Charging by Conduction

Physics Coach Stephens

Charging by Conduction

• • • • • • • Charging by conduction involves the contact of a charged object to a neutral object. Suppose that a positively charged aluminum plate is touched to a neutral metal sphere. – The neutral metal sphere becomes charged as the result of being contacted by the charged aluminum plate. Or suppose that a negatively charged metal sphere is touched to the top plate of a neutral needle electroscope . – The neutral electroscope becomes charged as the result of being contacted by the metal sphere. And finally, suppose that an uncharged physics student stands on an insulating platform and touches a negatively charged Van de Graaff generator. – The neutral physics student becomes charged as the result of contact with the Van de Graaff generator. Each of these examples involves contact between a charged object and a neutral object. In contrast to induction, where the charged object is brought near but never contacted to the object being charged, conduction charging involves making the physical connection of the charged object to the neutral object. Because charging by conduction involves contact, it is often called charging by contact.

• • • •

Charging by Conduction Using a Negatively Charged Object

To explain the process of charging by contact, we will first consider the case of using a negatively charged metal sphere to charge a neutral needle electroscope. Understanding the process demands that you understand that like charges repel and have an intense desire to reduce their repulsions by spreading about as far as possible. A negatively charged metal sphere has an excess of electrons; those electrons find each other repulsive and distance themselves from each other as far as possible. – The perimeter the sphere is the extreme to which they can go. – If there was ever a conducting pathway to a more spacious piece of real estate, one could be sure that the electrons would be on that pathway to the greener grass beyond. In human terms, electrons living in the same home despise each other and are always seeking a home of their own or at least a home with more rooms.

• • • •

Continued…

Given this understanding of electron-electron repulsions, it is not difficult to predict what excess electrons on the metal sphere would be inclined to do if the sphere were touched to the neutral electroscope. – Once the contact of the sphere to the electroscope is made, a countless number of excess electrons from the sphere move onto the electroscope and spread about the sphere-electroscope system. In general, the object that offers the most space in which to "hang out" will be the object that houses the greatest number of excess electrons. When the process of charging by conduction is complete, the electroscope acquires an excess negative charge due to the movement of electrons onto it from the metal sphere. The metal sphere is still charged negatively, only it has less excess negative charge than it had prior to the conduction charging process.

• • • • •

Charging by Conduction Using a Positively Charged Object

The previous example of charging by conduction involved touching a negatively charged object to a neutral object. Upon contact, electrons moved from the negatively charged object onto the neutral object. When finished, both objects were negatively charged. But what happens if a positively charged object is touched to a neutral object? To investigate this question, we consider the case of a positively charged aluminum plate being used to charge a neutral metal sphere by the process of conduction.

• • •

Continued…

The diagram below depicts the use of a positively charged aluminum plate being touched to a neutral metal sphere. – A positively charged aluminum plate has an excess of protons. – When looked at from an electron perspective, a positively charged aluminum plate has a shortage of electrons. In human terms, we could say that each excess proton is rather discontented. – It is not satisfied until it has found a negatively charged electron with which to co habitate. However, since a proton is tightly bound in the nucleus of an atom, it is incapable of leaving an atom in search of that longed-for electron. – It can however attract a mobile electron towards itself.

• • • • •

Continued…

And if a conducting pathway is made between a collection of electrons and an excess proton, one can be certain that there is likely an electron that would be willing to take the pathway. So when the positively charged aluminum plate is touched to the neutral metal sphere, countless electrons on the metal sphere migrate towards the aluminum plate. There is a mass migration of electrons until the positive charge on the aluminum plate metal sphere system becomes redistributed. Having lost electrons to the positively charged aluminum plate, there is a shortage of electrons on the sphere and an overall positive charge. The aluminum plate is still charged positively; only it now has less excess positive charge than it had before the charging process began.

Why Would the Electrons Move?

The above explanation might raise a rather difficult question: Why would an electron on the previously neutral metal sphere desire to move off the metal sphere in the first place? The metal sphere is neutral; every electron on it must be satisfied since there is a corresponding proton present. What would possibly induce an electron to go through the effort of migrating to a different territory in order to have what it already has?

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What Can an Electron do for his Country?

The best means of answering this question requires an understanding of the concept of electric potential. But since that concept does not arise until the next unit , a different approach to an answer will be taken. It ends up that electrons and protons are not as independent and individualized as we might think. From a human perspective, electrons and protons can't be thought of as independent citizens in a free enterprise system of government. – Electrons and protons don't actually do what is best for themselves, but must be more social-minded. – They must act like citizens of a state where the rule of law is to behave in a manner such that the overall repulsive affects within the society are reduced and the overall attractive affects are maximized. Electrons and protons will be motivated not by what is good for them, but rather by what is good for the country. – And in this sense, a country's boundary extends to the perimeter of the conductor material that an excess electron is within. – And in this case, an electron in the metal sphere is part of a country that extends beyond the sphere itself and includes the entire aluminum plate. So by moving from the metal sphere to the aluminum plate, an electron is able to reduce the total amount of repulsive affects within that country. – It serves to spread the excess positive charge over a greater surface area, thus reducing the total amount of repulsive forces between excess protons.

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Law of Conservation of Charge

In each of the other methods of charging discussed charging by induction charging by friction - the law of conservation of charge was illustrated. and – The law of conservation of charge states that charge is always conserved. – When all objects involved are considered prior to and after a given process, we notice that the total amount of charge among the objects is the same before the process starts as it is after the process ends. The same conservation law is observed during the charging by conduction process. – If a negatively charged metal sphere is used to charge a neutral electroscope, the overall charge before the process begins is the same as the overall charge when the process ends. • So if before the charging process begins, the metal sphere has 1000 units of negative charge and the electroscope is neutral, the overall charge of the two objects in the system is -1000 units. • Perhaps during the charging process, 600 units of negative charge moved from the metal sphere to the electroscope. • When the process is complete, the electroscope would have 600 units of negative charge and the metal sphere would have 400 units of negative charge (the original 1000 units minus the 600 units it transferred to the electroscope). – The overall charge of the two objects in the system is still -1000 units. – The overall charge before the process began is the same as the overall charge when the process is completed. Charge is neither created nor destroyed; it is simply transferred from one object to another object in the form of electrons.

Conduction Charging Requires a Conductor

• • • • In all of these examples, the charging by conduction process involved the touching of two conductors. Does contact charging have to occur through the contact of two conductors? Can an insulator conduct a charge to another object upon touching? And can an insulator be charged by conduction? – A complete discussion of these questions can get messy and quite often leads to a splitting of hairs over the definition of conduction and the distinction between conductors and insulators. The belief is taken here that only a conductor can conduct charge to another conductor. The process of noticeably charging an object by contact involves the two contacting objects momentarily sharing the net excess charge. – The excess charge is simply given a larger area over which to spread in order to reduce the total amount of repulsive forces between them. – This process demands that the objects be conductors in order for electrons to move about and redistribute themselves.

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Continued…

An insulator hinders such a movement of electrons between touching objects and about the surfaces of the objects. This is observed if an aluminum pie plate is placed upon a charged foam plate. When the neutral aluminum plate is placed upon the charged foam plate, the foam plate does not conduct its charge to the aluminum. Despite the fact that the two surfaces were in contact, charging by contact or conduction did not occur. (Or at least whatever charge transfer might have occurred was not noticeable by the customary means of using an electroscope, using a charge testing bulb or testing for its repulsion with a like-charged object.)

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Insulators Charge by Lightning

The charging of an electroscope by contact with a negatively charged insulating object would best be described as charging by lightning. Rather than being a process in which the two objects act together to share the excess charge, the process could best be described as the successful effort of electrons to burst through the space (air) between objects. The presence of a negatively charged plastic tube is capable of ionizing the air surrounding the tube and allowing excess electrons on the plastic tube to be conducted through the air to the electroscope.

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Continued…

This transfer of charge can happen with or without touching. – In fact, on a dry winter day the process of charging the metal electroscope with the charged insulator often occurs while the insulator is some distance away. – The dry air is more easily ionized and a greater quantity of electrons are capable of bursting through the space between the two objects. • On such occasions, a crackling sound is often heard and a flash of light is seen if the room is darkened. This phenomenon, occurring from several centimeters away, certainly does not fit the description of contact charging.

The two materials do not make any effort to share charge nor to act as a single object in an effort to reduce repulsive affects.

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Splitting Hairs??

Is this distinction between charging by conduction and charging by lightning a splitting of hairs? Perhaps. For certain, each process involves a transfer of charge from one object to another object, yielding the same result - two like-charged objects. – Yet, distinguishing between the two forms of charging is more consistent with the customary view that insulators are not conductors of charge. It also serves to explain why some insulators clearly do not always transfer their charge upon contact. This phenomenon of charging by lightning will be revisited during a discussion of electric fields and lightning discharges.

•

Check Your Understanding

A neutral metal sphere is touched by a negatively charged metal rod. As a result, the sphere will be ____ and the metal rod will be ____. Select the two answers in their respective order.

a. positively charged b. negatively charged c. neutral d. much more massive e. ... not enough information to tell

Answer #1

• • Answer: B & B This is a case of charging by conduction. When a charged object is used to charge a neutral object by conduction, the previously neutral object acquires the same type of charge as the charged object. The charge object maintains the same type of charge that it originally had. So in this case, both objects have a negative charge.

CYU #2

A neutral metal sphere is touched by a negatively charged metal rod. During the process, electrons are transferred from the _____ to the _____ and the sphere acquires a _____ charge.

a. neutral sphere, charged rod, negative b. neutral sphere, charged rod, positive c. charged rod, neutral sphere, negative d. charged rod, neutral sphere, positive e. ... nonsense! None of these describe what occurs.

Answer #2

• • Answer: C During charging by conduction, both objects acquire the same type of charge. If a negative object is used to charge a neutral object, then both objects become charged negatively. In order for the neutral sphere to become negative, it must gain electrons from the negatively charged rod.

CYU #3

A neutral metal sphere is touched by a positively charged metal rod. During the process, protons are transferred from the _____ to the _____ and the sphere acquires a _____ charge.

a. charged rod, neutral sphere, negative b. charged rod, neutral sphere, positive c. neutral sphere, charged rod, negative d. neutral sphere, charged rod, positive e. ... nonsense! None of these describe what occurs.

Answer #3

• • Answer: E Protons are never transferred in electrostatic activities. In this case, electrons are transferred from the neutral object to the positively charged rod and the sphere becomes charged positively.

CYU #4

A metal sphere is electrically neutral. It is touched by a positively charged metal rod. As a result, the metal sphere becomes charged positively. Which of the following occur during the process? List all that apply.

a. The metal sphere gains some protons. b. Electrons are transferred from the sphere to the rod.

c. The metal sphere loses electrons.

d. The overall charge of the system is conserved.

e. Protons are transferred from the rod to the sphere.

f. Positive electrons are moved between the two objects.

Answer #4

• • Answer: BCD In electrostatic activities, protons are never transferred (which rules out choices a and e). Electrons are not positively charged (ruling out choice e). Choices B, C and D are all true and explain the essential nature of the conduction charging process.

Grounding – The Removal of a Charge

Physics Coach Stephens

• • • • • • •

Grounding – The Removal of a Charge

The previous three sections discussed the three common methods of charging charging by friction , charging by induction , and charging by conduction . A discussion of charging would not be complete without a discussion of uncharging. Objects with an excess of charge - either positive or negative - can have this charge removed by a process known as grounding. Grounding is the process of removing the excess charge on an object by means of the transfer of electrons between it and another object of substantial size. When a charged object is grounded, the excess charge is balanced by the transfer of electrons between the charged object and a ground. A ground is simply an object that serves as a seemingly infinite reservoir of electrons; the ground is capable of transferring electrons to or receiving electrons from a charged object in order to neutralize that object. In this last section, the process of grounding will be discussed.

Grounding of a Negatively Charged Electroscope

• • To begin our discussion of grounding, we will consider the grounding of a negatively charged electroscope. Any negatively charged object has an excess of electrons. – If it is to have its charge removed, then it will have to lose its excess electrons. – Once the excess electrons are removed from the object, there will be equal numbers of protons and electrons within the object and it will have a balance of charge. – To remove the excess of electrons from a negatively charged electroscope, the electroscope will have to be connected by a conducting pathway to another object that is capable of receiving those electrons. • The other object is the ground. • In typical electrostatic experiments and demonstrations, this is simply done by touching the electroscope with one's hand. • Upon contact, the excess electrons leave the electroscope and enter the person who touches it. • These excess electrons subsequently spread about the surface of the person.

• •

Grounding

This process of grounding works because excess electrons find each other repulsive. – As is always the case, repulsive affects between like-charged electrons forces them to look for a means of spatially separating themselves from each other. – This spatial separation is achieved by moving to a larger object that allows a greater surface area over which to spread. – Because of the relative size of a person compared to a typical electroscope, the excess electrons (nearly all of them) are capable of reducing the repulsive forces by moving into the person (i.e., the ground). Like – contact charging discussed earlier , grounding is simply another example of charge sharing between two objects. The extent to which an object is willing to share excess charge is proportional to its size. – So an effective ground is simply an object with significant enough size to share the overwhelming majority of excess charge.

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Grounding of a Positively Charged Electroscope

The previous discussion describes the grounding of a negatively charged electroscope. – Electrons were transferred from the electroscope to the ground. But what if the electroscope is positively charged? How does electron transfer allow an object with an excess of protons to become neutralized? – To explore these questions, we will consider the grounding of a positively charged electroscope. • A positively charged electroscope must gain electrons in order to acquire an equal number of protons and electrons. • By gaining electrons from the ground, the electroscope will have a balance of charge and therefore be neutral. Thus, the grounding of a positively charged electroscope involves the transfer of electrons from the ground into the electroscope.

• • •

Continued…

This process works because excess positive charge on the electroscope attracts electrons from the ground (in this case, a person). – While this may disrupt any balance of charge present on the person, the significantly larger size of the person allows for the excess charge to distance itself further from each other. As in the case of grounding a negatively charged electroscope, the grounding of a positively charged electroscope involves charge sharing. – The excess positive charge is shared between the electroscope and the ground. And once again, the extent to which an object is willing to share excess charge is proportional to its size.

– The person is an effective ground because it has enough size to share the overwhelming majority of excess positive charge.

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The Need for a Conducting Pathway

Any object can be grounded provided that the charged atoms of that object have a conducting pathway between the atoms and the ground. A common lab activity involves taping two straws to a charged aluminum plate. – One straw is covered with aluminum foil and the other straw is bare plastic. – When the aluminum-covered straw is touched, the aluminum plate loses its charge. • It is grounded by means of the movement of electrons from the ground to the aluminum plate. – When the plastic straw is touched, grounding does not occur. • The plastic serves as an insulator and prevents the flow of electrons from the ground to the aluminum plate. Grounding requires a conducting pathway between the ground and the object to be grounded. Electrons will travel along that pathway.

Check Your Understanding

A positively charged pop can is touched by a person standing on the ground. The pop can subsequently becomes neutral. The pop can becomes neutral during this process because ______.

a. electrons pass from the pop can to the person (ground) b. electrons pass from the person (ground) to the pop can c. protons pass from the pop can to the person (ground) d. protons pass from the person (ground) to the pop can

Answer #1

• • Answer: B Protons do NOT move during electrostatic activities, so choices c and d can be ruled out. To ground a positively charged object, electrons must be added to it in order neutralize its excess positive charge. So electrons must move from the ground into the pop can.

CYU #2

A physics student, standing on the ground, touches an uncharged plastic baseball bat to a negatively charged electroscope. This will cause ___.

a. the electroscope to be grounded as electrons flow out of the electroscope. b. the electroscope to be grounded as electrons flow into the electroscope.

c. the electroscope to be grounded as protons flow out of the electroscope.

d. the electroscope to be grounded as protons flow into the electroscope.

e. the baseball bat to acquire an excess of protons.

f. absolutely nothing (or very little) to happen since the plastic bat does not conduct.

Answer #2

• • Answer: F In order to ground an electroscope, electrons must have a conducting pathway between the ground and the object. In this case, a piece of plastic is part of the pathway connecting the ground (the student) and the charged object. Since plastic is an insulator, electrons are incapable of moving through the baseball bat. Grounding does not occur in this instance. Were there a conducting pathway available, choice a would be the proper choice.

CYU #3

• • TRUE or FALSE: An object that becomes grounded gains neutrons during the grounding process.

Answer #3

• • Answer: False Neutrons are positioned in the nucleus of an atom. And like protons, neutrons are never transferred in electrostatic experiments. They are bound in the nucleus and cannot escape by ordinary electrostatic methods.

Charge Interactions Revisited

Physics Coach Stephens

• • • • • • • • •

Charge Interactions are Forces

It is possible that you might have watched two balloons repel each other a dozen or more times and never even thought of the balloon interaction as being a force. Or perhaps you have used a plastic golf tube or other object to raise small paper bits off the lab table and never thought of Newton's laws of motion. Perhaps even now you're thinking, "Why should I? That was the Newton's Laws unit and this is the Static Electricity unit." True. However, the physical world that we study does not separate itself into separate topics, as we teachers and students are prone to do. Physics has an amazing way of fitting together in a seamless fashion. The information that you have forgotten about from the Newton's laws unit has a mischievous way of creeping up on you in other units. That forgetfulness (or negligence or mere ignorance) will haunt you as you try to learn new physics. The more physics that you learn (as in really learn), the more that you come to recognize that the pieces of the physics puzzle fit together to form a unified picture of the world of sight, sound, touch and feel. Here we will explore how Newton's laws of motion interaction of charged objects.

fit together with the

• • •

F

elect

Suppose that you hold a charged plastic golf tube above a handful of paper bits at rest on the table. – The presence of the charged tube is likely to polarize a few bits of paper and then begin to exert an upward pull upon them. – The attraction between a charged tube and a polarized (yet neutral) paper bit is an electrical force - F

elect

. Like all the forces that we’ve studied, the electrical force is a push or pull exerted upon an object as a result of an interaction with another object. The interaction is the result of electrical charges and thus it is called an electrical force.

• • • •

Electrical Force is a Non-Contact Force

Unlike many forces that we study, the electrical force is a non-contact force - it exists despite the fact that the interacting objects are not in physical contact with each other. – The two objects can act over a separation distance and exert an influence upon each other. In this case, the plastic golf tube pulls upward upon the paper bit and a paper bit pulls downward upon the golf tube. – The force is significantly small. – If you were holding the golf tube, you would not likely sense the downward pull exerted upon it by the paper bit. On the other hand, the force is often large enough to either balance or even overwhelm the downward pull of gravity (F grav ) upon the paper bit and cause it to be elevated or even accelerated off the table. Of course the actual result of the force upon the paper bit is related to Newton's laws and a free-body analysis. – If at any moment, the electrical force were greater in magnitude than the gravitational force, the paper bit would be accelerated upward. – And if at any moment, the electrical force is equal in magnitude to the gravitational force, the paper bit will be suspended (or levitated) in midair. – The paper bit would be said to be at equilibrium.

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Balloons at Equilibrium

Now consider the case of the rubber balloons hanging by light threads from the ceiling. – If each balloon is rubbed in the same manner (with animal fur), they each become negatively charged and exert a repulsive affect upon each other. • This charge interaction results in a force upon each balloon that is directed away from the balloon with which it interacts. Once more, we can identify this repulsive affect as an electrical force. – This electrical force joins two other forces that act upon the balloon - the tension force and a force of gravity. • Since the balloons are at rest, the three forces must balance each other such that the net force is zero. A more in-depth analysis of this force cancellation requires a discussion of vectors and is saved for the end of the lesson.

Coulomb’s Laws

• • • • • • Both of these examples illustrate how the interaction between two charges results in a mutual force acting upon the charged objects. An electrical interaction is a force which, like any force, can be analyzed using a free-body diagram and Newton's laws. But what factors affect the magnitude of this force? Is there an equation that can be used to quantify it in the same manner as was done for the force of gravity (F grav = m•g) and the force of friction (F frict = mu•F norm )? The answer is Yes! Coulomb's law holds the key to understanding the answer to these questions. It is the topic of the next section of Lesson 3 .

Coulomb’s Law

Physics Coach Stephens

Coulomb’s Law

• • • • The interaction between charged objects is a non-contact force that acts over some distance of separation. Charge, charge and distance. – Every electrical interaction involves a force that highlights the importance of these three variables. Whether it is a plastic golf tube attracting paper bits, two like-charged balloons repelling or a charged Styrofoam plate interacting with electrons in a piece of aluminum, there is always two charges and a distance between them as the three critical variables that influence the strength of the interaction. In this section of Lesson 3, we will explore the importance of these three variables.

Force as a Vector Quantity

• • • The electrical force, like all forces, is typically expressed using the unit Newton.

Being a force, the strength of the electrical interaction is a vector quantity that has both magnitude and direction. The direction of the electrical force is dependent upon whether the charged objects are charged with like charge or opposite charge and upon their spatial orientation. – By knowing the type of charge on the two objects, the direction of the force on either one of them can be determined with a little reasoning.

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Continued…

In the diagram below, objects A and B have like charge causing them to repel each other. – Thus, the force on object A is directed away from B and the force on object B is directed away from A. On the other hand, objects C and D have opposite charge causing them to attract each other. – Thus, the force on object C is directed toward object D and the force on object D is directed toward object C. When it comes to the electrical force vector, perhaps the best way to determine the direction of it is to apply the fundamental rules of charge interaction (opposites attract and likes repel) using a little reasoning.

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Magnitude or Strength

Electrical force also has a magnitude or strength. – Like most types of forces, there are a variety of factors that influence the magnitude of the electrical force. Two like-charged balloons will repel each other and the strength of their repulsive force can be altered by changing three variables. – First, the quantity of charge on one of the balloons will affect the strength of the repulsive force. • The more charged a balloon is, the greater the repulsive force. – Second, the quantity of charge on the second balloon will affect the strength of the repulsive force. • Gently rub two balloons with animal fur and they repel a little. • Rub the two balloons vigorously to impart more charge to both of them, and they repel a lot. – Finally, the distance between the two balloons will have a significant and noticeable effect upon the repulsive force. • The electrical force is strongest when the balloons are closest together. • Decreasing the separation distance increases the force. The magnitude of the force and the distance between the two balloons is said to be inversely related.

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Coulomb’s Law Equation

The quantitative expression for the effect of these three variables on electric force is known as Coulomb's law. Coulomb's law states that the electrical force between two charged objects is directly proportional to the product of the quantity of charge on the objects and inversely proportional to the square of the separation distance between the two objects. In equation form, Coulomb's law can be stated as:

Continued...

• • • • • where Q

1

represents the quantity of charge on object 1 (in Coulombs),

Q 2

represents the quantity of charge on object 2 (in Coulombs), and d represents the distance of separation between the two objects (in meters). The symbol k is a proportionality constant known as the Coulomb's law constant. – The value of this constant is dependent upon the medium that the charged objects are immersed in. – In the case of air, the value is approximately 9.0 x 10 9 N • m 2 / C 2 . – If the charged objects are present in water, the value of k can be reduced by as much as a factor of 80. It is worthwhile to point out that the units on k are such that when substituted into the equation the units on charge (Coulombs) and the units on distance (meters) will be canceled, leaving a Newton as the unit of force.

• •

Continued…

The Coulomb's law equation provides an accurate description of the force between two objects whenever the objects act as point charges. A charged conducting sphere interacts with other charged objects as though all of its charge were located at its center. – While the charge is uniformly spread across the surface of the sphere, the center of charge can be considered to be the center of the sphere. – The sphere acts as a point charge with its excess charge located at its center. – Since Coulomb's law applies to point charges, the distance d in the equation is the distance between the centers of charge for both objects (not the distance between their nearest surfaces).

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Continued…

The symbols Q

1

and Q

2

in the Coulomb's law equation represent the quantities of charge on the two interacting objects. Since an object can be charged positively or negatively, these quantities are often expressed as "+" or "-" values. – The sign on the charge is simply representative of whether the object has an excess of electrons (a negatively charged object) or a shortage of electrons (a positively charged object). – – It might be tempting to utilize the "+" and "-" signs in the calculations of force. While the practice is not recommended, there is certainly no harm in doing so. When using the "+" and "-" signs in the calculation of force, the result will be that a "-" value for force is a sign of an attractive force and a "+" value for force signifies a repulsive force. Mathematically, the force value would be found to be positive when Q are of like charge - either both "+" or both "-".

1

and Q

2

And the force value would be found to be negative when Q

1

opposite charge - one is "+" and the other is "-". and Q

2

are of This is consistent with the concept that oppositely charged objects have an attractive interaction and like charged objects have a repulsive interaction. In the end, if you're thinking conceptually (and not merely mathematically), you would be very able to determine the nature of the force - attractive or repulsive - without the use of "+" and "-" signs in the equation.

Calculations Using Coulomb’s Law

• In physics courses, Coulomb's law is often used as a type of algebraic recipe to solve physics word problems. Three such examples are shown here.

Example A

Suppose that two point charges, each with a charge of +1.00 Coulomb are separated by a distance of 1.00 meter. Determine the magnitude of the electrical force of repulsion between them. This is not the most difficult mathematical problem that could be selected. It certainly was not chosen for its mathematical rigor. The problem-solving strategy utilized here may seem unnecessary given the simplicity of the given values. Nonetheless, the strategy will be used to illustrate its usefulness to any Coulomb's law problem.

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Continued…

The first step of the strategy is the identification and listing of known information in variable form. Here we know the charges of the two objects (Q separation distance between them (d). 1 and Q 2 ) and the The next step of the strategy involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the force. So F elect is the unknown quantity. The results of the first two steps are shown below.

Given:

Q 1 Q 2 = 1.00 C = 1.00 C d = 1.00 m

Find:

F elect = ???

•

Continued…

The next and final step of the strategy involves substituting known values into the Coulomb's law equation and using proper algebraic steps to solve for the unknown information. This step is shown below.

F elect F elect

F elect

= k • Q 1 • Q 2 / d 2 = (9.0 x 10 9 N•m 2 /C 2 ) • (1.00 C) • (1.00 C) / (1.00 m) 2

= 9.0 x 10 9 N

• • The force of repulsion of two +1.00 Coulomb charges held 1.00 meter apart is 9 billion Newton. This is an incredibly large force that compares in magnitude to the weight of more than 2000 jetliners.

• • •

Micro & Nano

This problem was chosen primarily for its conceptual message. Objects simply do not acquire charges on the order of 1.00 Coulomb. – In fact, more likely Q values are on the order of 10 -9 or possibly 10 -6 Coulombs. – For this reason, a Greek prefix is often used in front of the Coulomb as a unit of charge. – Charge is often expressed in units of microCoulomb (µC) and nanoCoulomb (nC). – If a problem states the charge in these units, it is advisable to first convert to Coulombs prior to substitution into the Coulomb's law equation. The following unit equivalencies will assist in such conversions.

– 1 Coulomb = 10 6 microCoulomb – 1 Coulomb = 10 9 nanoCoulomb

Problem-Solving Strategy

• The problem-solving strategy used in Example A included three steps: – Identify and list known information in variable form.

– List the unknown (or desired) information in variable form.

– Substitute known values into the Coulomb's law equation and using proper algebraic steps to solve for the unknown information. (In some cases and for some students, it might be easier to first do the algebra using the variables and then perform the substitution as the last step.)

• • • • • • • •

Example B

Two balloons are charged with an identical quantity and type of charge: -6.25 nC. They are held apart at a separation distance of 61.7 cm. Determine the magnitude of the electrical force of repulsion between them.

The problem states the value of Q 1 and Q 2 . Since these values are expressed in units of nanoCoulombs (nC), the conversion to Coulombs must be made. The problem also states the separation distance (d). Since distance is given in units of centimeters (cm), the conversion to meters must also be made. These conversions are required since the units of charge and distance in the Coulomb's constant are Coulombs and meters. The unknown quantity is the electrical force (F). The results of the first two steps are shown below.

Given:

Q 1 = -6.25 nC = -6.25 x 10 d = 61.7 cm = 0.617 m -9 C Q 2

Find:

F elect = ???

= -6.25 nC = -6.25 x 10 -9 C

•

Continued…

The final step of the strategy involves substituting known values into the Coulomb's law equation and using proper algebraic steps to solve for the unknown information. This substitution and algebra is shown below.

F elect F elect = k • Q 1 = (9.0 x 10 9 • Q 2 / d 2 N•m 2 /C 2 ) • (6.25 x 10 -9

F elect = 9.23 x 10 -7 N

C) • (6.25 x 10 -9 C) / (0.617 m) 2 • Note that the "-" sign was dropped from the Q 1 and Q 2 values prior to substitution into the Coulomb's law equation. – As mentioned above , the use of "+" and "-" signs in the equation would result in a positive force value if Q 1 and Q 2 are like charged and a negative force value if Q 1 and Q 2 are oppositely charged. – The resulting "+" and "-" signs on F signifies whether the force is attractive (a "-" F value) or repulsive (a "+" F value).

• • • • • •

Example C

Two balloons with charges of +3.37 µC and -8.21 µC attract each other with a force of 0.0626 Newton. Determine the separation distance between the two balloons.

The problem states the value of Q 1 and Q 2 . Since these values are in units of microCoulombs (µC), the conversion to Coulombs will be made. The problem also states the electrical force (F). The unknown quantity is the separation distance (d). The results of the first two steps are shown in the table below.

Given:

Q 1 = +3.37 µC = +3.37 x 10 F elect -6 C Q 2 = -8.21 µC = -8.21 x 10 = -0.0626 N (use a - force value since it is repulsive) -6

Find:

d = ???

C

• • • • • •

Continued…

As mentioned above , the use of the "+" and "-" signs is optional. However, if they are used, then they have to be used consistently for the Q values and the F values. Their use in the equation is illustrated in this problem.

The final step of the strategy involves substituting known values into the Coulomb's law equation and using proper algebraic steps to solve for the unknown information. In this case, the algebra is done first and the substitution is performed last. This algebra and substitution is shown below.

F elect d 2 • F = k • Q elect 1 • Q = k • Q 1 2 / d • Q 2 2 d 2 = k • Q 1 • Q 2 / F elect d = SQRT(k • Q 1 • Q 2 ) / F elect d = SQRT [(9.0 x 10 9 N•m 2 /C 2 ) • (-8.21 x 10 -6 d = Sqrt [ +3.98 m 2 ]

d = +1.99 m

C) • (+3.37 x 10 -6 C) / (-0.0626 N)]

Comparing Electrical & Gravitational Forces

• • Electrical force and gravitational force are the two non-contact forces discussed in The Physics Classroom tutorial . Coulomb's law equation for electrical force bears a strong resemblance to Newton's equation for universal gravitation .

• • • •

Continued…

The two equations have a very similar form. – Both equations show an inverse square relationship between force and separation distance. – And both equations show that the force is proportional to the product of the quantity that causes the force - charge in the case of electrical force and mass in the case of gravitational force. Yet there are some striking differences between these two forces. – First, a comparison of the proportionality constants - k versus G - reveals that the Coulomb's law constant (k) is significantly greater than Newton's universal gravitation constant (G). • Subsequently a unit of charge will attract a unit of charge with significantly more force than a unit of mass will attract a unit of mass. – Second, gravitational forces are only attractive; electrical forces can be either attractive or repulsive.

The inverse square relationship between force and distance that is woven into the equation is common to both non-contact forces. – This relationship highlights the importance of separation distance when it comes to the electrical force between charged objects. It is the focus of the next section.

Check Your Understanding

The Q in Coulomb's law equation stands for the _____.

a. mass of a charged object b. # of excess electrons on the object c. the current of a charged object d. the distance between charged objects e. charge of a charged object

Answer #1

• • Answer: E In the equation F symbol F (9 x 10 9

elect

N • m 2

elect

/ C 2

= k • Q 1

), Q

1

and Q quantity of charge on object 1 and object 2, and d represents the separation distance between the objects' centers.

• Q 2 / d 2 2

, the represents the electrostatic force of attraction or repulsion between objects 1 and 2. The symbol k is Coulomb's law constant represent the

CYU #2

The symbol d in Coulomb's law equation represents the distance from ___.

a. A to B b. A to D c. B to C d. B to D e. C to D f. A to G g. B to F h. C to E

Answer #2

• • Answer: G In the equation F symbol F (9 x 10 9

elect

N • m 2 / C

elect

2

= k • Q 1

), Q

1

and Q quantity of charge on object 1 and object 2, and d represents the separation distance between the objects' centers.

• Q 2 / d 2 2

, the represents the electrostatic force of attraction or repulsion between objects 1 and 2. The symbol k is Coulomb's law constant represent the

CYU #3

Determine the electrical force of attraction between two balloons with separate charges of +3.5 x 10 -8 C and -2.9 x 10 -8 C when separated a distance of 0.65 m.

•

Answer #3

Answer: 2.16 x 10

-5 N, attractive

• Step 1: Identify known values in variable form.

Q 1 Q 2 = +3.5 x 10 -8 C = -2.9 x 10 -8 C d = 0.65 m • Step 2: Identify the requested value F = ??? • Step 3: Substitute and solve

CYU #4

Determine the electrical force of attraction between two balloons that are charged with the opposite type of charge but the same quantity of charge. The charge on the balloons is 6.0 x 10 -7 C and they are separated by a distance of 0.50 m.

•

Answer #4

Answer: 0.0130 N • Step 1: Identify known values in variable form.

Q Q 2 1 = -6.0 x 10 -7 = +6.0 x 10 -7 C C d = 0.50 m.

• Step 2: Identify requested information F = ??? • Step 3: Substitute and solve.

CYU #5

Joann has rubbed a balloon with wool to give it a charge of -1.0 x 10 -6 C. She then acquires a plastic golf tube with a charge of +4.0 x 10 -6 C localized at a given position. She holds the location of charge on the plastic golf tube a distance of 50.0 cm above the balloon. Determine the electrical force of attraction between the golf tube and the balloon.

• Answer: 0.144 N

Answer #5

• Step 1: Identify known values in variable form.

Q 1 = -1.0 x 10^-6 C Q2 = +4.0 x 10 -6 C d = 50.0 cm = 0.50 m.

• Step 2: Identify requested information F = ??? • Step 3: Substitute and solve.

CYU #6

A balloon with a charge of 4.0 µC is held a distance of 0.70 m from a second balloon having the same charge. Calculate the magnitude of the repulsive force.

• Answer: 0.29 N

Answer #6

• Step 1: Identify known values in variable form.

Q 1 Q 2 = -4.0 x 10^-6 C = +4.0 x 10 -6 C d = 0.70 m.

• Step 2: Identify requested information F = ??? • Step 3: Substitute and solve.

CYU #7

At what distance of separation must two 1.00 microCoulomb charges be positioned in order for the repulsive force between them to be equivalent to the weight (on Earth) of a 1.00 kg mass?

•

Answer #7

Answer: 0.030 m or 3.0 cm • Step 1: Identify known values in variable form.

Q 1 Q 2 = 1.0 x 10 = 1.0 x 10 F elect = F grav -6 -6 C C = mg = 1.0 • 9.8 m/s/s = 9.8 N • Step 2: Identify requested information d = ??? • Step 3: Substitute and solve.

Inverse Square Law

Physics Coach Stephens

Inverse Square Law

• • • • • • Science in general and Physics in particular are concerned with relationships. Cause and effect is the focus of science. Nature is probed in order to find relationships and mathematical patterns. Scientists modify a set of conditions to see if there is a pattern of behavior in another set of measurable quantities. The goal is to answer the question of how does a change in a set of variables or conditions causally effect an observable outcome? In Physics, this search for cause and effect leads to questions like: – How does a force affect the acceleration of an object? How does the mass of an object affect its acceleration?

– How does the speed of a falling object affect the amount of air resistance that it experiences?

– How does the distance from a page to a light bulb affect the amount of light that illuminates the paper's surface?

– How does the frequency of a sound wave affect the speed at which the sound wave moves?

– How does the distance between two charged objects affect the force of attraction or repulsion that they encounter?

Direct & Inverse Relationships

• • • This search for cause and effect often leads to conclusive evidence that two variables are causally related (or not causally related). Careful observation and measurement might indicate that a pattern exists in which an increase in one variable always causes another measurable quantity to increase. – This type of cause-effect relationship is described as being a direct relationship. Observation might also indicate that an increase in one variable always causes another measurable quantity to decrease. – This type of cause-effect relationship is described as being an inverse relationship.

• • • • •

Inverse Relationships

Inverse relationships are common in nature. – In electrostatics, the electrical force between two charged objects is inversely related to the distance of separation between the two objects. • Increasing the separation distance between objects decreases the force of attraction or repulsion between the objects. • And decreasing the separation distance between objects increases the force of attraction or repulsion between the objects. – Electrical forces are extremely sensitive to distance. These observations are commonly made during demonstrations and lab experiments. Consider a charged plastic golf tube being brought near a collection of paper bits at rest upon a table. – The electrical interaction is so small at large distances that the golf tube does not seem to exert an influence upon the paper bits. – Yet if the tube is brought closer, an attractive interaction is observed and the strength is so significant that the paper bits are lifted off the table. In a similar manner, charged balloons are observed to exert their greatest influence upon other charged objects when the separation distance is reduced. Electrostatic force and distance are inversely related.

• • •

Inverse Square Relationship

The pattern between electrostatic force and distance can be further characterized as an inverse square relationship. Careful observations show that the electrostatic force between two point charges varies inversely with the square of the distance of separation between the two charges. – That is, the factor by which the electrostatic force is changed is the inverse of the square of the factor by which the separation distance is changed. • So if the separation distance is doubled (increased by a factor of 2), then the electrostatic force is decreased by a factor of four (2 raised to the second power). • And if the separation distance is tripled (increased by a factor of 3), then the electrostatic force is decreased by a factor of nine (3 raised to the second power). This square effect makes distance of double importance in its impact upon electrostatic force.

Continued…

• The inverse square relationship between electrostatic force and separation distance is illustrated in the table below.

Row

1 2 3 4 5

Separation Distance

20 m 40 m 60 m 80 m 100 m

Electrostatic Force

0.1280 N 0.0320 N 0.0142 N 0.0080 N 0.0051 N

•

Patterns

The above values illustrate a pattern: as the separation distance is doubled, the electrostatic force is decreased by a factor of four. – For instance, the distance in Row 2 is twice the distance of Row 1; and the electrostatic force in Row 2 is one-fourth the electrostatic force of Row 1. – A comparison of Row 1 and Row 3 illustrate that as the distance is increased by a factor three, the force is decreased by a factor of nine. – The distance in Row 3 is three times that of Row 1 and the force in Row 3 is one ninth that of Row 1. – A similar comparison of Rows 1 and Row 4 illustrates that as the distance is increased by a factor of four, the electrostatic force is decreased by a factor of 16. – The distance in Row 4 is four times that of Row 1 and the force in Row 4 is one sixteenth that of Row 1.

Row

1 2 3 4 5

Separation Distance

20 m 40 m 60 m 80 m 100 m

Electrostatic Force

0.1280 N 0.0320 N 0.0142 N 0.0080 N 0.0051 N

Continued…

• The equation shows that the distance squared term is in the denominator of the equation, opposite the force. This illustrates that force is inversely proportional to the square of the distance.

Check Your Understanding

• Alteration in the quantity of charge: • Two charged objects have a repulsive force of 0.080 N. If the charge of one of the objects is doubled, then what is the new force?

Answer #1

• • Answer: 0.160 N Explanation: Electrostatic force is directly related to the charge of each object. So if the charge of one object is doubled, then the force will become two times greater. Two times 0.080 N is 0.160 N.

CYU #2

• Alteration of the distance between two charged objects: • Two charged objects have a repulsive force of 0.080 N. If the distance separating the objects is doubled, then what is the new force?

Answer #2

• • Answer: 0.020 N Explanation: The electrostatic force is inversely related to the square of the separation distance. So if d is two times larger, then F is four times smaller - that is, one fourth the original value. One-fourth of 0.080 N is 0.020 N.

CYU #3

• Alteration in both the quantity of charge and the distance: • Two charged objects have a repulsive force of 0.080 N. If the charge of one of the objects is doubled, and the distance separating the objects is doubled, then what is the new force?

Answer #3

• • Answer: 0.040 N Explanation: The electrostatic force is directly related to the product of the charges and inversely related to the square of the separation distance. Doubling one of the charges would serve to double the force. Doubling the distance would serve to reduce the force by a factor of four. The combined affect of these two variations would be to decrease the force by a factor of two - changing it from 0.080 N to 0.040 N.