C H A P T E R 2 1 Coulomb’s Law
21-1 COULOMB’S LAW
Leraning Objectives
After reading this module, you should be able to …
21.01 Distinguish between being electrically neutral, negatively charged, and positively charged and identify excess charge.
21.02 Distinguish between conductors, nonconductors (insu- lators), semiconductors, and superconductors.
21.03 Describe the electrical properties of the particles in- side an atom.
21.04 Identify conduction electrons and explain their role in making a conducting object negatively or positively charged.
21.05 Identify what is meant by “electrically isolated” and by “grounding.”
21.06 Explain how a charged object can set up induced charge in a second object.
21.07 Identify that charges with the same electrical sign repel each other and those with opposite electrical signs attract each other.
21.08 For either of the particles in a pair of charged particles, draw a free-body diagram, showing the electrostatic force (Coulomb force) on it and anchoring the tail of the force vector on that particle.
21.09 For either of the particles in a pair of charged particles, apply Coulomb’s law to relate the magnitude of the electro- static force, the charge magnitudes of the particles, and the separation between the particles.
21.10 Identify that Coulomb’s law applies only to (point-like) particles and objects that can be treated as particles.
21.11 If more than one force acts on a particle, find the net force by adding all the forces as vectors, not scalars.
21.12 Identify that a shell of uniform charge attracts or repels a charged particle that is outside the shell as if all the shell’s charge were concentrated as a particle at the shell’s center.
21.13 Identify that if a charged particle is located inside a shell of uniform charge, there is no net electrostatic force on the particle from the shell.
21.14 Identify that if excess charge is put on a spherical conduc- tor, it spreads out uniformly over the external surface area.
21.15 Identify that if two identical spherical conductors touch or are connected by conducting wire, any excess charge will be shared equally.
21.16 Identify that a nonconducting object can have any given distribution of charge, including charge at interior points.
21.17 Identify current as the rate at which charge moves through a point.
21.18 For current through a point, apply the relationship be- tween the current, a time interval, and the amount of charge that moves through the point in that time interval.
Key Ideas
The strength of a particle’s electrical interaction with objects around it depends on its electric charge (usually repre-
sented as q), which can be either positive or negative. Particles with the same sign of charge repel each other, and particles with opposite signs of charge attract each other.
An object with equal amounts of the two kinds of charge is
electrically neutral, whereas one with an imbalance is electri- cally charged and has an excess charge.
Conductors are materials in which a significant number of
electrons are free to move. The charged particles in noncon- ductors (insulators) are not free to move.
Electric current i is the rate dq/dt at which charge passes a
point:
Coulomb’s law describes the electrostatic force (or electric force) between two charged particles. If the particles have
charges \(q_1\) and \(q_2\), are separated by distance r, and are at rest (or moving only slowly) relative to each other, then the magnitude of the force acting on each due to the other is given by
(Coulomb’s law),
where \(8.85 \times 10^{-12} c^2/N \cdot m^2\) is the permittivity constant. The ratio 1/4p´0 is often replaced with the electrostatic constant (or Coulomb constant) \(8.99 \times 10^{9} N \cdot m^2/C^2\) .
The electrostatic force vector acting on a charged particle
due to a second charged particle is either directly toward the second particle (opposite signs of charge) or directly away from it (same sign of charge).
If multiple electrostatic forces act on a particle, the net force
is the vector sum (not scalar sum) of the individual forces.
Shell theorem 1: A charged particle outside a shell with charge
uniformly distributed on its surface is attracted or repelled as if the shell’s charge were concentrated as a particle at its center.
Shell theorem 2: A charged particle inside a shell with
charge uniformly distributed on its surface has no net force acting on it due to the shell.
Charge on a conducting spherical shell spreads uniformly
over the (external) surface.
What Is Physics?
You are surrounded by devices that depend on the physics of electromagnetism, which is the combination of electric and magnetic phenomena. This physics is at the root of computers, television, radio, telecommunications, household lighting, and even the ability of food wrap to cling to a container. This physics is also the basis of the natural world. Not only does it hold together all the atoms and molecules in the world, it also produces lightning, auroras, and rainbows.
The physics of electromagnetism was first studied by the early Greek philosophers, who discovered that if a piece of amber is rubbed and then brought near bits of straw, the straw will jump to the amber. We now know that the attrac- tion between amber and straw is due to an electric force. The Greek philosophers also discovered that if a certain type of stone (a naturally occurring magnet) is brought near bits of iron, the iron will jump to the stone. We now know that the attraction between magnet and iron is due to a magnetic force.
From these modest origins with the Greek philosophers, the sciences of electricity and magnetism developed separately for centuries—until 1820, in fact, when Hans Christian Oersted found a connection between them: an electric cur- rent in a wire can deflect a magnetic compass needle. Interestingly enough, Oersted made this discovery, a big surprise, while preparing a lecture demonstra- tion for his physics students.
The new science of electromagnetism was developed further by workers in many countries. One of the best was Michael Faraday, a truly gifted experimenter with a talent for physical intuition and visualization. That talent is attested to by the fact that his collected laboratory notebooks do not contain a single equation. In the mid-nineteenth century, James Clerk Maxwell put Faraday’s ideas into mathematical form, introduced many new ideas of his own, and put electromag- netism on a sound theoretical basis.
Our discussion of electromagnetism is spread through the next 16 chapters. We begin with electrical phenomena, and our first step is to discuss the nature of electric charge and electric force.
Electric Charge
Here are two demonstrations that seem to be magic, but our job here is to make sense of them. After rubbing a glass rod with a silk cloth (on a day when the humidity is low), we hang the rod by means of a thread tied around its center (Fig. 21-la). Then we rub a second glass rod with the silk cloth and bring it near the hanging rod. The hanging rod magically moves away. We can see that a force repels it from the second rod, but how? There is no contact with that rod, no breeze to push on it, and no sound wave to disturb it.
In the second demonstration we replace the second rod with a plastic rod that has been rubbed with fur. This time, the hanging rod moves toward the nearby rod (Fig. 21-1b). Like the repulsion, this attraction occurs without any contact or obvious communication between the rods.
In the next chapter we shall discuss how the hanging rod knows of the pres- ence of the other rods, but in this chapter let’s focus on just the forces that are in- volved. In the first demonstration, the force on the hanging rod was repulsive, and in the second, attractive. After a great many investigations, scientists figured out that the forces in these types of demonstrations are due to the electric charge that we set up on the rods when they are in contact with silk or fur. Electric charge is an intrinsic property of the fundamental particles that make up objects such as the rods, silk, and fur. That is, charge is a property that comes automatically with those particles wherever they exist.
Two Types. There are two types of electric charge, named by the American scientist and statesman Benjamin Franklin as positive charge and negative charge. He could have called them anything (such as cherry and walnut), but us- ing algebraic signs as names comes in handy when we add up charges to find the net charge. In most everyday objects, such as a mug, there are about equal num- bers of negatively charged particles and positively charged particles, and so the net charge is zero, the charge is said to be balanced, and the object is said to be electrically neutral (or just neutral for short).
Excess Charge. Normally you are approximately neutral. However, if you live in regions where the humidity is low, you know that the charge on your body can be- come slightly unbalanced when you walk across certain carpets. Either you gain neg- ative charge from the carpet (at the points of contact between your shoes with the carpet) and become negatively charged, or you lose negative charge and become pos- itively charged. Either way, the extra charge is said to be an excess charge.You proba- bly don’t notice it until you reach for a door handle or another person. Then, if your excess charge is enough, a spark leaps between you and the other object, eliminating your excess charge. Such sparking can be annoying and even somewhat painful. Such charging and discharging does not happen in humid conditions because the water in the air neutralizes your excess charge about as fast as you acquire it.
Two of the grand mysteries in physics are (1) why does the universe have par- ticles with electric charge (what is it, really?) and (2) why does electric charge come in two types (and not, say, one type or three types). We just do not know. Nevertheless, with lots of experiments similar to our two demonstrations scien- tists discovered that
Particles with the same sign of electrical charge repel each other, and particles with opposite signs attract each other.
In a moment we shall put this rule into quantitative form as Coulomb’s law of electrostatic force (or electric force) between charged particles. The term electro- static is used to emphasize that, relative to each other, the charges are either sta- tionary or moving only very slowly.
Demos. Now let’s get back to the demonstrations to understand the motions of the rod as being something other than just magic. When we rub the glass rod with a silk cloth, a small amount of negative charge moves from the rod to the silk (a transfer like that between you and a carpet), leaving the rod with a small amount of excess positive charge. (Which way the negative charge moves is not obvious and requires a lot of experimentation.) We rub the silk over the rod to in- crease the number of contact points and thus the amount, still tiny, of transferred charge. We hang the rod from the thread so as to electrically isolate it from its sur- roundings (so that the surroundings cannot neutralize the rod by giving it enough negative charge to rebalance its charge). When we rub the second rod with the silk cloth, it too becomes positively charged. So when we bring it near the first rod, the two rods repel each other (Fig. 21-2a).
Next, when we rub the plastic rod with fur, it gains excess negative charge from the fur. (Again, the transfer direction is learned through many experiments.) When we bring the plastic rod (with negative charge) near the hanging glass rod (with positive charge), the rods are attracted to each other (Fig. 21-2b). All this is subtle.You cannot see the charge or its transfer, only the results.
Conductors and Insulators
We can classify materials generally according to the ability of charge to move through them. Conductors are materials through which charge can move rather freely; examples include metals (such as copper in common lamp wire), the human body, and tap water. Nonconductors — also called insulators — are materials through which charge cannot move freely; examples include rubber (such as the insulation on common lamp wire), plastic, glass, and chemically pure water. Semiconductors are materials that are intermediate between conductors and insulators; examples include silicon and germanium in computer chips. Super- conductors are materials that are perfect conductors, allowing charge to move with- out any hindrance. In these chapters we discuss only conductors and insulators.
Conducting Path. Here is an example of how conduction can eliminate excess charge on an object. If you rub a copper rod with wool, charge is transferred from the wool to the rod. However, if you are holding the rod while also touching a faucet, you cannot charge the rod in spite of the transfer. The reason is that you, the rod, and the faucet are all conductors connected, via the plumbing, to Earth’s surface, which is a huge conductor. Because the excess charges put on the rod by the wool repel one an- other, they move away from one another by moving first through the rod, then through you, and then through the faucet and plumbing to reach Earth’s surface, where they can spread out.The process leaves the rod electrically neutral.
In thus setting up a pathway of conductors between an object and Earth’s surface, we are said to ground the object, and in neutralizing the object (by elimi- nating an unbalanced positive or negative charge), we are said to discharge the object. If instead of holding the copper rod in your hand, you hold it by an insulating handle, you eliminate the conducting path to Earth, and the rod can then be charged by rubbing (the charge remains on the rod), as long as you do not touch it directly with your hand.
Charged Particles. The properties of conductors and insulators are due to the structure and electrical nature of atoms. Atoms consist of positively charged protons, negatively charged electrons, and electrically neutral neutrons. The pro- tons and neutrons are packed tightly together in a central nucleus.
The charge of a single electron and that of a single proton have the same magnitude but are opposite in sign. Hence, an electrically neutral atom contains equal numbers of electrons and protons. Electrons are held near the nucleus because they have the electrical sign opposite that of the protons in the nucleus and thus are attracted to the nucleus. Were this not true, there would be no atoms and thus no you.
When atoms of a conductor like copper come together to form the solid, some of their outermost (and so most loosely held) electrons become free to wander about within the solid, leaving behind positively charged atoms ( positive ions). We call the mobile electrons conduction electrons. There are few (if any) free electrons in a nonconductor.
Induced Charge. The experiment of Fig. 21-3 demonstrates the mobility of charge in a conductor. A negatively charged plastic rod will attract either end of an isolated neutral copper rod. What happens is that many of the conduction electrons in the closer end of the copper rod are repelled by the negative charge on the plastic rod. Some of the conduction electrons move to the far end of the copper rod, leaving the near end depleted in electrons and thus with an unbal- anced positive charge. This positive charge is attracted to the negative charge in the plastic rod. Although the copper rod is still neutral, it is said to have an induced charge, which means that some of its positive and negative charges have been separated due to the presence of a nearby charge.
Similarly, if a positively charged glass rod is brought near one end of a neutral copper rod, induced charge is again set up in the neutral copper rod but now the near end gains conduction electrons, becomes negatively charged, and is attracted to the glass rod, while the far end is positively charged.
Note that only conduction electrons, with their negative charges, can move; positive ions are fixed in place. Thus, an object becomes positively charged only through the removal of negative charges.
Blue Flashes from a Wintergreen LifeSaver
Indirect evidence for the attraction of charges with opposite signs can be seen with a wintergreen LifeSaver (the candy shaped in the form of a marine lifesaver). If you adapt your eyes to darkness for about 15 minutes and then have a friend chomp on a piece of the candy in the darkness, you will see a faint blue flash from your friend’s mouth with each chomp. Whenever a chomp breaks a sugar crystal into pieces, each piece will probably end up with a different number of electrons. Suppose a crystal breaks into pieces A and B, with A ending up with more electrons on its surface than B (Fig. 21-4). This means that B has positive ions (atoms that lost electrons to A) on its surface. Because the electrons on A are strongly attracted to the positive ions on B, some of those electrons jump across the gap between the pieces.
As A and B move away from each other, air (primarily nitrogen, N 2 ) flows into the gap, and many of the jumping electrons collide with nitrogen molecules in the air, causing the molecules to emit ultraviolet light. You cannot see this type of light. However, the wintergreen molecules on the surfaces of the candy pieces absorb the ultraviolet light and then emit blue light, which you can see — it is the blue light coming from your friend’s mouth.
Checkpoint 1
The figure shows five pairs of plates: A, B, and D are charged plastic plates and C is an elec- trically neutral copper plate.The electrostatic forces between the pairs of plates are shown for three of the pairs. For the remaining two pairs, do the plates repel or attract each other?
Coulomb’s Law
Now we come to the equation for Coulomb’s law, but first a caution. This equa- tion works for only charged particles (and a few other things that can be treated as particles). For extended objects, with charge located in many different places, we need more powerful techniques. So, here we consider just charged particles and not, say, two charged cats.
If two charged particles are brought near each other, they each exert an elec- trostatic force on the other. The direction of the force vectors depends on the signs of the charges. If the particles have the same sign of charge, they repel each other. That means that the force vector on each is directly away from the other particle (Figs. 21-5a and b). If we release the particles, they accelerate away from each other. If, instead, the particles have opposite signs of charge, they attract each other. That means that the force vector on each is directly toward the other particle (Fig. 21-5c). If we release the particles, they accelerate toward each other.
The equation for the electrostatic forces acting on the particles is called Coulomb’s law after Charles-Augustin de Coulomb, whose experiments in 1785 led him to it. Let’s write the equation in vector form and in terms of the particles shown in Fig. 21-6, where particle 1 has charge q1 and particle 2 has charge q2. (These sym- bols can represent either positive or negative charge.) Let’s also focus on particle 1 and write the force acting on it in terms of a unit vector that points along a radial axis extending through the two particles, radially away from particle 2. (As with other unit vectors, has a magnitude of exactly 1 and no unit; its purpose is to point, like a direction arrow on a street sign.) With these decisions, we write the electrostatic force as
(Coulomb’s law), (21-1)
where r is the separation between the particles and k is a positive constant called the electrostatic constant or the Coulomb constant. (We’ll discuss k below.)
Let’s first check the direction of the force on particle 1 as given by Eq. 21-1. If q1 and q2 have the same sign, then the product q1q2 gives us a positive result. So, Eq. 21-1 tells us that the force on particle 1 is in the direction of . That checks, be- cause particle 1 is being repelled from particle 2. Next, if q1 and q2 have opposite signs, the product q1q2 gives us a negative result. So, now Eq. 21-1 tells us that the force on particle 1 is in the direction opposite . That checks because particle 1 is being attracted toward particle 2.
An Aside. Here is something that is very curious. The form of Eq. 21-1 is the same as that of Newton’s equation (Eq. 13-3) for the gravitational force between two particles with masses m1 and m2 and separation r:
(Newton’s law), (21-2)
where G is the gravitational constant. Although the two types of forces are wildly different, both equations describe inverse square laws (the \(1/r^2\) dependences) that involve a product of a property of the interacting particles—the charge in one case and the mass in the other. However, the laws differ in that gravitational forces are always attractive but electrostatic forces may be either attractive or re- pulsive, depending on the signs of the charges. This difference arises from the fact that there is only one type of mass but two types of charge.
Unit. The SI unit of charge is the coulomb. For practical reasons having to do with the accuracy of measurements, the coulomb unit is derived from the SI unit am- pere for electric current i. We shall discuss current in detail in Chapter 26, but here let’s just note that current i is the rate dq/dt at which charge moves past a point or through a region:
(electric current). (21-3)
Rearranging Eq. 21-3 and replacing the symbols with their units (coulombs C, amperes A, and seconds s) we see that
1 C = (1 A)(1 s).
Force Magnitude. For historical reasons (and because doing so simplifies many other formulas), the electrostatic constant k in Eq. 21-1 is often written as 1/4p´0. Then the magnitude of the electrostatic force in Coulomb’s law becomes
(Coulomb’s law). (21-4)
The constants in Eqs. 21-1 and 21-4 have the value
(21-5)
The quantity \(\varepsilon_0\), called the permittivity constant, sometimes appears separately in equations and is
. (21-6)
Working a Problem. Note that the charge magnitudes appear in Eq. 21-4, which gives us the force magnitude. So, in working problems in this chapter, we use Eq. 21-4 to find the magnitude of a force on a chosen particle due to a second particle and we separately determine the direction of the force by considering the charge signs of the two particles.
Multiple Forces. As with all forces in this book, the electrostatic force obeys the principle of superposition. Suppose we have n charged particles near a chosen particle called particle 1; then the net force on particle 1 is given by the vector sum
(21-7)
in which, for example, \(\overrightarrow{F}_{14}\) is the force on particle 1 due to the presence of particle 4.
This equation is the key to many of the homework problems, so let’s state it in words. If you want to know the net force acting on a chosen charged particle that is surrounded by other charged particles, first clearly identify that chosen particle and then find the force on it due to each of the other particles. Draw those force vectors in a free-body diagram of the chosen particle, with the tails anchored on the particle. (That may sound trivial, but failing to do so easily leads to errors.) Then add all those forces as vectors according to the rules of Chapter 3, not as scalars. (You cannot just willy-nilly add up their magnitudes.) The result is the net force (or resultant force) acting on the particle.
Although the vector nature of the forces makes the homework problems harder than if we simply had scalars, be thankful that Eq. 21-7 works. If two force vectors did not simply add but for some reason amplified each other, the world would be very difficult to understand and manage.
Shell Theories. Analogous to the shell theories for the gravitational force (Module 13-1), we have two shell theories for the electrostatic force:
Shell theory 1. A charged particle outside a shell with charge uniformly distrib- uted on its surface is attracted or repelled as if the shell’s charge were concentrated as a particle at its center.
Shell theory 2. A charged particle inside a shell with charge uniformly distributed on its surface has no net force acting on it due to the shell.
(In the first theory, we assume that the charge on the shell is much greater than the particle’s charge. Thus the presence of the particle has negligible effect on the distribution of charge on the shell.)
Spherical Conductors
If excess charge is placed on a spherical shell that is made of conducting material, the excess charge spreads uniformly over the (external) surface. For example, if we place excess electrons on a spherical metal shell, those electrons repel one another and tend to move apart, spreading over the available surface until they are uniformly dis- tributed. That arrangement maximizes the distances between all pairs of the excess electrons. According to the first shell theorem, the shell then will attract or repel an external charge as if all the excess charge on the shell were concentrated at its center.
If we remove negative charge from a spherical metal shell, the resulting pos- itive charge of the shell is also spread uniformly over the surface of the shell. For example, if we remove n electrons, there are then n sites of positive charge (sites missing an electron) that are spread uniformly over the shell. According to the first shell theorem, the shell will again attract or repel an external charge as if all the shell’s excess charge were concentrated at its center.
Checkpoint 2
The figure shows two protons (symbol p) and one electron (symbol e) on an axis. On the central proton, what is the direction of (a) the force due to the electron, (b) the force due to the other proton, and (c) the net force?
Sample Problem 21.01 Finding the net force due to two other particles
This sample problem actually contains three examples, to build from basic stuff to harder stuff. In each we have the same charged particle 1. First there is a single force acting on it (easy stuff). Then there are two forces, but they are just in opposite directions (not too bad). Then there are again two forces but they are in very different directions (ah, now we have to get serious about the fact that they are vectors). The key to all three examples is to draw the forces correctly before you reach for a calculator, otherwise you may be cal- culating nonsense on the calculator. (Figure 21-7 is available in WileyPLUS as an animation with voiceover.)
(a) Figure 21-7a shows two positively charged particles fixed in place on an x axis. The charges are \(q_1 =1.60 \times 10^{-19} C\) and \(q_2 = 3.20 \times 10^{-19} C\), and the particle separation is R= 0.0200 m. What are the magnitude and direction of the electrostatic force overrightarrow{F}_{12}` on particle 1 from particle 2?
KEY IDEAS
Because both particles are positively charged, particle 1 is repelled by particle 2, with a force magnitude given by Eq. 21-4. Thus, the direction of force \(\overrightarrow{F}_{12}\) on particle 1 is away from parti- cle 2, in the negative direction of the x axis, as indicated in the free-body diagram of Fig. 21-7b.
Two particles: Using Eq. 21-4 with separation R substituted for r, we can write the magnitude \(F_{12}\) of this force as
Thus, force \(\overrightarrow{F}_{12}\) has the following magnitude and direction (relative to the positive direction of the x axis):
We can also write \(\overrightarrow{F}_{12}\) in unit-vector notation as
. (Answer)
(b) Figure 21-7c is identical to Fig. 21-7a except that particle 3 now lies on the x axis between particles 1 and 2. Particle 3 has charge \(q_3 = -3.20 \times 10^{-19} C\) and is at a distance from \(\frac{3}{4}R\) from article 1.What is the net electrostatic force \(\overrightarrow{F}_{1,net}\) on particle 1 due to particles 2 and 3?
KEY IDEA
The presence of particle 3 does not alter the electrostatic force on particle 1 from particle 2.Thus, force still acts on particle 1. Similarly, the force that acts on particle 1 due to particle 3 is not affected by the presence of particle 2. Because particles 1
Figure 21-7 (a) Two charged particles of charges q1 and q2 are fixed in place on an x axis. (b) The free-body diagram for particle 1, showing the electrostatic force on it from particle 2. (c) Particle 3 included. (d) Free-body diagram for particle 1. (e) Particle 4 included. (f ) Free-body diagram for particle 1.
and 3 have charge of opposite signs, particle 1 is attracted to particle 3.Thus, force is directed toward particle 3, as indicated in the free-body diagram of Fig. 21-7d.
9999
Three particles: To find the magnitude of :math:` , we can rewrite Eq. 21-4 as We can also write in unit-vector notation: The net force on particle 1 is the vector sum of F : 12F : 1,net F : 13 ␄ (2.05 ␈ 10 ␅24 N)iˆ . F : 13 ␄ 2.05 ␈ 10 ␅24 N. ␈ (1.60 ␈ 10 ␅19 C)(3.20 ␈ 10 ␅19 C) (3 4 )2 (0.0200 m) 2 ␄ (8.99 ␈ 10 9 N␚m2 /C2) F13 ␄ 1 4p´0 ␂q1␂␂q3␂ (3 4R)2 F : 13 F : 13 and ; that is, from Eq. 21-7, we can write the net force on particle 1 in unit-vector notation as
hus, has the following magnitude and direction (relative to the positive direction of the x axis): 9.00 ␈ 10␅25 N and 0␎. (Answer) (c) Figure 21-7e is identical to Fig. 21-7a except that particle 4 is now included. It has charge q4 ␄ ␅3.20 ␈ 10␅19 C, is at a distance from particle 1, and lies on a line that makes an3 4R F : 1,net ␄ (9.00 ␈ 10 ␅25 N)iˆ ␄ ␅(1.15 ␈ 10 ␅24 N)iˆ ␃ (2.05 ␈ 10 ␅24 N)iˆ F : 1,net ␄ F : 12 ␃ F : 13 F : 1,net F : 13 angle u ␄ 60␎ with the x axis. What is the net electrostatic force on particle 1 due to particles 2 and 4? KEY IDEA The net force is the vector sum of and a new forceF : 12F : 1,net F : 1,net acting on particle 1 due to particle 4. Because particles 1 and 4 have charge of opposite signs, particle 1 is attracted to particle 4. Thus, force on particle 1 is directed towardF : 14 F : 14 particle 4, at angle 60␎, as indicated in the free-body dia- gram of Fig. 21-7f. Four particles: We can rewrite Eq. 21-4 as
Then from Eq. 21-7, we can write the net force on parti- cle 1 as Because the forces and are not directed along the same axis, we cannot sum simply by combining their mag- nitudes. Instead, we must add them as vectors, using one of the following methods
Method 1. Summing directly on a vector-capable calculator. For , we enter the magnitude and the angle
80␎. For , we enter the magnitude and the2.05 ␈ 10␅24 F : 14 angle 60␎.Then we add the vectors. Method 2. Summing in unit-vector notation. First we rewrite as Substituting N for F14 and 60␎ for u, this becomes . Then we sum: (Answer) Method 3. Summing components axis by axis. The sum of the x components gives us The sum of the y components gives us The net force has the magnitude (Answer) To find the direction of , we take u ␄ tan␅1 F1,net,y F1,net,x ␄ ␅86.0␎. F : 1,net F1,net ␄ 2F 2 1,net,x ␃ F 2 1,net,y ␄ 1.78 ␈ 10 ␅24 N. F : 1,net ␄ 1.78 ␈ 10 ␅24 N. ␄ (2.05 ␈ 10 ␅24 N)(sin 60␎) F1,net,y ␄ F12,y ␃ F14,y ␄ 0 ␃ F14 sin 60␎ ␄ ␅1.25 ␈ 10 ␅25 N. ␄ ␅1.15 ␈ 10 ␅24 N ␃ (2.05 ␈ 10 ␅24 N)(cos 60␎) F1,net,x ␄ F12,x ␃ F14,x ␄ F12 ␃ F14 cos 60␎ ␆ (␅1.25 ␈ 10 ␅25 N)iˆ ␃ (1.78 ␈ 10 ␅24 N)jˆ. ␃ (1.025 ␈ 10 ␅24 N)iˆ ␃ (1.775 ␈ 10 ␅24 N)jˆ ␄ ␅(1.15 ␈ 10 ␅24 N)iˆ F : 1,net ␄ F : 12 ␃ F : 14 F : 14 ␄ (1.025 ␈ 10 ␅24 N)iˆ ␃ (1.775 ␈ 10 ␅24 N)jˆ 2.05 ␈ 10␅24 F : 14 ␄ (F14 cos u)iˆ ␃ (F14 sin u)jˆ .
Method 3. Summing components axis by axis. The sum of the x components gives us The sum of the y components gives us The net force has the magnitude (Answer) To find the direction of , we take u ␄ tan␅1 F1,net,y F1,net,x ␄ ␅86.0␎. F : 1,net F1,net ␄ 2F 2 1,net,x ␃ F 2 1,net,y ␄ 1.78 ␈ 10 ␅24 N. F : 1,net ␄ 1.78 ␈ 10 ␅24 N. ␄ (2.05 ␈ 10 ␅24 N)(sin 60␎) F1,net,y ␄ F12,y ␃ F14,y ␄ 0 ␃ F14 sin 60␎ ␄ ␅1.25 ␈ 10 ␅25 N. ␄ ␅1.15 ␈ 10 ␅24 N ␃ (2.05 ␈ 10 ␅24 N)(cos 60␎) F1,net,x ␄ F12,x ␃ F14,x ␄ F12 ␃ F14 cos 60␎ ␆ (␅1.25 ␈ 10 ␅25 N)iˆ ␃ (1.78 ␈ 10 ␅24 N)jˆ. ␃ (1.025 ␈ 10 ␅24 N)iˆ ␃ (1.775 ␈ 10 ␅24 N)jˆ ␄ ␅(1.15 ␈ 10 ␅24 N)iˆ F : 1,net ␄ F : 12 ␃ F : 14 F : 14 ␄ (1.025 ␈ 10 ␅24 N)iˆ ␃ (1.775 ␈ 10 ␅24 N)jˆ 2.05 ␈ 10␅24 F : 14 ␄ (F14 cos u)iˆ ␃ (F14 sin u)jˆ . F : 14 However, this is an unreasonable result because mustF : 1,net Additional examples, video, and practice available at WileyPLUS have a direction between the directions of and . To correct u, we add 180␎, obtaining ␅86.0␎ ␃ 180␎ ␄ 94.0␎. (Answer)
Checkpoint 3 The figure here shows three arrangements of an electron e and two protons p.(a) Rank the arrangements according to the magnitude of the net electrostatic force on the electron due to the protons,largest first.(b) In situation c,is the angle between the net force on the electron and the line labeled d less than or more than 45␎?
Sample Problem 21.02 Equilibrium of two forces on a particle
Figure 21-8a shows two particles fixed in place: a particle of charge q1 ␄ ␃8q at the origin and a particle of charge q2 ␄ ␅2q at x ␄ L. At what point (other than infinitely far away) can a proton be placed so that it is in equilibrium (the net force on it is zero)? Is that equilibrium stable or unstable? (That is, if the pro- ton is displaced, do the forces drive it back to the point of equi- librium or drive it farther away?) KEY IDEA If is the force on the proton due to charge q1 and is the force on the proton due to charge q2, then the point we seek is where Thus, (21-8) This tells us that at the point we seek, the forces acting on the proton due to the other two particles must be of equal magnitudes, F1 ␄ F2, (21-9) and that the forces must have opposite directions. Reasoning: Because a proton has a positive charge, the pro- ton and the particle of charge q1 are of the same sign, and force on the proton must point away from q1. Also, the proton and the particle of charge q2 are of opposite signs, so force on the proton must point toward q2. “Away from q1” and “toward q2” can be in opposite directions only if the pro- ton is located on the x axis. If the proton is on the x axis at any point between q1 and q2, such as point P in Fig. 21-8b, then and are in the same direction and not in opposite directions as required. If the proton is at any point on the x axis to the left of q1, such as point S in Fig. 21-8c, then and are in opposite directions. However, Eq. 21-4 tells us that and can- not have equal magnitudes there: F1 must be greater than F2, because F1 is produced by a closer charge (with lesser r) of greater magnitude (8q versus 2q). Finally, if the proton is at any point on the x axis to the right of q2, such as point R in Fig. 21-8d, then and are again in opposite directions. However, because now the charge of greater magnitude (q1) is farther away from the pro- ton than the charge of lesser magnitude, there is a point at which F1 is equal to F2. Let x be the coordinate of this point, and let qp be the charge of the proton
igure 21-8 (a) Two particles of charges q1 and q2 are fixed in place on an x axis, with separation L. (b) – (d) Three possible locations P, S, and R for a proton. At each location, is the force on the protonF : 1 from particle 1 and is the force on the proton from particle 2
Calculations: With Eq.21-4,we can now rewrite Eq.21-9: (21-10) (Note that only the charge magnitudes appear in Eq. 21-10. We already decided about the directions of the forces in drawing Fig. 21-8d and do not want to include any posi- tive or negative signs here.) Rearranging Eq. 21-10 gives us After taking the square roots of both sides, we find x ␄ 2L. (Answer) The equilibrium at x ␄ 2L is unstable; that is, if the proton is displaced leftward from point R, then F1 and F2 both increase but F2 increases more (because q2 is closer than q1), and a net force will drive the proton farther leftward. If the proton is dis- placed rightward, both F1 and F2 decrease but F2 decreases more, and a net force will then drive the proton farther right- ward. In a stable equilibrium, if the proton is displaced slightly, it returns to the equilibrium position
Sample Problem 21.03 Charge sharing by two identical conducting sphere
In Fig. 21-9a, two identical, electrically isolated conducting spheres A and B are separated by a (center-to-center) dis- tance a that is large compared to the spheres. Sphere A has a positive charge of ␃Q, and sphere B is electrically neutral. Initially, there is no electrostatic force between the spheres. (The large separation means there is no induced charge.) (a) Suppose the spheres are connected for a moment by a conducting wire. The wire is thin enough so that any net charge on it is negligible. What is the electrostatic force between the spheres after the wire is removed? KEY IDEAS (1) Because the spheres are identical, connecting them means that they end up with identical charges (same sign and same amount). (2) The initial sum of the charges (including the signs of the charges) must equal the final sum of the charges. Reasoning: When the spheres are wired together, the (nega- tive) conduction electrons on B, which repel one another, have a way to move away from one another (along the wire to positively charged A, which attracts them—Fig. 21-9b). As B loses negative charge, it becomes positively charged, and as A gains negative charge, it becomes less positively charged. The transfer of charge stops when the charge on B has in- creased to ␃Q/2 and the charge on A has decreased to ␃Q/2, which occurs when ␅Q/2 has shifted from B to A. After the wire has been removed (Fig. 21-9c), we can assume that the charge on either sphere does not disturb the uniformity of the charge distribution on the other sphere, because the spheres are small relative to their separation.Thus, we can apply the first shell theorem to each sphere. By Eq. 21-4 with q1 ␄ q2 ␄ Q/2 and r ␄ a
Figure 21-9 Two small conducting spheres A and B. (a) To start, sphere A is charged positively. (b) Negative charge is transferred from B to A through a connecting wire. (c) Both spheres are then charged posi- tively. (d) Negative charge is transferred through a grounding wire to sphere A. (e) Sphere A is then neutral
The spheres, now positively charged, repel each other. (b) Next, suppose sphere A is grounded momentarily, and then the ground connection is removed. What now is the electrostatic force between the spheres? Reasoning: When we provide a conducting path between a charged object and the ground (which is a huge conductor), we neutralize the object. Were sphere A negatively charged, the mutual repulsion between the excess electrons would cause them to move from the sphere to the ground. However, because sphere A is positively charged, electrons with a total charge of ␅Q/2 move from the ground up onto the sphere (Fig. 21-9d), leaving the sphere with a charge of 0 (Fig. 21-9e). Thus, the electrostatic force is again zero.
21-2 CHARGE IS QUANTIZED After reading this module, you should be able to … 21.19 Identify the elementary charge. 21.20 Identify that the charge of a particle or object must be a positive or negative integer times the elementary charge.
● Electric charge is quantized (restricted to certain values). ● The charge of a particle can be written as ne, where n is a positive or negative integer and e is the elementary charge, which is the magnitude of the charge of the electron and proton (␆ 1.602 ␈ 10␅19 C). Learning Objectives
harge Is Quantized In Benjamin Franklin’s day, electric charge was thought to be a continuous fluid — an idea that was useful for many purposes. However, we now know tha
luids themselves, such as air and water, are not continuous but are made up of atoms and molecules; matter is discrete. Experiment shows that “electrical fluid” is also not continuous but is made up of multiples of a certain elementary charge. Any positive or negative charge q that can be detected can be written as q ␄ ne, n ␄ ␂1, ␂2, ␂3, … , (21-11) in which e, the elementary charge, has the approximate value e ␄ 1.602 ␈ 10␅19 C. (21-12) The elementary charge e is one of the important constants of nature. The electron and proton both have a charge of magnitude e (Table 21-1). (Quarks, the con- stituent particles of protons and neutrons, have charges of ␂e/3 or ␂2e/3, but they apparently cannot be detected individually. For this and for historical reasons, we do not take their charges to be the elementary charge.) You often see phrases — such as “the charge on a sphere,” “the amount of charge transferred,” and “the charge carried by the electron” — that suggest that charge is a substance. (Indeed, such statements have already appeared in this chapter.) You should, however, keep in mind what is intended: Particles are the substance and charge happens to be one of their properties, just as mass is. When a physical quantity such as charge can have only discrete values rather than any value, we say that the quantity is quantized. It is possible, for example, to find a particle that has no charge at all or a charge of ␃10e or ␅6e, but not a parti- cle with a charge of, say, 3.57e. The quantum of charge is small. In an ordinary 100 W lightbulb, for example, about 10 19 elementary charges enter the bulb every second and just as many leave. However, the graininess of electricity does not show up in such large-scale phenomena (the bulb does not flicker with each electron).
Checkpoint 4 Initially, sphere A has a charge of ␅50e and sphere B has a charge of ␃20e. The spheres are made of conducting material and are identical in size. If the spheres then touch, what is the resulting charge on sphere A?
Sample Problem 21.04 Mutual electric repulsion in a nucleus
he nucleus in an iron atom has a radius of about 4.0 ␈ 10␅15 m and contains 26 protons. (a) What is the magnitude of the repulsive electrostatic force be- tween two of the protons that are separated by 4.0 ␈ 10␅15 m? KEY IDEA The protons can be treated as charged particles, so the mag- nitude of the electrostatic force on one from the other is given by Coulomb’s law. Calculation: Table 21-1 tells us that the charge of a proton is ␃e. Thus, Eq. 21-4 gives us . (Answer)␄ 14 N ␄ (8.99 ␈ 10 9 N␚m2/C2)(1.602 ␈ 10 ␅19 C)2 (4.0 ␈ 10
No explosion: This is a small force to be acting on a macro- scopic object like a cantaloupe, but an enormous force to be acting on a proton. Such forces should explode the nucleus of any element but hydrogen (which has only one proton in its nucleus). However, they don’t, not even in nuclei with a great many protons. Therefore, there must be some enor- mous attractive force to counter this enormous repulsive electrostatic force. (b) What is the magnitude of the gravitational force between those same two protons? KEY IDEA Because the protons are particles, the magnitude of the gravitational force on one from the other is given by Newton’s equation for the gravitational force (Eq. 21-2). Calculation: With mp (␄ 1.67 ␈ 10␅27 kg) representing the
mass of a proton, Eq. 21-2 gives us . (Answer) Weak versus strong: This result tells us that the (attractive) gravitational force is far too weak to counter the repulsive electrostatic forces between protons in a nucleus. Instead, the protons are bound together by an enormous force called
aptly) the strong nuclear force — a force that acts between protons (and neutrons) when they are close together, as in a nucleus. Although the gravitational force is many times weaker than the electrostatic force, it is more important in large- scale situations because it is always attractive.This means that it can collect many small bodies into huge bodies with huge masses, such as planets and stars, that then exert large gravita- tional forces. The electrostatic force, on the other hand, is re- pulsive for charges of the same sign, so it is unable to collect either positive charge or negative charge into large concen- trations that would then exert large electrostatic forces.
21-3 CHARGE IS CONSERVED
62121-3 CHARG E IS CONSE RVE D Additional examples, video, and practice available at WileyPLUS 21-3 CHARGE IS CONSERVED After reading this module, you should be able to … 21.21 Identify that in any isolated physical process, the net charge cannot change (the net charge is always conserved). 21.22 Identify an annihilation process of particles and a pair production of particles. 21.23 Identify mass number and atomic number in terms of the number of protons, neutrons, and electrons.
The net electric charge of any isolated system is always
conserved. ● If two charged particles undergo an annihilation process, they have opposite signs of charge. ● If two charged particles appear as a result of a pair produc- tion process, they have opposite signs of charge.
Charge Is Conserved If you rub a glass rod with silk, a positive charge appears on the rod. Measure- ment shows that a negative charge of equal magnitude appears on the silk. This suggests that rubbing does not create charge but only transfers it from one body to another, upsetting the electrical neutrality of each body during the process. This hypothesis of conservation of charge, first put forward by Benjamin Franklin, has stood up under close examination, both for large-scale charged bodies and for atoms, nuclei, and elementary particles. No exceptions have ever been found. Thus, we add electric charge to our list of quantities — including energy and both linear momentum and angular momentum — that obey a con- servation law. Important examples of the conservation of charge occur in the radioactive decay of nuclei, in which a nucleus transforms into (becomes) a different type of nucleus. For example, a uranium-238 nucleus ( 238 U) transforms into a thorium- 234 nucleus ( 234 Th) by emitting an alpha particle. Because that particle has the same makeup as a helium-4 nucleus, it has the symbol 4He. The number used in the name of a nucleus and as a superscript in the symbol for the nucleus is called the mass number and is the total number of the protons and neutrons in the nucleus. For example, the total number in 238 U is 238. The number of protons in a nucleus is the atomic number Z, which is listed for all the elements in Appendix F. From that list we find that in the decay 238 U : 234 Th ␃ 4 He, (21-13)
622 CHAPTE R 21 COU LOM B’S L AW the parent nucleus 238 U contains 92 protons (a charge of ␃92e), the daughter nucleus 234 Th contains 90 protons (a charge of ␃90e), and the emitted alpha parti- cle 4He contains 2 protons (a charge of ␃2e). We see that the total charge is ␃92e before and after the decay; thus, charge is conserved. (The total number of pro- tons and neutrons is also conserved: 238 before the decay and 234 ␃ 4 ␄ 238 after the decay.) Another example of charge conservation occurs when an electron e␅ (charge ␅e) and its antiparticle, the positron e␃ (charge ␃e), undergo an annihilation process, transforming into two gamma rays (high-energy light): e␅ ␃ e␃ : g ␃ g (annihilation). (21-14) In applying the conservation-of-charge principle, we must add the charges alge- braically, with due regard for their signs. In the annihilation process of Eq. 21-14 then, the net charge of the system is zero both before and after the event. Charge is conserved. In pair production, the converse of annihilation, charge is also conserved. In this process a gamma ray transforms into an electron and a positron: g : e␅ ␃ e␃ (pair production) . (21-15) Figure 21-10 shows such a pair-production event that occurred in a bubble cham- ber. (This is a device in which a liquid is suddenly made hotter than its boiling point. If a charged particle passes through it, tiny vapor bubbles form along the particle’s trail.) A gamma ray entered the chamber from the bottom and at one point transformed into an electron and a positron. Because those new particles were charged and moving, each left a trail of bubbles. (The trails were curved because a magnetic field had been set up in the chamber.) The gamma ray, being electrically neutral, left no trail. Still, you can tell exactly where it underwent pair production — at the tip of the curved V, which is where the trails of the electron and positron begin
Electric Charge The strength of a particle’s electrical interaction with objects around it depends on its electric charge (usually repre- sented as q), which can be either positive or negative. Particles with the same sign of charge repel each other, and particles with opposite signs of charge attract each other. An object with equal amounts of the two kinds of charge is electrically neutral, whereas one with an imbalance is electrically charged and has an excess charge. Conductors are materials in which a significant number of electrons are free to move. The charged particles in nonconductors (insulators) are not free to move. Electric current i is the rate dq/dt at which charge passes a point: (electric current). (21-3) Coulomb’s Law Coulomb’s law describes the electrostatic force (or electric force) between two charged particles. If the parti- cles have charges q1 and q2 , are separated by distance r, and are at rest (or moving only slowly) relative to each other, then the magni- tude of the force acting on each due to the other is given by (Coulomb’s law), (21-4) where is the permittivity constant. The ratio 1/4p´0 is often replaced with the electrostatic constant (or Coulomb constant) .k ␄ 8.99 ␈ 109 N␚m2 /C2 ´0 ␄ 8.85 ␈ 10␅12 C2 /N␚m2 F ␄ 1 4p´0 ␂q1␂ ␂q2␂ r2 i ␄ dq dt
The electrostatic force vector acting on a charged particle due to a second charged particle is either directly toward the second particle (opposite signs of charge) or directly away from it (same sign of charge).As with other types of forces, if multiple electrostatic forces act on a particle, the net force is the vector sum (not scalar sum) of the individual forces. The two shell theories for electrostatics are Shell theorem 1: A charged particle outside a shell with charge uniformly distributed on its surface is attracted or repelled as if the shell’s charge were concentrated as a particle at its center. Shell theorem 2: A charged particle inside a shell with charge uniformly distributed on its surface has no net force acting on it due to the shell. Charge on a conducting spherical shell spreads uniformly over the (external) surface. The Elementary Charge Electric charge is quantized (re- stricted to certain values). The charge of a particle can be written as ne, where n is a positive or negative integer and e is the elemen- tary charge, which is the magnitude of the charge of the electron and proton (␆ 1.602 ␈ 10␅19 C). Conservation of Charge The net electric charge of any iso- lated system is always conserved.
623QU ESTIONS Questions 1 Figure 21-11 shows four situations in which five charged particles are evenly spaced along an axis. The charge values are indicated except for the central particle, which has the same charge in all four situations. Rank the situations according to the magnitude of the net electrostatic force on the central particle, greatest first. 2 Figure 21-12 shows three pairs of identical spheres that are to be touched together and then separated. The initial charges on them are indicated. Rank the pairs according to (a) the magnitude of the charge transferred during touching and (b) the charge left on the positively charged sphere, greatest first. 4 Figure 21-14 shows two charged particles on an axis. The charges are free to move. However, a third charged particle can be placed at a certain point such that all three particles are then in equilibrium. (a) Is that point to the left of the first two particles, to their right, or between them? (b) Should the third particle be positively or negatively charged? (c) Is the equilibrium stable or unstable? 5 In Fig. 21-15, a central particle of charge ␅q is surrounded by two cir- cular rings of charged particles. What are the magnitude and direction of the net electrostatic force on the cen- tral particle due to the other parti- cles? (Hint: Consider symmetry.) 6 A positively charged ball is brought close to an electrically neu- tral isolated conductor. The conductor is then grounded while the ball is kept close. Is the conductor charged positively, charged neg- atively, or neutral if (a) the ball is first taken away and then the –e –e +e –e (1) +e +e +e –e (2) –e –e +e +e (3) –e +e +e –e (4) Figure 21-11 Question 1. +6e –4e (1) 0 +2e (2) –12e +14e (3) Figure 21-12 Question 2. 3 Figure 21-13 shows four situations in which charged particles are fixed in place on an axis. In which situations is there a point to the left of the particles where an e
round connection is removed and (b) the ground connection is first removed and then the ball is taken away? 7 Figure 21-16 shows three situations involving a charged parti- cle and a uniformly charged spherical shell. The charges are given, and the radii of the shells are indicated. Rank the situations ac- cording to the magnitude of the force on the particle due to the presence of the shell, greatest first. R 2R R/2 +8Q –q+2q +6q –4Q +5Q (a) (b) (c) d Figure 21-16 Question 7. 8 Figure 21-17 shows four arrangements of charged particles. Rank the arrangements according to the magnitude of the net electrostatic force on the particle with charge ␃Q, greatest first. +Q p p d 2d (a) +Q e p
Figure 21-18 shows four situations in which particles of
charge ␃q or ␅q are fixed in place. In each situation, the parti
les on the x axis are equidistant from the y axis. First, consider the middle particle in situation 1; the middle particle experiences an electrostatic force from each of the other two particles. (a) Are the magnitudes F of those forces the same or different? (b) Is the magnitude of the net force on the middle particle equal to, greater than, or less than 2F ? (c) Do the x components of the two forces add or cancel? (d) Do their y components add or cancel? (e) Is the direction of the net force on the middle particle that of the canceling components or the adding components? (f) What is the direction of that net force? Now consider the remaining situations: What is the direction of the net force on the middle parti- cle in (g) situation 2, (h) situation 3, and (i) situation 4? (In each situation, consider the symmetry of the charge distribution and determine the canceling components and the adding components.) 10 In Fig. 21-19, a central particle of charge ␅2q is surrounded by a square array of charged particles, separated by either distance d or d/2 along the perimeter of the square. What are the magni- tude and direction of the net electrostatic force on the cen- tral particle due to the other particles? (Hint: Consideration of symmetry can greatly re- duce the amount of work re- quired here.) 11 Figure 21-20 shows three identical conducting bubbles A, B, and C floating in a con- ducting container that is grounded by a wire. The bubbles ini- tially have the same charge. Bubble A bumps into the con- tainer’s ceiling and then into bubble B. Then bubble B bumps into bubble C, which then drifts to the container’s floor. When bubble C reaches the floor, a charge of ␅3e is transferred up- ward through the wire, from the ground to the container, as in- dicated. (a) What was the initial charge of each bubble? When (b) bubble A and (c) bubble B reach the floor, what is the charge transfer through the wire? (d) During this whole process, what is the total charge transfer through the wire? 12 Figure 21-21 shows four situations in which a central proton is partially surrounded by protons or electrons fixed in place along a half-circle. The angles u are identical; the angles f are also. (a) In each situation, what is the direction of the net force on the central proton due to the other particles? (b) Rank the four situations ac- cording to the magnitude of that net force on the central proton, greatest first. +2q –5q +3q
Tutoring problem available (at instructor’s discretion) in WileyPLUS and WebAssign SSM Worked-out solution available in Student Solutions Manual • – ••• Number of dots indicates level of problem difficulty Additional information available in The Flying Circus of Physics and at flyingcircusofphysics.com WWW Worked-out solution is at ILW Interactive solution is at
odule 21-1 Coulomb’s Law •1 Of the charge Q initially on a tiny sphere, a por- tion q is to be transferred to a second, nearby sphere. Both spheres
odule 21-1 Coulomb’s Law •1 Of the charge Q initially on a tiny sphere, a por- tion q is to be transferred to a second, nearby sphere. Both spheres
2 Identical isolated conducting spheres 1 and 2 have equal charges and are separated by a distance that is large compared with their diameters (Fig. 21-22a). The electrostatic force acting on sphere 2 due to sphere 1 is . Suppose now that a third identical sphere 3, having an insulating handle and initially neutral, is touched first to sphere 1 (Fig. 21-22b), then to sphere 2 (Fig. 21-22c), and finally removed (Fig. 21-22d). The electrostatic force that now acts on sphere 2 has magnitude F .What is the ratio F /F?
SSM static force between A and B at the end of experiment 2 to that at the end of experiment 1? ••9 Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of 0.108 N when their center-to-center separation is 50.0 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 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 posi- tive charge on the other? ••10 In Fig. 21-25, four particles form a square. The charges are WWWSSM 26.0 mC and point charge q2 ␄ ␅47.0 mC for the electrostatic force between them to have a magnitude of 5.70 N? •4 In the return stroke of a typical lightning bolt, a current of 2.5 ␈ 10 4 A exists for 20 ms. How much charge is transferred in this event? •5 A particle of charge ␃3.00 ␈ 10␅6 C is 12.0 cm distant from a second particle of charge ␅1.50 ␈ 10␅6 C. Calculate the magni- tude of the electrostatic force between the particles. •6 Two equally charged particles are held 3.2 ␈ 10␅3 m apart and then released from rest. The initial acceleration of the first particle is observed to be 7.0 m/s2 and that of the second to be 9.0 m/s2 . If the mass of the first particle is 6.3 ␈ 10␅7 kg, what are (a) the mass of the second particle and (b) the magnitude of the charge of each particle? ••7 In Fig. 21-23, three charged particles lie on an x axis. Particles 1 and 2 are fixed in place. Particle 3 is free to move, but the net elec- trostatic force on it from particles 1 and 2 happens to be zero. If L23 ␄ L12 , what is the ratio q1/q2? ••8 In Fig. 21-24, three identical conducting spheres initially have the following charges: sphere A, 4Q; sphere B, ␅6Q; and sphere C, 0. Spheres A and B are fixed in place, with a center-to-center separation that is much larger than the spheres. Two experiments are conducted. In experiment 1, sphere C is touched to sphere A and then (separately) to sphere B, and then it is removed. In experi- ment 2, starting with the same initial states, the procedure is re- versed: Sphere C is touched to sphere B and then (separately) to sphere A, and then it is removed. What is the ratio of the electro
•11 In Fig. 21-25, the particlesILW have charges q1 ␄ ␅q2 ␄ 100 nC and q3 ␄ ␅q4 ␄ 200 nC, and distance a ␄ 5.0 cm. What are the (a) x and (b) y components of the net electrostatic force on particle 3? ••12 Two particles are fixed on an x axis. Particle 1 of charge 40 mC is located at x ␄ ␅2.0 cm; particle 2 of charge Q is located at x ␄ 3.0 cm. Particle 3 of charge magnitude 20 mC is released from rest on the y axis at y ␄ 2.0 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? ••13 In Fig. 21-26, particle 1 of x y 1 2 L Figure 21-26 Problems 13, 19, 30, 58, and 67. charge ␃1.0 mC and particle 2 of charge ␅3.0 mC are held at separation L ␄ 10.0 cm on an x axis. If particle 3 of un- known charge q3 is to be located such that the net electrostatic force on it from particles 1 and 2 is zero, what must be the (a) x and (b) y coor- dinates of particle 3? ••14 Three particles are fixed on an x axis. Particle 1 of charge q1 is at x ␄ ␅a, and particle 2 of charge q2 is at x ␄ ␃a. If their net electro- static force on particle 3 of charge ␃Q is to be zero, what must be the ratio q1/q2 when particle 3 is at (a) x ␄ ␃0.500a and (b) x ␄ ␃1.50a? ••15 The charges and coordinates of two charged particles held fixed in an xy plane are q1 ␄ ␃3.0 mC, x1 ␄ 3.5 cm, y1 ␄ 0.50 cm, and q2 ␄ ␅4.0 mC, x2 ␄ ␅2.0 cm, y2 ␄ 1.5 cm. Find the (a) magni- tude and (b) direction of the electrostatic force on particle 2 due to particle 1. At what (c) x and (d) y coordinates should a third parti- cle of charge q3 ␄ ␃4.0 mC be placed such that the net electrostatic force on particle 2 due to particles 1 and 3 is zero? ••16 In Fig. 21-27a, particle 1 (of charge q1) and particle 2 (of charge q2) are fixed in place on an x axis, 8.00 cm apart. Particle 3 (of
charge q3 ␄ ␃8.00 ␈ 10␅19 C) is to be placed on the line between par- ticles 1 and 2 so that they produce a net electrostatic force on it. Figure 21-27b gives the x component of that force versus the coordi- nate x at which particle 3 is placed.The scale of the x axis is set by xs ␄ 8.0 cm.What are (a) the sign of charge q1 and (b) the ratio q2 /q1? ••17 In Fig. 21-28a, particles 1 and 2 have charge 20.0 mC each and are held at separation distance d ␄ 1.50 m. (a) What is the magnitude of the electrostatic force on particle 1 due to particle 2? In Fig. 21-28b, particle 3 of charge 20.0 mC is positioned so as to complete an equilateral trian- gle. (b) What is the magnitude of the net electrostatic force on parti- cle 1 due to particles 2 and 3? ••18 In Fig. 21-29a, three positively charged particles are fixed on an x axis. Particles B and C are so close to each other that they can be con- sidered to be at the same distance from particle A. The net force on particle A due to particles B and C is 2.014 ␈ 10␅23 N in the negative direction of the x axis. In Fig. 21- 29b, particle B has been moved to the opposite side of A but is still at the same distance from it. The net force on A is now 2.877 ␈ 10␅24 N in the negative direction of the x axis. What is the ratio qC/qB
cle 2 of charge ␃4.00q are held at separation L ␄ 9.00 cm on an x axis. If particle 3 of charge q3 is to be located such that the three particles remain in place when released, what must be the (a) x and (b) y coordinates of particle 3, and (c) the ratio q3 /q? •••20 Figure 21-30a shows an arrangement of three charged particles separated by distance d. Particles A and C are fixed on the x axis, but particle B can be moved along a circle centered on particle A. During the movement, a radial line between A and B makes an angle u relative to the positive direction of the x axis (Fig. 21-30b). The curves in Fig. 21-30c give, for two situations, the magnitude Fnet of the net electrostatic force on particle A due to the other particles. That net force is given as a function of angle u and as a multiple of a basic amount F0. For example on curve 1, at u ␄ 180␎, we see that Fnet ␄ 2F0. (a) For the situation corresponding to curve 1, what is the ratio of the charge of particle C to that of particle B (in- cluding sign)? (b) For the situation corresponding to curve 2, what is that ratio? 626 CHAPTE R 21 COU LOM B’S L AW charge q3 ␄ ␃8.00 ␈ 10␅19 C) is to be placed on the line between par- ticles 1 and 2 so that they produce a net electrostatic force on it. Figure 21-27b gives the x component of that force versus the coordi- nate x at which particle 3 is placed.The scale of the x axis is set by xs ␄ 8.0 cm.What are (a) the sign of charge q1 and (b) the ratio q2 /q1? ••17 In Fig. 21-28a, particles 1 and 2 have charge 20.0 mC each and are held at separation distance d ␄ 1.50 m. (a) What is the magnitude of the electrostatic force on particle 1 due to particle 2? In Fig. 21-28b, particle 3 of charge 20.0 mC is positioned so as to complete an equilateral trian- gle. (b) What is the magnitude of the net electrostatic force on parti- cle 1 due to particles 2 and 3? ••18 In Fig. 21-29a, three positively charged particles are fixed on an x axis. Particles B and C are so close to each other that they can be con- sidered to be at the same distance from particle A. The net force on particle A due to particles B and C is 2.014 ␈ 10␅23 N in the negative direction of the x axis. In Fig. 21- 29b, particle B has been moved to the opposite side of A but is still at the same distance from it. The net force on A is now 2.877 ␈ 10␅24 N in the negative direction of the x axis. What is the ratio qC/qB
nonconducting spherical shell, with an inner radius of
4.0 cm and an outer radius of 6.0 cm, has charge spread nonuni- formly through its volume between its inner and outer surfaces. The volume charge density r is the charge per unit volume, with the unit coulomb per cubic meter. For this shell r ␄ b/r, where r is the dis- tance in meters from the center of the shell and b ␄ 3.0 mC/m2 . What is the net charge in the shell? •••22 Figure 21-31 shows an 2 of charge q1 ␄ q2 ␄ ␃3.20 ␈ 10␅19 C are on a y axis at distance d ␄ 17.0 cm from the origin. Particle 3 of charge q3 ␄ ␃6.40 ␈ 10␅19 C is moved gradu- ally along the x axis from x ␄ 0 to x ␄ ␃5.0 m. At what values of x will the magnitude of the electrostatic force on the third particle from the other two particles be (a) minimum and (b) maximum? What are the (c) minimum and (d) maximum magnitudes? Module 21-2 Charge Is Quantized •24 Two tiny, spherical water drops, with identical charges of ␅1.00 ␈ 10␅16 C, have a center-to-center separation of 1.00 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? •25 How many electrons would have to be removed from aILW Figure 21-28 Problem 17. d (a ) (b ) 1 2 d d 3 A AB C B C x x (a) (b) Figure 21-29 Problem 18. •27 The magnitude of the electrostatic force between two iden-SSM Fnet 2 1 0 90°0° θ θ 180° 1 2(a) (b) (c) x x A C B BA C d d d Figure 21-30 Problem 20. arrangement of four charged parti- cles, with angle u ␂ 30.0␃ and dis- tance d ␂ 2.00 cm. Particle 2 has charge q2 ␂ ␄8.00 ␅ 10 ␆19 C; par- ticles 3 and 4 have charges q3 ␂ q4 ␂ ␆1.60 ␅ 10 ␆19 C. (a) What is dis- tance D between the origin and particle 2 if the net electrostatic force on particle 1 due to the other particles is zero? (b) If parti- cles 3 and 4 were moved closer to the x axis but maintained their symmetry about that axis, would the required value of D be greater than, less than, or the same as in part (a)? •••23 In Fig. 21-32, particles 1 and x y 1 2d D 3 4 θ θ Figure 21-31 Problem 22. Figure 21-32 Problem 23. x 1 3 d d 2 y coin to leave it with a charge of ␄1.0 ␅ 10␆7 C? •26 What is the magnitude of the electrostatic force between a singly charged sodium ion (Na␄, of charge ␄e) and an adjacent singly charged chlorine ion (Cl␆, of charge ␆e) in a salt crystal if their separation is 2.82 ␅ 10␆10 m? tical ions that are separated by a distance of 5.0 ␅ 10␆10 m is 3.7 ␅ 10␆9 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)? •28 A current of 0.300 A through your chest can send your x y 1 3 4 2 Figure 21-33 Problem 29. 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 min, how many conduction electrons pass
hrough your chest? ••29 In Fig. 21-33, particles 2 and 4, of charge ␆e, are fixed in place on a y axis, at y2 ␂ ␆10.0 cm
nd y4 ␂ 5.00 cm. Particles 1 and 3, of charge ␆e, can be moved along the x axis. Particle 5, of charge ␄e, is fixed at the origin. Initially particle 1 is at x1 ␂ ␆10.0 cm and particle 3 is at x3 ␂ 10.0 cm. (a) To what x value must particle 1 be moved to rotate the direction of the net electric force on particle 5 by 30␃ counter- clockwise? (b) With particle 1 fixed at its new position, to what x value must you move particle 3 to rotate back to its original direction? ••30 In Fig. 21-26, particles 1 and 2 are fixed in place on an x axis, at a separation of L ␂ 8.00 cm. Their charges are q1 ␂ ␄e and q2 ␂ ␆27e. Particle 3 with charge q3 ␂ ␄4e is to be placed on the line between particles 1 and 2, so that they produce a net electrostatic force on it. (a) At what coordinate should particle 3 be placed to minimize the magnitude of that force? (b) What is that minimum magnitude? ••31 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 sur- face 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? ••32 Figure 21-34a shows charged particles 1 and 2 that are fixed in place on an x axis. Particle 1 has a charge with a magnitude of |q1| ␂ 8.00e. Particle 3 of charge q3 ␂ ␄8.00e is initially on the x axis near particle 2. Then particle 3 is gradually moved in the posi- tive direction of the x axis. As a result, the magnitude of the net electrostatic force on particle 2 due to particles 1 and 3 changes. Figure 21-34b gives the x component of that net force as a function of the position x of particle 3. The scale of the x axis is set by xs ␂ 0.80 m. The plot has an asymptote of F2,net ␂ 1.5 ␅ 10␆25 N as x : ␇. As a multiple of e and including the sign, what is the charge q2 of particle 2?
Calculate the number of coulombs of positive charge in 250 cm 3 of (neutral) water. (Hint: A hydrogen atom contains one pro- ton; an oxygen atom contains eight protons.)
••34 Figure 21-35 shows electrons 1 and 2 on an x axis and charged ions 3 and 4 of iden- tical charge ␆q and at identical angles u. Electron 2 is free to move; the other three particles are fixed in place at horizontal distances R from electron 2 and are intended to hold electron 2 in place. For physically possible va
es of q ␈ 5e, what are the (a) smallest, (b) second smallest, and (c) third smallest values of u for which electron 2 is held in place?