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1/25/17 Electric Fields (1) A. B. Kaye, Ph.D. Associate Professor of Physics 24 January 2017 Electricity and Magnetism Why study electricity and magnetism? • The laws of electricity and magnetism play a central role in the operation of many modern devices • The interatomic and intermolecular forces responsible for the formation of solids and liquids are electric in nature • This is why you don’t fall through your chair when you sit on it – even though most of the chair is empty space Historically speaking… Chinese § Documents suggest that magnetism was observed as early as 2000 BC Greeks § Electrical and magnetic phenomena observed as early as 700 BC § In 580 BC, Thales showed that when you rub amber with a piece of cloth, it became endowed with the power of attracting light bodies to itself (remained unexplained) § Other experiments with magnetite 1600 § William Gilbert (physician to Queen Elizabeth) showed electrification effects were not confined to just amber. § The electrification effects were a general phenomena. 1 1/25/17 Historically speaking… 1733 § Charles Francois du Fay discovers that electricity comes in two kinds, which he called resinous (–) and vitreous (+) 1785 § Charles Augustin Coulomb confirmed inverse square law form for electric forces. He also proposes a combined fluid/action-at-adistance theory with two conducting fluids; fighting breaks out between single– and double-fluid partisans. 1792 § Alessandro Volta shows that when moisture comes between two different metals, electricity is created. • The first example of flowing electricity Historically speaking… 21 April 1820 § Hans Christian Ørsted found a compass needle deflected when near a wire carrying an electric current • Electricity and magnetism must have a direct relationship • Andre Marie Ampere shows that parallel currents attract or repel each other depending on the direction of current flow (opposite currents attract) 1826 § Georg Simon Ohm establishes the result now known as Ohm’s law: V=IR 1831 § Faraday invented a bell that could be rung at a distance using an electric wire in 1831 – a precursor to the modern doorbell § Michael Faraday and Joseph Henry (independently) showed that when a magnet is moved inside a coil of copper wire, a small electric current is generated • Faraday published his findings first Historically speaking… ~1870 § Thomas Alva Edison builds the first practical DC generator. § George Westinghouse purchased and develops Nikola Tesla’s patented motor for generating AC 1873 § James Clerk Maxwell used observations and other experimental facts as a basis for formulating the laws of electromagnetism in his Treatise on Electricity and Magnetism • Unified electricity and magnetism See class website for “A Brief History of Electromagnetism” and “Another Brief History of Electromagnetism” 2 1/25/17 ELECTRIC FIELDS Properties of Electric Charges Electric Charges • There are two kinds of electric charges Called “positive” and “negative” • • Negative charges are the type possessed by electrons • Positive charges are the type possessed by protons (or positrons) • Charges of the same sign repel one another and charges with opposite signs attract one another • Electric charge is always conserved in an isolated system • E.g., charge is not created in the process of rubbing two objects together; the electrification is due to the transfer of charge from one object to the other Conservation of Electric Charges • A glass rod is rubbed with silk • Electrons are transferred from the glass to the silk; and each electron adds a negative charge to the silk • An equal positive charge is left on the rod What does this mean? • “Charge” is not a substance, but is rather a property that is inferred from observed interactions • Right now, we don’t understand what “charge” is, but we can explain how it behaves 3 1/25/17 Formal Statement of Charge Conservation • We can express the conservation of charge using the charge density r (in Coulombs per cubic meter) and the electric current density J (in Amperes per square meter) as: • • • The 1st term on the LHS the time rate of change of the charge density at some point in space The 2nd term on the LHS is the divergence of the current density at the same point Because these two terms are equal, the only way for r to change if for a current to flow into or out of the point. • Can we prove this? Sure we can. The net current flowing into a volume can be written as: • • • Here, S = ∂V is the boundary of the volume V oriented by outward-pointing normal vectors Further, dS is a “shorthand” way of writing NdS – those outward-pointing normal vectors J is the current density (charge per unit area per unit time) at the surface of the volume; it points in the direction of the current flow Derivation (2) • From the divergence theorem, we can express the current as: • Conservation of charge means that the net current into the volume must equal the net change of current within the volume: (a) • The total charge q in the volume V is the integral (sum) of the charge density in that volume: • … which means that (b) Derivation (3) • Let’s now set equations (a) and (b) equal to each other (since they are both expressions for dq/dt): • Since this must be true for every volume V, we can write 4 1/25/17 Example Problem – Conservation of Charge • Two identical metal spheres, separated by some distance and labeled “A” and “B” are charged; sphere A carries a net charge of +7Q and sphere B carries of net charge of –3Q. • The spheres are brought together, allowed to touch, and then separated again. What is the charge on each of the spheres after the separation? Quantization of Electric Charges • In 1909, Robert Millikan discovered that charge always occurs as integral multiples of a fundamental quantity Therefore, the electric charge, q, is said to be quantized q is the standard symbol used for charge as a variable q = ±Ne • • • • N is an integer • e is the fundamental unit of charge • |e| = 1.6021766208 x 10–19 C • Electron: q = –e • Proton/Positron: q = +e ELECTRIC FIELDS Interesting Problems 5 1/25/17 Interesting Problem #1 • Compare the magnitudes of the gravitational force of attraction and of the electric force of attraction between the electron and the proton in a hydrogen atom. • According to Newtonian mechanics, what is the acceleration of the electron? • Assume that the distance between these particles in a hydrogen atom is 5.3 x 10–11 m. Interesting Problem #2 • A simple electroscope for the detection and measurement of electric charge consists of two small foil-covered cork balls of 1.5 x 10–4 kg each suspended by threads that are 10 cm long. • When equal electric charges are placed on the balls, the electric repulsive force pushes them apart, and the angle between the threads indicates the magnitude of the electric charge. If the equilibrium angle between the threads is 60º, what is the magnitude of the charge? Interesting Problem #3 • Two point charges +Q and –Q are separated by a distance d. • A positive point charge q is equidistant from these charges, at a distance x from their midpoint. • What is the electric force F on the point charge q? 6 1/25/17 ELECTRIC FIELDS Charging Objects by Induction Types of Materials If we classify materials by how easy it is to move electrons through them, then we end up with three different types: 1. Conductors 2. Insulators 3. Semiconductors Let’s look at each one of these in more detail Conductors • Electrical conductors are materials in which some of the electrons are free electrons • • • Free electrons are “not bound” to the atoms; the electrons can move relatively freely through the material When a good conductor is charged in a small region, the charge readily distributes itself over the entire surface of the material The best five conducting materials are: 1. Silver 2. Copper 3. Gold 4. Aluminum 5. Zinc 7 1/25/17 Insulators • Electrical insulators are materials in which all of the electrons are bound to atoms These electrons can not move relatively freely through the material When a good insulator is charged in a small region, the charge is unable to move to other regions of the material Examples of good insulators: • • • • Teflon • PVC and other plastics • Ceramics/porcelain • Paper/cardboard • Rubber • Air Semiconductors • The electrical properties of semiconductors are somewhere between those of insulators and conductors • Examples of semiconductor materials include silicon and germanium • Semiconductors made from these materials are commonly used in making electronic chips • The electrical properties of semiconductors can be changed by the addition of controlled amounts of certain atoms to the material in a process called “doping” There are three ways to give an object a net charge 1. Charging by friction – This is useful for charging insulators. If you rub one material with another (say, a plastic ruler with a piece of paper towel), electrons have a tendency to be transferred from one material to the other. For example, rubbing glass with silk or saran wrap generally leaves the glass with a positive charge; rubbing a PVC rod with fur generally gives the rod a negative charge. 2. Charging by conduction – Useful for charging metals and other conductors. If a charged object touches a conductor, some charge will be transferred between the object and the conductor, charging the conductor with the same sign as the charge on the object. 3. Charging by induction (example follows) 8 1/25/17 Charging by Induction • Charging by induction requires no contact with the object inducing the charge • Assume we start with a neutral metallic sphere. • • The sphere has the same number of positive and negative charges A charged rubber rod is placed near the sphere • • It does not touch the sphere The electrons in the neutral sphere are redistributed Charging by Induction (con’t) • The sphere is grounded • • Some electrons can leave the sphere through the ground wire The ground wire is removed • • There will now be more positive charges that are not uniformly distributed The positive charge has been induced in the sphere Charging by Induction (con’t) • The rod is removed • • The electrons remaining on the sphere redistribute themselves and there is still a net positive charge on the sphere The charge is now uniformly distributed 9 1/25/17 Charge Rearrangement in Insulators • A process similar to induction can take place in insulators • The charges within the molecules of the material are rearranged Why does static electricity seem more apparent in winter? • In the winter, we see static electricity in many places: • • • • Taking clothes out of the dryer Taking a sweater off Combing/brushing your hair Getting a shock from a doorknob (or something else) after walking on carpet • Why is this more obvious in winter? • • • The air is much drier, and dry air is a good insulator That means once you charge something it will tend to stay charged (until it finds a way to discharge!) In more humid conditions, the water molecules in the air (which are polarized) will tend to remove excess charge quickly 10