CK-12 People's Physics Book Version 2

by James H. Dann

  a. The tube closed at one end will have a longer fundamental wavelength and a lower frequency. b. If the temperature increases the wavelength will not change, but the frequency will increase accordingly.

  struck by bullet first.

  a. b.

  c.

  Chapter 13: Electricity Version 2

  The Big Idea

  Conservation of charge is the fourth of the five conservation laws in physics. There are two types of charge: positive and negative; the law of conservation of electric charge states that the net charge of the universe remains constant. As with momentum and energy, in any closed system charge can be transferred from one body to another and can move within the system but cannot leave the system.

  Electromagnetism is associated with charge and is a fundamental force of nature, like gravity (which for us is associated with mass). If charges are static, the only manifestation of electromagnetism is the Coulomb electric force. In the same way the gravitational force that an object exerts upon other objects, and that other objects exert on it, depends on the amount of mass it possesses, the Coulomb electric force that an object experiences depends on the amount of electric charge the object possesses. Like gravity, the Coulomb electric force decreases with the square of the distance. The Coulomb electric force is responsible for many of the forces we discussed previously: the normal force, contact forces such as friction, and so on --- all of these forces arise in the mutual attraction and repulsion of charged particles.

  Although the law determining the magnitude of the Coulomb electric force has the same form as the law of gravity, the electric constant is 20 orders of magnitude greater than the gravitational constant. That is why electricity normally dominates gravity at the atomic and molecular level. Since there is only one type of mass but two opposite types of electric charge, gravity will dominate in large bodies unless there is a separation of charge.

  Electric Forces and Fields

  The Coulomb Force Law

  The Coulomb Force Law states that any two charged particles --- with charge measured in units of Coulombs --- at a distance from each other will experience a force of repulsion or attraction along the line joining them equal to: This looks a lot like the Law of Universal Gravitation, which deals with attraction between objects with mass. The big difference is that while any two masses experience mutual attraction, two charges can either attract or repel each other, depending on whether the signs of their charges are alike:

  Like gravitational (and all other) forces, Coulomb forces add as vectors. Thus to find the force on a charge from an arrangement of charges, one needs to find the vector sum of the force from each charge in the arrangement.

  Example 1

  Question: Two negatively charged spheres (one with ; the other with ) are apart. Where could you place an electron so that it will be suspended in space between them with a net force of zero (for this problem we will ignore the force of repulsion between the two charges because they are held in place)?

  Answer: Consider the diagram above; here is the distance between the electron and the small charge, while is the force the electron feels due to it. For the electron to be balanced in between the two charges, the forces of repulsion caused by the two charges on the electron would have to be balanced. To do this, we will set the equation for the force exerted by two charges on each other equal and solve for a distance ratio. We will denote the difference between the charges through the subscripts "s" for the smaller charge, "e" for the electron, and "l" for the larger charge.

  Now we can cancel. The charge of the electron cancels. The constant also cancels. We can then replace the large and small charges with the numbers. This leaves us with the distances. We can then manipulate the equation to produce a ratio of the distances.

  Given this ratio, we know that the electron is twice as far from the large charge () as from the small charge (). Given that the distance between the small and large charges is , we can determine that the electron must be located away from the large charge and away from the smaller charge.

  Electric Fields and Electric Forces

  Gravity and the Coulomb force have a nice property in common: they can be represented by fields. Fields are a kind of bookkeeping tool used to keep track of forces. Take the electromagnetic force between two charges given above: If we are interested in the acceleration of the first charge only --- due to the force from the second charge --- we can rewrite this force as the product of and . The first part of this product only depends on properties of the object we're interested in (the first charge), and the second part can be thought of as a property of the point in space where that object is.

  In fact, the quantity captures everything about the electromagnetic force on any object possible at a distance r from . If we had replaced with a different charge, , we would simply multiply by to find the new force on the new charge. Such a quantity, here, is referred to as the electric field from charge at that point: in this case, it is the electric field due to a single charge:

  The electric field is a vector quantity, and points in the direction that a force felt by a positive charge at that point would. If we are given the electric field at some point, it is just a matter of multiplication --- as illustrated above --- to find the force any charge would feel at that point: Note that this is true for all electric fields, not just those from point charges. In general, the electric field at a point is the force a positive test charge of magnitude 1 would feel at that point. Any other charge will feel a force along the same line (but possibly in the other direction) in proportion to its magnitude. In other words, the electric field can be though of as "force per unit charge".

  In the case given above, the field was due to a single charge. Such a field is shown in the figure below. Notice that this a field due to a positive charge, since the field arrows are pointing outward. The field produced by a point charge will be radially symmetric i.e., the strength of the field only depends on the distance, , from the charge, not the direction; the lengths of the arrows represent the strength of the field.

  Example 2

  Question: Calculate the electric field a distance of away from a charge. Then, calculate the force on a charge placed at this point.

  Answer: To calculate the electric field we will use the equation Before we solve for the electric field by plugging in the values, we convert all of the values to the same units. Now that we have consistent units we can solve the problem. To solve for the force at the point we will use the equation We already know all of the values so all we have to do is convert all of the values to the same units and then plug in the values.

  Fields Due to Several Charges

  To find the field at a point due to an arrangement of charges --- in fact, all electric fields arise due to some arrangement of charges --- we find the vector sum of the individual fields: Electric fields are used more frequently than gravitational ones because there are two types of charge, which makes electric force and potential energy harder to keep track of than their gravitational counterparts. To apply this approach to gravitational forces --- that is, to find a net gravitational field --- one needs to repeat the steps above, with mass in place of charge (left for the reader).

  Example 3

  Question: For the diagram above, draw (qualitatively) the electric field vectors at the points shown using the test charge method.

  Answer: We will start with Test Charge 1. Test charges are always positive and have magnitude 1. Therefore we know that the test charge will want to go toward the negative charge and away from the positive charge (like charges repel and opposite charges attract). The strength of the electric field felt by the test charge is dependent on the inverse square of the distance of the charges as shown by the equation The farther away from the source of the field, the weaker the field becomes. Therefore Test Charge 1 will experience a stronger field from the 1C charge. Because the distance from Test Charge 1 to the is only slightly longer than the distance from Test Charge 1 to the 1C charge, the vectors
will be similar in length. Once we have determined the relative scale of each vector, we can add them using the parallelogram method. The resultant vector is the electric field at that point.

  Finding the electric field at Test Charge 2 will involve all of the same steps. First we must determine which charge Test Charge 2 is closer to. Like Test Charge 1, Test Charge 2 is closer to the 1C charge. However, Test Charge 2 is drastically closer whereas Test Charge 1 was only slightly closer. Therefore, the electric field that Test Charge 2 experiences as a result of the 1C charge will be strong, thus resulting in a longer arrow. The distance between the and Test Charge 2 is large and therefore the electric field experienced by Test Charge 2 as a result of the charge will be small. The resultant vectors will look something like this.

  Electric Potential

  Like gravity, the electric force can do work and has a potential energy associated with it. But like we use fields to keep track of electromagnetic forces, we use electric potential, or voltage to keep track of electric potential energy. So instead of looking for the potential energy of specific objects, we define it in terms of properties of the space where the objects are.

  The electric potential difference, or voltage difference (often just called voltage) between two points (A and B) in the presence of an electric field is defined as the work it would take to move a positive test charge of magnitude 1 from the first point to the second against the electric force provided by the field. For any other charge , then, the relationship between potential difference and work will be: The energy that the object gains or loses when traveling through a potential difference is supplied (or absorbed) by the electric field --- there is nothing else there. Therefore, it follows that electric fields contain energy.

  To summarize: just as an electric field denotes force per unit charge, so electric potential differences represent potential energy differences per unit charge. A useful mnemonic is to consider a cell phone: the battery has the potential to do work for you, but it needs a charge! Actually, the analogy there is much more rigorous than it at first seems; we'll see why in the chapter on current. Since voltage is a quantity proportional to work it is a scalar, and can be positive or negative.

  Electric Field of a Parallel Plate Capacitor

  Suppose we have two parallel metal plates set a distance from one another. We place a positive charge on one of the plates and a negative charge on the other. In this configuration, there will be a uniform electric field between the plates pointing from, and normal to, the plate carrying the positive charge. The magnitude of this field is given by where V is the potential difference (voltage) between the two plates.

  The amount of charge, , held by each plate is given by where again is the voltage difference between the plates and is the capacitance of the plate configuration. Capacitance can be thought of as the capacity a device has for storing charge . In the parallel plate case the capacitance is given by where is the area of the plates, is the distance between the plates, and is the permittivity of free space whose value is .

  The electric field between the capacitor plates stores energy. The electric potential energy, , stored in the capacitor is given by

  Where does this energy come from? Recall, that in our preliminary discussion of electric forces we assert that "like charges repel one another". To build our initial configuration we had to place an excess of positive and negative charges, respectively, on each of the metal plates. Forcing these charges together on the plate had to overcome the mutual repulsion that the charges experience; this takes work. The energy used in moving the charges onto the plates gets stored in the field between the plates. It is in this way that the capacitor can be thought of as an energy storage device. This property will become more important when we study capacitors in the context of electric circuits in the next chapter.

  Note: Many home-electronic circuits include capacitors; for this reason, it can be dangerous to mess around with old electronic components, as the capacitors may be charged even if the unit is unplugged. For example, old computer monitors (not flat screens) and TVs have capacitors that hold dangerous amounts of charge hours after the power is turned off.

  More on Electric and Gravitational Potential

  There are several differences between our approach to gravity and electricity that could cause confusion. First, with gravity we usually used the concept of "energy", rather than "energy difference". Second, we spoke about it in absolute terms, rather than "per unit mass".

  To address the first issue: when we dealt with gravitational potential energy we had to set some reference height where it is equal to . In this sense, we were really talking about potential energy differences rather than absolute levels then also: at any point, we compared the gravitational potential energy of an object to the energy it would have had at the reference level . When we used the formula we implicitly set the initial point as the zero: no free lunch! For the same reason, we use the concept of electric potential difference between two points --- or we need to set the potential at some point to 0, and use it as a reference. This is not as easy in this case though; usually a point very far away ("infinitely" far) is considered to have 0 electric potential.

  Regarding the second issue: in the chapter on potential energy, we could have gravitational potential difference between two points at different heights as . This, of course, is the work required to move an object of mass one a height against gravity. To find the work required for any other mass, we would multiply this by its magnitude. In other words, Which is exactly analogous to the equation above.

  Summary of Relationships

  The following table recaps the relationships discussed in this chapter.

  Relationship between "per Coulomb" and absolute quantities. Property of Object. Property of Space. Combine Into:

  Charge (Coulombs) Field* (Newtons/Coulomb) Force (Newtons)

  Charge (Coulombs) Potential* (Joules/C) Potential Energy (Joules)

  An advanced note: for a certain class of forces called conservative forces e.g., gravity and the electromagnetic force, a specific potential distribution corresponds to a unique field. Conversely, a field corresponds to a unique potential distribution up to an additive constant. Remember though, it's relative potential between points not absolute potential that is physically relevant. In effect the field corresponds to a unique potential. In particular, we see that in the case of conservative forces the scalar potential (one degree of freedom per point) carries all information needed to determine the vector electric field (three degrees of freedom per point. The potential formulation is even more useful than it at first seems.

  Key Concepts

  Electrons have negative charge and protons have positive charge. The magnitude of the charge is the same for both: .

  In any closed system, electric charge is conserved. The total electric charge of the universe does not change. Therefore, electric charge can only be transferred not lost from one body to another.

  Normally, electric charge is transferred when electrons leave the outer orbits of the atoms of one body (leaving it positively charged) and move to the surface of another body (causing the new surface to gain a negative net charge). In a plasma, the fourth state of matter, all electrons are stripped from the atoms, leaving positively charged ions and free electrons.

  Similarly-charged objects have a repulsive force between them. Oppositely charged objects have an attractive force between them.

  The value of the electric field tells you the force that a charged object would feel if it entered this field. Electric field lines tell you the direction a positive charge would go if it were placed in the field.

  Electric potential is measured in units of Volts (V) thus electric potential is often referred to as voltage. Electric potential is the source of the electric potential energy.

  Positive charges move towards lower electric potential; negative charges move toward higher electric potential. If you are familiar with a contour map then positive charges go 'downhill' and nega
tive charges go 'uphill'.

  Faraday cages consist of a metal box. All of your sensitive electronics are encased in a metal box called a Faraday cage. The Faraday cage protects everything inside from external electric fields. Basically the electrons in the metal box move around to cancel out the electric field, thus preventing it from coming inside the box and thus preventing movement of charge and possible blown out electronic chips. Cars and airplanes, being enclosed in metal, are also Faraday cages and thus the safest place to be in a lightning storm.

  Boeing 747 Gets Hit By Lightning. From live leak(Watch on Youtube)

  Key Applications

  In problems that ask for excess negative or positive charge, remember that each electron has one unit of the fundamental charge .

  To find the speed of a particle after it traverses a voltage difference, use the equation for the conservation of energy:

  Force and electric field are vectors. Use your vector math skills (i.e. keep the x and y directions separate) when solving two-dimensional problems.

  Electricity Problem Set

  After sliding your feet across the rug, you touch the sink faucet and get shocked. Explain what is happening.