Chapter 18

How many electrons must be removed from an electrically neutral silver dollar to give it a charge of \(+2.4 \mu \mathrm{C} ?\)

A plate carries a charge of \(-3.0 \mu \mathrm{C}\), while a rod carries a charge of \(+2.0 \mu \mathrm{C}\) How many electrons must be transferred from the plate to the rod, so that both objects have the same charge?

A metal sphere has a charge of \(+8.0 \mu \mathrm{C}\). What is the net charge after \(6.0 \times 10^{13}\) electrons have been placed on it?

Four identical metallic objects carry the following charges: \(+1.6,+6.2,-4.8,\) and \(-9.4 \mu \mathrm{C} .\) The objects are brought simultaneously into contact, so that each touches the others. Then they are separated, (a) What is the final charge on each object? (b) How many electrons (or protons) make up the final charge on each object?

Consider three identical metal spheres, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Sphere A carries a charge of \(+5 q .\) Sphere \(\mathrm{B}\) carries a charge of \(-q\). Sphere \(\mathrm{C}\) carries no net charge. Spheres \(\mathrm{A}\) and \(\mathrm{B}\) are touched together and then separated. Sphere \(\mathrm{C}\) is then touched to sphere \(A\) and separated from it. Last, sphere \(C\) is touched to sphere \(B\) and separated from it. (a) How much charge ends up on sphere \(\mathrm{C}\) ? What is the total charge on the three spheres (b) before they are allowed to touch each other and (c) after they have touched?

Consider three identical metal spheres, \(A, B,\) and \(C .\) Sphere A carries a charge of \(+5 q .\) Sphere \(B\) carries a charge of \(-q\). Sphere \(\mathrm{C}\) carries no net charge. Spheres \(\mathrm{A}\) and \(\mathrm{B}\) are touched together and then separated. Sphere \(\mathrm{C}\) is then touched to sphere \(A\) and separated from it. Last, sphere \(C\) is touched to sphere \(B\) and separated from it. (a) How much charge ends up on sphere \(\mathrm{C}\) ? What is the total charge on the three spheres (b) before they are allowed to touch each other and (c) after they have touched?

Water has a mass per mole of \(18.0 \mathrm{~g} / \mathrm{mol}\), and each water molecule \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) has 10 electrons. (a) How many electrons are there in one liter \(\left(1.00 \times 10^{-3} \mathrm{~m}^{3}\right)\) of water? (b) What is the net charge of all these electrons?

Two charges attract each other with a force of \(1.5 \mathrm{~N}\). What will be the force if the distance between them is reduced to one-ninth of its original value?

Two spherical objects are separated by a distance of \(1.80 \times 10^{-3} \mathrm{~m}\). The objects are initially electrically neutral and are very small compared to the distance between them. Each object acquires the same negative charge due to the addition of electrons. As a result, each object experiences an electrostatic force that has a magnitude of \(4.55 \times 10^{-21} \mathrm{~N}\). How many electrons did it take to produce the charge on one of the objects?

Two very small spheres are initially neutral and separated by a distance of \(0.50 \mathrm{~m}\). Suppose that \(3.0 \times 10^{13}\) electrons are removed from one sphere and placed on the other. (a) What is the magnitude of the electrostatic force that acts on each sphere? (b) Is the force attractive or repulsive? Why?

A charge \(+q\) is located at the origin, while an identical charge is located on the \(x\) axis at \(x=+0.50 \mathrm{~m}\). A third charge of \(+2 q\) is located on the \(x\) axis at such a place that the net electrostatic force on the charge at the origin doubles, its direction remaining unchanged. Where should the third charge be located?

Consult Concept Simulation 18.1 at for insight into this problem. Three charges are fixed to an \(x, y\) coordinate system. A charge of \(+18 \mu \mathrm{C}\) is on the \(y\) axis at \(y=+3.0 \mathrm{~m}\). A charge of \(-12 \mu \mathrm{C}\) is at the origin. Last, a charge of \(+45 \mu \mathrm{C}\) is on the \(x\) axis at \(x=+3.0 \mathrm{~m}\). Determine the magnitude and direction of the net electrostatic force on the charge at \(x=+3.0 \mathrm{~m}\). Specify the direction relative to the \(-x\) axis.

A charge of \(-3.00 \mu \mathrm{C}\) is fixed at the center of a compass. Two additional charges are fixed on the circle of the compass (radius \(=0.100 \mathrm{~m}\) ). The charges on the circle are \(-4.00 \mu \mathrm{C}\) at the position due north and \(+5.00 \mu \mathrm{C}\) at the position due east. What is the magnitude and direction of the net electrostatic force acting on the charge at the center? Specify the direction relative to due east.

Two particles, with identical positive charges and a separation of \(2.60 \times 10^{-2} \mathrm{~m}\), are released from rest. Immediately after the release, particle 1 has an acceleration \(\overrightarrow{\mathbf{a}},\) whose magnitude is \(4.60 \times 10^{3} \mathrm{~m} / \mathrm{s}^{2}\), while particle 2 has an acceleration \(\overrightarrow{\mathbf{a}}_{2}\) whose magnitude is \(8.50 \times 10^{3} \mathrm{~m} / \mathrm{s}^{2}\). Particle 1 has a mass $$ \text { of } 6.00 \times 10^{-6} \mathrm{~kg} . \text { Find } $$ (a) the charge on each particle and (b) the mass of particle 2 .

Two tiny conducting spheres are identical and carry charges of \(-20.0 \mu \mathrm{C}\) and \(+50.0 \mu \mathrm{C}\). They are separated by a distance of \(2.50 \mathrm{~cm}\). (a) What is the magnitude of the force that each sphere experiences, and is the force attractive or repulsive? (b) The spheres are brought into contact and then separated to a distance of \(2.50 \mathrm{~cm}\). Determine the magnitude of the force that each sphere now experiences, and state whether the force is attractive or repulsive.

Interactive Solution \(\underline{18.15}\) at provides a model for solving this type of problem. Two small objects, \(\mathrm{A}\) and \(\mathrm{B},\) are fixed in place and separated by \(3.00 \mathrm{~cm}\) in a vacuum. Object \(\mathrm{A}\) has a charge of \(+2.00 \mu \mathrm{C},\) and object \(\mathrm{B}\) has a charge of \(-2.00 \mu \mathrm{C}\). How many electrons must be removed from \(\mathrm{A}\) and put onto \(\mathrm{B}\) to make the electrostatic force that acts on each object an attractive force whose magnitude is \(68.0 \mathrm{~N} ?\)

Two spheres are mounted on identical horizontal springs and reston a frictionless table, as in the drawing. When the spheres are uncharged, the spacing between them is \(0.0500 \mathrm{~m},\) and the springs are unstrained. When each sphere has a charge of \(+1.60 \mu \mathrm{C},\) the spacing doubles. Assuming that the spheres have a negligible diameter, determine the spring constant of the springs.

Multiple-Concept Example 3 illustrates several of the concepts used in this problem. A single electron orbits a lithium nucleus that contains three protons \((+3 e)\). The radius of the orbit is \(1.76 \times 10^{-11} \mathrm{~m}\). Determine the kinetic energy of the electron.

An electrically neutral model airplane is flying in a horizontal circle on a 3.0 -m guideline, which is nearly parallel to the ground. The line breaks when the kinetic energy of the plane is \(50.0 \mathrm{~J}\). Reconsider the same situation, except that now there is a point charge of \(+q\) on the plane and a point charge of \(-q\) at the other end of the guideline. In this case, the line breaks when the kinetic energy of the plane is \(51.8 \mathrm{~J}\). Find the magnitude of the charges.

Two identical small insulating balls are suspended by separate \(0.25-\mathrm{m}\) threads that are attached to a common point on the ceiling. Each ball has a mass of \(8.0 \times 10^{-4} \mathrm{~kg}\). Initially the balls are uncharged and hang straight down. They are then given identical positive charges and, as a result, spread apart with an angle of \(36^{\circ}\) between the threads. Determine (a) the charge on each ball and (b) the tension in the threads.

There are four charges, each with a magnitude of \(2.0 \mu \mathrm{C}\). Two are positive and two are negative. The charges are fixed to the corners of a 0.30 -m square, one to a corner, in such a way that the net force on any charge is directed toward the center of the square. Find the magnitude of the net electrostatic force experienced by any charge.

A tiny ball (mass \(=0.012 \mathrm{~kg}\) ) carries a charge of \(-18 \mu \mathrm{C}\). What electric field (magnitude and direction) is needed to cause the ball to float above the ground?

Four point charges have the same magnitude of \(2.4 \times 10^{-12} \mathrm{C}\) and are fixed to the corners of a square that is 4.0 \(\mathrm{cm}\) on a side. Three of the charges are positive and one is negative. Determine the magnitude of the net electric field that exists at the center of the square.

Two charges, \(-16\) and \(+4.0 \mu \mathrm{C}\), are fixed in place and separated by \(3.0 \mathrm{~m}\). (a) At what spot along a line through the charges is the net electric field zero? Locate this spot relative to the positive charge. (Hint: The spot does not necessarily lie between the two charges.) (b) What would be the force on a charge of \(+14 \mu \mathrm{C}\) placed at this spot?

The membrane surrounding a living cell consists of an inner and an outer wall that are separated by a small space. Assume that the membrane acts like a parallel plate capacitor in which the effective charge density on the inner and outer walls has a magnitude of \(7.1 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2} \cdot\) (a) What is the magnitude of the electric field within the cell membrane? (b) Find the magnitude of the electric force that would be exerted on a potassium ion (K \(+\); charge \(=+e\) ) placed inside the membrane.

Two charges are placed on the \(x\) axis. One of the charges \(\left(q_{1}=+8.5 \mu \mathrm{C}\right)\) is at \(x_{1}=+3.0 \mathrm{~cm}\) and the other \(\left(q_{2}=-21 \mu \mathrm{C}\right)\) is at \(x_{1}=+9.0 \mathrm{~cm} .\) Find the net electric field (magnitude and direction) at (a) \(x=0 \mathrm{~cm}\) and (b) \(x=+6.0 \mathrm{~cm}\).

A long, thin rod (length \(=4.0 \mathrm{~m}\) ) lies along the \(x\) axis, with its midpoint at the origin. In a vacuum, \(\mathrm{a}+8.0 \mu \mathrm{C}\) point charge is fixed to one end of the rod, and \(a-8.0 \mu C\) point charge is fixed to the other end. Everywhere in the \(x, y\) plane there is a constant external electric field (magnitude \(\left.=5.0 \times 10^{3} \mathrm{~N} / \mathrm{C}\right)\) that is perpendicular to the rod. With respect to the \(z\) axis, find the magnitude of the net torque applied to the rod.

A small drop of water is suspended motionless in air by a uniform electric field that is directed upward and has a magnitude of \(8480 \mathrm{~N} / \mathrm{C}\). The mass of the water drop is \(3.50 \times 10^{-9} \mathrm{~kg} .\) (a) Is the excess charge on the water drop positive or negative? Why? (b) How many excess electrons or protons reside on the drop?

Two parallel plate capacitors have circular plates. The magnitude of the charge on these plates is the same. However, the electric field between the plates of the first capacitor is \(2.2 \times 10^{5} \mathrm{~N} / \mathrm{C},\) while the field within the second capacitor is \(3.8 \times 10^{5} \mathrm{~N} / \mathrm{C}\). Determine the ratio \(r_{2} / r_{1}\) of the plate radius for the second capacitor to the plate radius for the first capacitor.

Interactive Solution \(18.37\) at provides a model for problems of this kind. A small object has a mass of \(3.0 \times 10^{-3} \mathrm{~kg}\) and a charge of \(-34 \mu \mathrm{C}\). It is placed at a certain spot where there is an electric field. When released, the object experiences an acceleration of \(2.5 \times 10^{3} \mathrm{~m} / \mathrm{s}^{2}\) in the direction of the \(+x\) axis. Determine the magnitude and direction of the electric field.

Interactive Solution \(\underline{18.37}\) at provides a model for problems of this kind. A small object has a mass of \(3.0 \times 10^{-3} \mathrm{~kg}\) and a charge of \(-34 \mu \mathrm{C}\). It is placed at a certain spot where there is an electric field. When released, the object experiences an acceleration of \(2.5 \times 10^{3} \mathrm{~m} / \mathrm{s}^{2}\) in the direction of the \(+x\) axis. Determine the magnitude and direction of the electric field.

A rectangle has a length of \(2 d\) and a height of \(d\). Each of the following three charges is located at a corner of the rectangle: \(+q_{1}\) (upper left corner), \(+q_{2}\) (lower right corner), and \(-q\) (lower left corner). The net electric field at the (empty) upper right corner is zero. Find the magnitudes of \(q_{1}\) and \(q_{2}\). Express your an swers in terms of \(q\).

Two particles are in a uniform electric field whose value is \(+2500 \mathrm{~N} / \mathrm{C}\). The mass and charge of particle 1 are \(m_{1}=1.4 \times 10^{-5} \mathrm{~kg}\) and \(q_{1}=-7.0 \mu \mathrm{C},\) while the corresponding values for particle 2 are \(m_{2}=2.6 \times 10^{-5} \mathrm{~kg}\) and \(q_{2}=+18 \mu \mathrm{C} .\) Initially the particles are at rest. The particles are both located on the same electric field line, but are separated from each other by a distance \(d\). When released, they accelerate, but always remain at this same distance from each other. Find \(d\).

A spherical surface completely surrounds a collection of charges. Find the electric flux through the surface if the collection consists of (a) a single \(+3.5 \times 10^{-6} \mathrm{C}\) charge, (b) a single \(-2.3 \times 10^{-6} \mathrm{C}\) charge, and (c) both of the charges in (a) and (b).

A rectangular surface \((0.16 \mathrm{~m} \times 0.38 \mathrm{~m})\) is oriented in a uniform electric field of \(580 \mathrm{~N} / \mathrm{C}\). What is the maximum possible electric flux through the surface?

A vertical wall \((5.9 \mathrm{~m} \times 2.5 \mathrm{~m})\) in a house faces due east. A uniform electric field has a magnitude of \(150 \mathrm{~N} / \mathrm{C}\). This field is parallel to the ground and points \(35^{\circ}\) north of east. What is the electric flux through the wall?

A charge \(Q\) is located inside a rectangular box. The electric flux through each of the six surfaces of the box is \(\Phi_{1}=+1500 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}, \Phi_{2}=+2200 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}, \Phi_{3}=+4600 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}, \Phi_{4}=-1800 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}, \Phi_{5}=-3500 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}\), and \(\Phi_{6}=-540 \mathrm{c}\) Wha

\(\operatorname{ssm}\) A cube is located with one corner at the origin of an \(x, y, z\) coordinate system. One of the cube's faces lies in the \(x, y\) plane, another in the \(y, z\) plane, and another in the \(x, z\) plane. In other words, the cube is in the first octant of the coordinate system. The edges of the cube are \(0.20 \mathrm{~m}\) long. A uniform electric field is parallel to the \(x, y\) plane and points in the direction of the \(+y\) axis. The magnitude of the field is \(1500 \mathrm{~N} / \mathrm{C}\). (a) Find the electric flux through each of the six faces of the cube. (b) Add the six values obtained in part (a) to show that the electric flux through the cubical surface is zero, as Gauss' law predicts, since there is no net charge within the cube

A cube is located with one corner at the origin of an \(x, y, z\) coordinate system. One of the cube's faces lies in the \(x, y\) plane, another in the \(y, z\) plane, and another in the \(x, z\) plane. In other words, the cube is in the first octant of the coordinate system. The edges of the cube are \(0.20 \mathrm{~m}\) long. A uniform electric field is parallel to the \(x, y\) plane and points in the direction of the \(+y\) axis. The magnitude of the field is \(1500 \mathrm{~N} / \mathrm{C}\). (a) Find the electric flux through each of the six faces of the cube. (b) Add the six values obtained in part (a) to show that the electric flux through the cubical surface is zero, as Gauss' law predicts, since there is no net charge within the cube.

A long, thin, straight wire of length \(L\) has a positive charge \(Q\) distributed uniformly along it. Use Gauss' law to show that the electric field created by this wire at a radial distance \(r\) has a magnitude of \(E=\lambda /\left(2 \pi \epsilon_{0} r\right)\), where \(\lambda=Q / L .\) (Hint: For a Gaussian surface, use a cylinder aligned with its axis along the wire and note that the cylinder has a flat surface at either end, as well as a curved surface.)

An electric field of \(260000 \mathrm{~N} / \mathrm{C}\) points due west at a certain spot. What are the magnitude and direction of the force that acts on a charge of \(-7.0 \mu \mathrm{C}\) at this spot?

The force of repulsion that two like charges exert on each other is \(3.5 \mathrm{~N}\). What will the force be if the distance between the charges is increased to five times its original value?

Conceptual Example 14 deals with the hollow spherical conductor in Figure \(18-31\). The conductor is initially electrically neutral, and then a charge \(+q\) is placed at the center of the hollow space. Suppose the conductor initially has a net charge of \(+2 q\) instead of being neutral. What is the total charge on the interior and on the exterior surface when the \(+q\) charge is placed at the center?

At a distance \(r_{1}\) from a point charge, the magnitude of the electric field created by the charge is \(248 \mathrm{~N} / \mathrm{C}\). At a distance \(r_{2}\) from the charge, the field has a magnitude of \(132 \mathrm{~N} /\) C. Find the ratio \(r_{2} / r_{1}\)

In a vacuum, two particles have charges of \(q_{1}\) and \(q_{2}\), where \(q_{1}=+3.5 \mu \mathrm{C}\). They are separated by a distance of \(0.26 \mathrm{~m}\), and particle 1 experiences an attractive force of \(3.4 \mathrm{~N}\). What is \(q_{2}\) (magnitude and sign)?

Interactive LearningWare 18.1 at offers some perspective on this problem. Two tiny spheres have the same mass and carry charges of the same magnitude. The mass of each sphere is \(2.0 \times 10^{-6} \mathrm{~kg} .\) The gravitational force that each sphere exerts on the other is balanced by the electric force. (a) What algebraic signs can the charges have? (b) Determine the charge magnitude.

A charge of \(q=+7.50 \mu \mathrm{C}\) is located in an electric field. The \(x\) and \(y\) components of the electric field are \(E_{x}=6.00 \times 10^{3} \mathrm{~N} / \mathrm{C}\) and \(E_{y}=8.00 \times 10^{3} \mathrm{~N} / \mathrm{C}\), respectively. (a) What is the magnitude of the force on the charge? (b) Determine the angle that the force makes with the \(+x\) axis.

Two charges are located along the \(x\) axis: \(q_{1}=+6.0 \mu \mathrm{C}\) at \(x_{1}=+4.0 \mathrm{~cm}\), and \(q_{2}=+6.0 \mu \mathrm{C}\) at \(x_{2}=-4.0 \mathrm{~cm}\). Two other charges are located on the \(y\) axis: \(q_{3}=+3.0 \mu \mathrm{C}\) at \(y_{3}=+5.0 \mathrm{~cm}\), and \(q_{4}=-8.0 \mu \mathrm{C}\) at \(y_{4}=+7.0 \mathrm{~cm} .\) Find the net electric field (magnitude and direction) at the origin.

Four point charges have equal magnitudes. Three are positive, and one is negative, as the drawing shows. They are fixed in place on the same straight line, and adjacent charges are equally separated by a distance \(d\). Consider the net electrostatic force acting on each charge. Calculate the ratio of the largest to the smallest net force.

A proton is moving parallel to a uniform electric field. The electric field accelerates the proton and thereby increases its linear momentum to \(5.0 \times 10^{-23} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) from \(1.5 \times 10^{-23} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) in a time of \(6.3 \times 10^{-6} \mathrm{~s}\). What is the magnitude of the electric field?