \(\lambda = +\)3.00 \(\mu\)C\(/\)m. What is the kinetic energy of the sphere when it is 4.50 cm from the line of charge if the only force on it is the force exerted by the line of charge?

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 47.0 nC\(/m^2\), what is the magnitude of \(\overrightarrow{E}\) 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 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 force this field exerts on a particle with charge \(+\)2.40 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.

Certain sharks can detect an electric field as weak as 1.0 \(\mu\)V\(/\)m. To grasp how weak this field is, if you wanted to produce it between two parallel metal plates by connecting an ordinary 1.5V AA battery across these plates, how far apart would the plates have to be?

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) 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 3.75 kV? (b) What is the potential of the sphere 's surface relative to infinity?

A metal sphere with radius \(r_a\) 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) \(r < r_a ;\) (ii) \(r_a < r < r_b ;\) (iii) \(r > r_b .\) (\(Hint\): The net potential is the sum of the potentials due to the individual spheres.) Take \(V\) to be zero when \(r\) is infinite. (b) Show that the potential of the inner sphere with respect to the outer is $$V_{ab} = \frac{q} {4\pi\epsilon_0} ( \frac{1} {r_a} - \frac{1} {r_b} )$$ (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the spheres has magnitude $$E(r) = \frac {V_{ab}} {(1/r_a - 1/r_b)}\frac {1} {r_2}$$ (d) Use Eq. (23.23) and the result from part (a) to find the electric field at a point outside the larger sphere at a distance \(r\) from the center, where \(r > r_b\) . (e) Suppose the charge on the outer sphere is not \(-q\) but a negative charge of different magnitude, say \(-Q\). Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different.

A very large plastic sheet carries a uniform charge density of \(-\)6.00 nC\(/\)m\(^2\) 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 reference point for potential? (b) Find the spacing between equipotential surfaces that differ from each other by 1.00 V. What type of surfaces are these?

In a certain region of space, the electric potential is \(V(x, y, z) = Axy - Bx^2 + Cy,\) 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?

In a certain region of space the electric potential is given by \(V = +Ax^2y - Bxy^2,\) where \(A =\) 5.00 \(V/m^3\) and \(B =\) 8.00 \(V/m^3\). Calculate the magnitude and direction of the electric field at the point in the region that has coordinates \(x =\) 2.00 m, \(y =\) 0.400 m, and \(z = 0\).

A point charge \(q_1 = +\)5.00 \(\mu\)C is held fixed in space. From a horizontal distance of 6.00 cm, a small sphere with mass 4.00 \(\times 10^{-3}\) kg and charge \(q2 = +\)2.00 \(\mu\)C is fired toward the fixed charge with an initial speed of 40.0 m\(/\)s. Gravity can be neglected. What is the acceleration of the sphere at the instant when its speed is 25.0 m\(/\)s?

A point charge \(q_1 =\) 4.00 nC is placed at the origin, and a second point charge \(q_2 = -\)3.00 nC is placed on the \(x\)-axis at \(x = +\)20.0 cm. A third point charge \(q_3 =\) 2.00 nC is to be placed on the \(x\)-axis between \(q_1\) and \(q_2\) . (Take as zero the potential energy of the three charges when they are infinitely far apart.) (a) What is the potential energy of the system of the three charges if \(q_3\) is placed at \(x = +\)10.0 cm? (b) Where should \(q_3\) be placed to make the potential energy of the system equal to zero?

A positive point charge \(q_1 = +5.00 \times 10^{-4}\) C is held at a fixed position. A small object with mass 4.00 \(\times 10^{-3}\) kg and charge \(q_2 = -3.00 \times 10^{-4}\) C is projected directly at \(q_1\) . Ignore gravity. When \(q_2\) is 0.400 m away, its speed is 800 m\(/\)s. What is its speed when it is 0.200 m from \(q_1\) ?

A gold nucleus has a radius of 7.3 \(\times 10^{-15}\) m and a charge of \(+79e\). Through what voltage must an alpha particle, with charge \(+2e\), be accelerated so that it has just enough energy to reach a distance of 2.0 \(\times 10^{-14}\) m from the surface of a gold nucleus? (Assume that the gold nucleus remains stationary and can be treated as a point charge.)

A small sphere with mass 5.00 \(\times 10^{-7}\) kg and charge \(+\)7.00 \(\mu\)C is released from rest a distance of 0.400 m above a large horizontal insulating sheet of charge that has uniform surface charge density \(\sigma = +\)8.00 pC\(/\)m\(^2\). Using energy methods, calculate the speed of the sphere when it is 0.100 m above the sheet.

When radium-226 decays radioactively, it emits an alpha particle (the nucleus of helium), and the end product is radon-222. We can model this decay by thinking of the radium-226 as consisting of an alpha particle emitted from the surface of the spherically symmetric radon-222 nucleus, and we can treat the alpha particle as a point charge. The energy of the alpha particle has been measured in the laboratory and has been found to be 4.79 MeV when the alpha particle is essentially infinitely far from the nucleus. Since radon is much heavier than the alpha particle, we can assume that there is no appreciable recoil of the radon after the decay. The radon nucleus contains 86 protons, while the alpha particle has 2 protons and the radium nucleus has 88 protons. (a) What was the electric potential energy of the alpha\(-\)radon combination just before the decay, in MeV and in joules? (b) Use your result from part (a) to calculate the radius of the radon nucleus.

A particle with charge \(+\)7.60 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 cm, the additional force has done 6.50 \(\times 10^{-5}\) J of work and the particle has 4.35 \(\times 10^{-5}\) 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?

Identical charges \(q = +\)5.00 \(\mu\)C are placed at opposite corners of a square that has sides of length 8.00 cm. Point \(A\) is at one of the empty corners, and point \(B\) is at the center of the square. A charge \(q_0 = -\)3.00 \(\mu\)C is placed at point \(A\) and moves along the diagonal of the square to point \(B\). (a) What is the magnitude of the net electric force on \(q_0\) when it is at point \(A\)? Sketch the placement of the charges and the direction of the net force. (b) What is the magnitude of the net electric force on \(q_0\) when it is at point \(B\)? (c) How much work does the electric force do on \(q_0\) during its motion from \(A\) to \(B\)? Is this work positive or negative? When it goes from \(A\) to \(B\), does \(q_0\) move to higher potential or to lower potential?

A vacuum tube diode 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 given by $$V(x) = Cx^{{4}/{3}}$$ where \(x\) is the distance from the cathode and \(C\) is a constant, characteristic of a particular diode and operating conditions. Assume that the distance between the cathode and anode is 13.0 mm and the potential difference between electrodes is 240 V. (a) Determine the value of \(C\). (b) Obtain a formula for the electric field between the electrodes as a function of \(x\). (c) Determine the force on an electron when the electron is halfway between the electrodes.

(a) Calculate the potential energy of a system of two small spheres, one carrying a charge of 2.00 \(\mu\)C and the other a charge of \(-\)3.50 \(\mu\)C, with their centers separated by a distance of 0.180 m. Assume that \(U = 0\) when the charges are infinitely separated. (b) Suppose that one sphere is held in place; the other sphere, with mass 1.50 g, is shot away from it. What minimum initial speed would the moving sphere need 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 is infinitely far from the fixed sphere.)

Two spherical shells have a common center. The inner shell has radius \(R_1 =\) 5.00 cm and charge \(q1 = +3.00 \times 10^{-6}\) C; the outer shell has radius \(R_2 =\) 15.0 cm and charge \(q2 = -5.00 \times 10^{-6}\) C. Both charges are spread uniformly over the shell surface. What is the electric potential due to the two shells at the following distances from their common center: (a) \(r =\) 2.50 cm; (b) \(r =\) 10.0 cm; (c) \(r =\) 20.0 cm? Take \(V = 0\) at a large distance from the shells.