Chapter 21

Of the charge \(Q\) initially on a tiny sphere, a portion \(q\) is to be transferred to a second, nearby sphere. Both spheres can be treated as particles and are fixed with a certain separation. For what value of \(q / Q\) will the electrostatic force between the two spheres be maximized?

Identical isolated conducting spheres 1 and 2 have equal charges and are separated by a distance that is large compared with their diameters (Fig. \(21-22 a\) ). The electrostatic force acting on sphere 2 due to sphere 1 is \(\vec{F}\). Suppose now that a third identical sphere 3 , having an insulating handle and initially neutral, is touched first to sphere 1 (Fig. \(21-22 b\) ), then to sphere 2 (Fig. \(21-22 c\) ), and finally removed (Fig. \(21-22 d\) ). The electrostatic force that now acts on sphere 2 has magnitude \(F^{\prime} .\) What is the ratio \(F^{\prime} / F ?\)

What must be the distance between point charge \(q_{1}=26.0 \mu \mathrm{C}\) and point charge \(q_{2}=-47.0 \mu \mathrm{C}\) for the electrostatic force between them to have a magnitude of \(5.70 \mathrm{~N} ?\)

In the return stroke of a typical lightning bolt, a current of \(2.5 \times 10^{4}\) A exists for \(20 \mu\) s. How much charge is transferred in this event?

A particle of charge \(+3.00 \times 10^{-6} \mathrm{C}\) is \(12.0 \mathrm{~cm}\) distant from a second particle of charge \(-1.50 \times 10^{-6} \mathrm{C}\). Calculate the magnitude of the electrostatic force between the particles.

Two equally charged particles are held \(3.2 \times 10^{-3} \mathrm{~m}\) apart and then released from rest. The initial acceleration of the first particle is observed to be \(7.0 \mathrm{~m} / \mathrm{s}^{2}\) and that of the second to be \(9.0 \mathrm{~m} / \mathrm{s}^{2} .\) If the mass of the first particle is \(6.3 \times 10^{-7} \mathrm{~kg},\) what are (a) the mass of the second particle and (b) the magnitude of the charge of each particle?

Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of \(0.108 \mathrm{~N}\) when their center-to-center separation is \(50.0 \mathrm{~cm}\). The spheres are then connected by a thin conducting wire. When the wire is removed, the spheres repel each other with an electrostatic force of \(0.0360 \mathrm{~N}\). Of the initial charges on the spheres, with a positive net charge, what was (a) the negative charge on one of them and (b) the positive charge on the other?

Two particles are fixed on an \(x\) axis. Particle 1 of charge \(40 \mu \mathrm{C}\) is located at \(x=-2.0 \mathrm{~cm} ;\) particle 2 of charge \(Q\) is located at \(x=3.0 \mathrm{~cm} .\) Particle 3 of charge magnitude \(20 \mu \mathrm{C}\) is released from rest on the \(y\) axis at \(y=2.0 \mathrm{~cm} .\) What is the value of \(Q\) if the initial acceleration of particle 3 is in the positive direction of (a) the \(x\) axis and (b) the \(y\) axis?

Three particles are fixed on an \(x\) axis. Particle 1 of charge \(q_{1}\) is at \(x=-a,\) and particle 2 of charge \(q_{2}\) is at \(x=+a .\) If their net electrostatic force on particle 3 of charge \(+Q\) is to be zero, what must be the ratio \(q_{1} / q_{2}\) when particle 3 is at (a) \(x=+0.500 a\) and (b) \(x=+1.50 a ?\)

The charges and coordinates of two charged particles held fixed in an \(x y\) plane are \(q_{1}=+3.0 \mu \mathrm{C}, x_{1}=3.5 \mathrm{~cm}, y_{1}=0.50 \mathrm{~cm}\) and \(q_{2}=-4.0 \mu \mathrm{C}, x_{2}=-2.0 \mathrm{~cm}, y_{2}=1.5 \mathrm{~cm} .\) Find the (a) magnitude and (b) direction of the electrostatic force on particle 2 due to particle \(1 .\) At what \((\mathrm{c}) x\) and (d) \(y\) coordinates should a third particle of charge \(q_{3}=+4.0 \mu \mathrm{C}\) be placed such that the net electrostatic force on particle 2 due to particles 1 and 3 is zero?

A nonconducting spherical shell, with an inner radius of \(4.0 \mathrm{~cm}\) and an outer radius of \(6.0 \mathrm{~cm},\) has charge spread nonuniformly through its volume between its inner and outer surfaces. The volume charge density \(\rho\) is the charge per unit volume, with the unit coulomb per cubic meter. For this shell \(\rho=b / r,\) where \(r\) is the distance in meters from the center of the shell and \(b=3.0 \mu \mathrm{C} / \mathrm{m}^{2}\). What is the net charge in the shell?

Two tiny, spherical water drops, with identical charges of \(-1.00 \times 10^{-16} \mathrm{C},\) have a center-to-center separation of \(1.00 \mathrm{~cm} .\) (a) What is the magnitude of the electrostatic force acting between them? (b) How many excess electrons are on each drop, giving it its charge imbalance?

How many electrons would have to be removed from a coin to leave it with a charge of \(+1.0 \times 10^{-7} \mathrm{C} ?\)

What is the magnitude of the electrostatic force between a singly charged sodium ion \(\left(\mathrm{Na}^{+},\right.\) of charge \(\left.+e\right)\) and an adjacent singly charged chlorine ion \(\left(\mathrm{Cl}^{-},\right.\) of charge \(\left.-e\right)\) in a salt crystal if their separation is \(2.82 \times 10^{-10} \mathrm{~m} ?\)

The magnitude of the electrostatic force between two identical ions that are separated by a distance of \(5.0 \times 10^{-10} \mathrm{~m}\) is \(3.7 \times 10^{-9} \mathrm{~N}\). (a) What is the charge of each ion? (b) How many electrons are "missing" from each ion (thus giving the ion its charge imbalance)?

A current of 0.300 A through your chest can send your heart into fibrillation, ruining the normal rhythm of heartbeat and disrupting the flow of blood (and thus oxygen) to your brain. If that current persists for \(2.00 \mathrm{~min}\), how many conduction electrons pass through your chest?

Earth’s atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. What would be the electric current intercepted by the total surface area of the planet?

Calculate the number of coulombs of positive charge in \(250 \mathrm{~cm}^{3}\) of (neutral) water. (Hint: A hydrogen atom contains one proton; an oxygen atom contains eight protons.)

In crystals of the salt cesium chloride, cesium ions \(\mathrm{Cs}^{+}\) form the eight corners of a cube and a chlorine ion \(\mathrm{Cl}^{-}\) is at the cube's center (Fig. \(21-36\) ). The edge length of the cube is \(0.40 \mathrm{nm}\). The \(\mathrm{Cs}^{+}\) ions are each deficient by one electron (and thus each has a charge of \(+e\) ), and the \(\mathrm{Cl}^{-}\) ion has one excess electron (and thus has a charge of \(-e\) ). (a) What is the magnitude of the net electrostatic force exerted on the \(\mathrm{Cl}^{-}\) ion by the eight \(\mathrm{Cs}^{+}\) ions at the corners of the cube? (b) If one of the Cs \(^{+}\) ions is missing, the crystal is said to have a defect; what is the magnitude of the net electrostatic force exerted on the \(\mathrm{Cl}^{-}\) ion by the seven remaining \(\mathrm{Cs}^{+}\) ions?

Electrons and positrons are produced by the nuclear transformations of protons and neutrons known as beta decay. (a) If a proton transforms into a neutron, is an electron or a positron produced? (b) If a neutron transforms into a proton, is an electron or a positron produced?

Figure 21-37 shows four identical conducting spheres that are actually well separated from one another. Sphere \(W\) (with an initial charge of zero) is touched to sphere \(A\) and then they are separated. Next, sphere \(W\) is touched to sphere \(B\) (with an initial charge of \(-32 e\) ) and then they are separated. Finally, sphere \(W\) is touched to sphere \(C\) (with an initial charge of \(+48 e\) ), and then they are separated. The final charge on sphere \(W\) is \(+18 e .\) What was the initial charge on sphere \(A ?\)

(a) What equal positive charges would have to be placed En Earth and on the Moon to neutralize their gravitational attraction? (b) Why don't you need to know the lunar distance to solve this problem? (c) How many kilograms of hydrogen ions (that is, protons) would be needed to provide the positive charge calculated in (a)?

How many megacoulombs of positive charge are in \(1.00 \mathrm{~mol}\) of neutral molecular-hydrogen gas \(\left(\mathrm{H}_{2}\right) ?\)

Point charges of \(+6.0 \mu \mathrm{C}\) and \(-4.0 \mu \mathrm{C}\) are placed on an \(x\) axis, at \(x=8.0 \mathrm{~m}\) and \(x=16 \mathrm{~m},\) respectively. What charge must be placed at \(x=24 \mathrm{~m}\) so that any charge placed at the origin would experience no electrostatic force?

A neutron consists of one "up" quark of charge \(+2 e / 3\) and two "down" quarks each having charge \(-e / 3 .\) If we assume that the down quarks are \(2.6 \times 10^{-15} \mathrm{~m}\) apart inside the neutron, what is the magnitude of the electrostatic force between them?

A charged nonconducting rod, with a length of \(2.00 \mathrm{~m}\) and a cross- sectional area of \(4.00 \mathrm{~cm}^{2}\), lies along the positive side of an \(x\) axis with one end at the origin. The volume charge density \(\rho\) is charge per unit volume in coulombs per cubic meter. How many excess electrons are on the rod if \(\rho\) is (a) uniform, with a value of \(-4.00 \mu \mathrm{C} / \mathrm{m}^{3},\) and \((\mathrm{b})\) nonuniform, with a value given by \(\rho=b x^{2}\) where \(b=-2.00 \mu \mathrm{C} / \mathrm{m}^{5} ?\)

What would be the magnitude of the electrostatic force between two \(1.00 \mathrm{C}\) point charges separated by a distance of (a) \(1.00 \mathrm{~m}\) and (b) \(1.00 \mathrm{~km}\) if such point charges existed (they do not) and this configuration could be set up?

A charge of \(6.0 \mu \mathrm{C}\) is to be split into two parts that are then separated by \(3.0 \mathrm{~mm}\). What is the maximum possible magnitude of the electrostatic force between those two parts?

Of the charge \(Q\) on a tiny sphere, a fraction \(\alpha\) is to be transferred to a second, nearby sphere. The spheres can be treated as particles. (a) What value of \(\alpha\) maximizes the magnitude \(F\) of the electrostatic force between the two spheres? What are the (b) smaller and (c) larger values of \(\alpha\) that put \(F\) at half the maximum magnitude?

If a cat repeatedly rubs against your cotton slacks on a dry day, the charge transfer between the cat hair and the cotton can leave you with an excess charge of \(-2.00 \mu \mathrm{C}\). (a) How many electrons are transferred between you and the cat? You will gradually discharge via the floor, but if instead of waiting, you immediately reach toward a faucet, a painful spark can suddenly appear as your fingers near the faucet. (b) In that spark, do electrons flow from you to the faucet or vice versa? (c) Just before the spark appears, do you induce positive or negative charge in the faucet? (d) If, instead, the cat reaches a paw toward the faucet, which way do electrons flow in the resulting spark? (e) If you stroke a cat with a bare hand on a dry day, you should take care not to bring your fingers near the cat's nose or you will hurt it with a spark. Considering that cat hair is an insulator, explain how the spark can appear.

We know that the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that these magnitudes differ from each other by \(0.00010 \%\). With what force would two copper coins, placed \(1.0 \mathrm{~m}\) apart, repel each other? Assume that each coin contains \(3 \times 10^{22}\) copper atoms. (Hint: A neutral copper atom contains 29 protons and 29 electrons.) What do you conclude?

Three charged particles form a triangle: particle 1 with charge \(Q_{1}=80.0 \mathrm{nC}\) is at \(x y\) coordinates \((0,3.00 \mathrm{~mm}),\) particle 2 with charge \(Q_{2}\) is at \((0,-3.00 \mathrm{~mm}),\) and particle 3 with charge \(q=18.0 \mathrm{nC}\) is at \((4.00 \mathrm{~mm}, 0) .\) In unit-vector notation, what is the electrostatic force on particle 3 due to the other two particles if \(Q_{2}\) is equal to (a) \(80.0 \mathrm{nC}\) and (b) \(-80.0 \mathrm{nC} ?\)

Two point charges of \(30 \mathrm{nC}\) and \(-40 \mathrm{nC}\) are held fixed on an \(x\) axis, at the origin and at \(x=72 \mathrm{~cm},\) respectively. A particle with a charge of \(42 \mu \mathrm{C}\) is released from rest at \(x=28 \mathrm{~cm} .\) If the initial acceleration of the particle has a magnitude of \(100 \mathrm{~km} / \mathrm{s}^{2},\) what is the particle's mass?

Two small, positively charged spheres have a combined charge of \(5.0 \times 10^{-5} \mathrm{C}\). If each sphere is repelled from the other by an electrostatic force of \(1.0 \mathrm{~N}\) when the spheres are \(2.0 \mathrm{~m}\) apart, what is the charge on the sphere with the smaller charge?

An electron is in a vacuum near Earth's surface and located at \(y=0\) on a vertical \(y\) axis. At what value of \(y\) should a second electron be placed such that its electrostatic force on the first electron balances the gravitational force on the first electron?

In a spherical metal shell of radius \(R,\) an electron is shot from the center directly toward a tiny hole in the shell, through which it escapes. The shell is negatively charged with a surface charge density (charge per unit area) of \(6.90 \times 10^{-13} \mathrm{C} / \mathrm{m}^{2} .\) What is the magnitude of the electron's acceleration when it reaches radial distances (a) \(r=0.500 R\) and (b) \(2.00 R ?\)

An electron is projected with an initial speed \(v_{i}=3.2 \times 10^{5} \mathrm{~m} / \mathrm{s}\) directly toward a very distant proton that is at rest. Because the proton mass is large relative to the electron mass, assume that the proton remains at rest. By calculating the work done on the electron by the electrostatic force, determine the distance between the two particles when the electron instantaneously has speed \(2 v_{i}\).

A \(100 \mathrm{~W}\) lamp has a steady current of \(0.83 \mathrm{~A}\) in its filament. How long is required for \(1 \mathrm{~mol}\) of electrons to pass through the lamp?

The charges of an electron and a positron are \(-e\) and \(+e .\) The mass of each is \(9.11 \times 10^{-31} \mathrm{~kg} .\) What is the ratio of the electrical force to the gravitational force between an electron and a positron?