PHYS 202 Notes, Week 1 Greg Christian January 19 & 21, 2016 This week we introduce the fundamental quantities of electric charge and the laws governing the forces and fields generated by electric charges. Electric Charge Electric charge is a fundamental property of matter. It s something that is intrinsic to everything, similar to mass. Only two types of charge exist: positive (+) and negative ( ) 1. Opposite charges (+/+ or / ) repel each other, while like charges (+/ ) attract. Electric charge is conserved, that is, it cannot be created or destroyed. However, we can move charges around to create a net charge on a given object 2. The total amount of net charge on an object is called its magnitude, while the sign tells whether it s positive or negative. Where does electric charge come from? Electric charge originates in atoms, from the three types of particles of which they are composed: protons (positive charge), neutrons (no charge), and electrons (negative charge). Protons and electrons have the same magnitude of charge but opposite signs. As pictured in Figure 1, these particle are arranged to make atoms in the following way: A nucleus made of protons and neutrons at the center with net positive charge. Electrons surrounding the nucleus with net negative charge. 1 As named by Benjamin Franklin in the 1700s. 2 A classic example is rubbing a glass rod with silk, giving it a net positive charge, or a plastic rod with wool giving a net negative charge. 7 Li Atom Net charge zero electrons (charge 3) n + n n + + n nucleus (charge +3) Figure 1: An atom, made up of a positivelycharged nucleus and negativelycharged surrounding electrons. In normal atoms, the number of protons and electrons is the same, giving the total atom a net charge of zero. In other words, ordinary atoms are charge neutral. Likewise, so is most ordinary matter. The electrons in atoms are only weakly bound to the nucleus. As a result, it is possible to add or remove them, creating ions, which do have a net charge. There are two types of ions: 1. Positive ions created by removing electrons. 2. Negative ions created by adding electrons. Drawings of each are shown in Figure 2. In a normalsized (or macroscopic) object, made up of many atoms, 3 large numbers of electrons can 3 On the order of 10 23. be added or removed. This gives the object a net charge. In the classic example of rubbing a glass rod with silk or a plastic rod with wool, this is what s going on:
phys 202 notes, week 1 2 (a) 7 Li negative ion Net charge 1 nucleus n + n (charge +3) n + + n electrons (charge 4) (b) 7 Li positive ion Net charge +1 nucleus n + (charge +3) n n + + n Figure 2: a) A positively charged ion, with electrons removed; b) A negatively charged ion, with electrons added. electrons (charge 2) Rubbing the glass rod with silk removes electrons, giving a net positive charge. Rubbing the plastic rod with wool adds electrons, giving a net negative charge. Conductors and Insulators In general, materials can be divided into two types with respect to whether or not charges can move through them 4 : 1. Insulators, which do not allow charges to move. 2. Conductors, which do allow charges to move. When a conductor connects two objects with different charges, it allows that charge to move between the two objects. In contrast, when an insulator connects two differentlycharged objects, the charges stay where they are. A good example of this is a wire connected to a battery. The wire is a conductor and allows charge to move freely from the positive terminal of the battery to the negative one. This moving charge is what powers things like a lightbulb or a smartphone or a Tesla car. However, the plastic surrounding the wire is an insulator that does not allow charge to move through it. In conductors, the moving charge results from either electrons or ions moving through the material. Whatever is moving through the material is sometimes referred to as the charge carrier. Most conductors are metals, such as copper wire, whose charge carriers are electrons 5. The free movement of charge through conductors can lead to some notable effects. Induction The free movement of electrons in a metal 6 leads to a phenomena called induction. This results from the electrons moving from one area of the conductor to another, leading to local excesses of positive and 4 There is also a third type, the semiconductor, which can behave like an insulator or a conductor depending on the situation 5 The other common types of conductors are liquids or gasses which allow ions (which can be either positively or negatively charged) to move. A good example is seawater, whose charge carrier is the dissolved salt ions 6 Or ions in some other conductor.
phys 202 notes, week 1 3 negative charge. Figure 3 shows an example of this. Here s a breakdown of what happens: The negatively charged rod repels the electrons on the surface of the metal, creating a force towards the right side of the sphere. The electrons move towards the right side of the sphere, leaving a net positive charge on the left side. The positive charge attracts electrons back towards the left. Things end up in equilibrium with the force to the right from the rod exactly balancing the force to the left from the positive charges. Figure 3: Example of induced charge on a metal sphere. Conservation and Quantization of Charge There are two fundamental principles relating to charges: 1. The conservation of charge: charge cannot be created or destroyed. It can only be moved around to create local excesses of positive or negative charge. 2. The quantization of charge: charge comes in discrete packets, with the fundamental unit being the charge on an electron (or proton). Any charge you see is some combination of these packets. Compare with currency, which comes in discrete packets of 1. Coulomb s Law Coulomb s law 7 describes the force F arising from two charges q 1 and q 2 some distance r apart: 8 It says that the magnitude of the force is F = k q 1q 2 r 2. (1) 7 Discovered in 1784 by Charles Augustin de Coulomb using a torsion balance. 8 Note the similarity to the law of gravity, F = Gm 1 m 2 /r 2. The only difference is that mass is always positive, making gravity always attractive. a) Proportional to the product of the magnitude of the charges; and b) Inversely proportional to the square of the distance between them. The direction of the force is determined by the signs of the charges: opposite signs attract while like signs repel. In Eq. (1), the term k is a fundamental constant of nature. Using SI units, it s equal to k = 8.99 10 9 N m 2 /C 2. This introduces the SI unit of charge, the coulomb, denoted as C. The fundamental unit of charge is denoted by the symbol e and has a value of 9 e = 1.60217653 10 19 C. 9 As alluded to in the last section, the charge on an electron is e and that on a proton is +e.
phys 202 notes, week 1 4 constant ɛ 0 : where ɛ 0 = 8.854 10 12 C 2 / ( N m 2). k = 1 4πɛ 0, (2) As you can see, this is a very small quantity in terms of Coulombs. Even compared to charges encountered in everyday life, the Coulomb is quite large. Charges on the order of microcoulombs (1 µc = 10 6 C) or nanocoulombs (1 nc = 10 9 C) are more common. The constant k is often expressed 10 in terms of another fundamental 10 This may seem like a silly overcomplication at the moment, but later on you ll learn that expressing things in terms of ɛ 0 makes sense when studying different topics. Superposition Coulomb s law obeys the principle of superposition, that is, the total force on a given charge is the vector sum of all the acting forces from other charges. Figure 4 illustrates an example of this. q 1 = +1 C q 2 = +1 C F 12 F23 F tot q 3 = 1 C Electric Fields When charges interact by Coulomb s law, what is there to tell them about each other? To answer this question, we can introduce the concept of an electric field. The electric field is a quantity describing the magnitude and direction of the force that would act on a charge q if it were at a particular location in space. Again, the electric field is an abstract quantity a concept that we use to describe how real forces are transmitted. To define the electric field, pretend there is a test charge q at some location P in space. The electric field is given by the force that would act on the test charge divided by its magnitude: Figure 4: Illustration of the superposition principle. The total force on q 1 is the vector sum of the forces from q 2 (repulsive) and q 3 (attractive). E = F/q. (3) Both the force and the electric field are vector quantities, meaning that they have both a magnitude and a direction. In Eq. (3), the direction of F and E is the same if q is positive and opposite if q is negative. The electric field is different at every point in space, just how the force on q would be different if you moved it around. This is illustrated in Figure 5. Because it has magnitude and direction, the electric field is referred to as a vector field. One thing that is important to remember is that the actual electric field has nothing to do with the test charge q and does not depend on it in any way. It is simply a device that we made up to illustrate the concept. Electric fields are, however, generated by other collections of charges, called the source of the electric field. Figure 5: Illustration of the electric field generated by a point charge. The field gets weaker as you move away from the charge, just like the force on a test charge would get weaker as it was moved away (c.f. Coulomb s law, Eq. (1)).
phys 202 notes, week 1 5 Calculating Electric Fields The magnitude and direction of an electric field can be modeled with mathematical equations. Before getting into this, let s introduce two concepts: 1. The source point, S: the location of the charges generating the field. 2. The field point, P: the location where we want to calculate the field. In general, electric fields can be generated by multiple sources. In this case, the total field is the vector sum of the individual fields, 11 Etotal = E1 + E2 + E3 +... (4) The actual equations describing electric fields depend on the geometry of the source. Next, we ll introduce the equations for some of the more common source shapes. 11 What we mean by vector sum, is that both the magnitude and the direction of the individual fields are added together. Think of stacking a bunch of arrows up together headtotail. Point Charge This electric field due to a point charge q is given by E = k q r 2. (5) This follows naturally from Coulomb s law, Eq. (1), and the electric field definition, Eq. (3). Spherical Charge Distribution Spherical charge distributions have a unique and useful property: The electric field outside a sphere containing charge q is exactly the same as the field produced by a point charge at the center of the sphere. In practice, what this means is that we can treat a spherical charge distribution just like a point charge, using Eq. (5) to calculate the electric field outside it. One thing to be aware of, however, is that this principle only applies to points outside the sphere. Example Problems Coulomb s Law
Electric Fields phys 202 notes, week 1 9
Opposite of force (b/c negative charge):: UP DOWN