Electromagnetism 1 PDF

Physics 1 Electrostatic Field. Electric Current. Assoc. prof. dr. Kristina Bočkutė Minutes from the last lecture o Ideal gas model, 𝑝𝑉 = 𝑛𝑅𝑇. o Isothermal, isochoric, isobaric processes, pV diagram. o First law is a generalization of the principle of conservation of energy to include energy tr...

Physics 1 Electrostatic Field. Electric Current. Assoc. prof. dr. Kristina Bočkutė Minutes from the last lecture o Ideal gas model, 𝑝𝑉 = 𝑛𝑅𝑇. o Isothermal, isochoric, isobaric processes, pV diagram. o First law is a generalization of the principle of conservation of energy to include energy transfer through heat as well as mechanical work. o Second law: it is impossible for any system to undergo a process in which it absorbs heat from a reservoir at a single temperature and converts the heat completely into mechanical work, with the system ending in the same state in which it began. No transfer of heat from cold body to hot body is possible. o Third law: A system's entropy (measure of disorder) approaches a constant value as its temperature approaches absolute zero. 2  Minutes from the last lecture According to the 1st law of thermodynamics, applied to a gas, the increase in the internal energy during any process: A) equals the heat input minus the work done on the gas B) equals the heat input plus the work done on the gas C) equals the work done on the gas minus the heat input D) is independent of the heat input E) is independent of the work done on the gas 3  Minutes from the last lecture The heat capacity at constant volume of an ideal gas depends on: A) the temperature B) the pressure C) the volume D) the number of molecules E) none of above 4  Minutes from the last lecture The pressure of an ideal gas is doubled during a process in which the heat given up by the gas equals the work done on the gas. As a result, the volume is: A) doubled B) halved C) unchanged D) need more information to answer E) the process is impossible 5 Outline Electric current: o Electric charges. Coulomb’s law. o Drift velocity, current, current density. o Electrostatic field. o Resistivity and resistance. o Electric dipole. o Electromotive force. o Electric flux. o Energy and power. o Gauss’s law. Direct-current circuits: o Electric potential. o Resistor connections: series, parallel. o Capacitors. o Kirchhoff’s Rules. o Dielectrics. Polarization. o RC circuits. o Summary. Other processes. o Summary. o Next lecture. 6 Electric Charge The ancient Greeks discovered 600 B.C. that after they rubbed amber with wool, the amber could attract other objects. Today we say that the amber has acquired a net electric charge or has become charged. There are two types of charges: Negative; Positive. Two positive charges or two negative charges repel each other. A positive charge and a negative charge attract each other. 7 Electric Charge The structure of atoms can be described in terms of three particles: the negatively charged electron, the positively charged proton, the uncharged neutron. Nucleus of atom is made up of protons and neutrons. It has a positive charge and attracts electrons (attractive electric forces). 𝑚𝑒 = 9.10938356(11) × 10−31kg 𝑚𝑝 = 1.672621898(21) × 10−27kg 𝑚𝑛 = 1.674927471(21) × 10−27kg 8  Electric Charge The negative charge of the electron has (within experimental error) exactly the same magnitude as the positive charge of the proton. In a neutral atom the number of electrons equals the number of protons in the nucleus, and the net electric charge is exactly zero. The number of protons or electrons in a neutral atom of an element is called the atomic number of the element. Ionization: Positive ion – one or more electrons removed; Negative ion – one or more electrons gained. 6 9 Electric Charge Principle of conservation of charge: The algebraic sum of all the electric charges in any closed system is constant. In any charging process, charge is not created or destroyed; it is merely transferred from one object to another. The magnitude of charge of the electron or proton is a natural unit of charge 𝒆. 𝑒 = 1.602176565(35) × 10−19C Every observable amount of electric charge is always an integer multiple of this basic unit. We say that charge is quantized. Charge of a macroscopic object is 𝑞 = 𝑁𝑝𝑒 − 𝑁𝑒𝑒 = 𝑁𝑝 − 𝑁𝑒 𝑒 Np and Ne are the number of protons and electrons contained in the object. 10  Electric Charge o Conductors are those materials through, or along which charge easily moves. o Insulators are those materials on or in which charges remain immobile. o Semiconductors are intermediate in their properties between good conductors and good insulators. Both insulators and conductors can be charged. They differ in the mobility of the charge. 11 Charging Insulators are often charged by rubbing. Charges on the rod are on the surface and that charge is conserved. Charge can be transferred upon contact, but it does not move around. Metals (conductors) can’t be charged by rubbing, but charge can be transferred by contact. 12 Discharging Touching a charged object discharges it. Humans are reasonably good conductors (full of salty water). Touching of metal with a hand discharges the metal (almost no charge is left on metal). The earth itself is a giant conductor because of its water and a variety of ions. Any object that is physically connected to the earth through a conductor is said to be grounded. The purpose of grounding objects, such as circuits and appliances, is to prevent the buildup of any charge on the objects. 13 Coulomb’s Law The magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. 𝑞1𝑞2 𝐹=𝑘 2 𝑟 k is a proportionality constant, r – distance between two point charges q1 and q2. Force magnitude is always positive. The SI unit of electric charge is coulomb (1 C). 𝑘 = 8.987551787 × 109N ∙ m2/C2 ≅ 8.988 × 109N ∙ m2/C2 14 Coulomb’s Law In principle we can measure the electric force F between two equal charges q at a measured distance r and use Coulomb’s law to determine the charge. One coulomb (C) is that amount of charge such that 1.602176634 x 10-19 C is equal to the magnitude of the charge on an electron or proton. Constant k can be expressed via electric constant (or permittivity constant) 𝜖0 and Coulomb’s law becomes: 1 𝑞1𝑞2 𝐹= 4𝜋𝜖0 𝑟2 1 −12 2 𝜖0 = 8.854 × 10 C /N ∙ m 2 = 9.0 × 109N ∙ m2/C2 4𝜋𝜖0 Typically, the range of charge is from microcoulomb (1𝜇C = 10−6C) to nanocoulomb (1nC = 10−9C) 15  Electric Field Electric force is long-range force, no contact is required for one charged particle to exert a force on another. Michael Faraday analysed magnets and came with the idea that the space itself around magnet is altered. This alteration is mechanism by which long-range force is exerted. A first alters or modifies the space around it, and particle B then comes along and interacts with this altered space. Faraday called this interaction a field (the physical entity exists at every point in space). 16 Electric Field The charge makes an alteration everywhere in space. Other charges then respond to the alteration at their position. The alteration of the space around a mass is called the gravitational field. Similarly, the space around a charge is altered to create the electric field. The electric field is the agent that exerts an electric force on a charged particle. In SI units, the unit of electric field magnitude is 1 newton per coulomb (1N/C). The force exerted on a point charge q0 by an electric field 𝐸: This equation is valid only for point charges. 17 Electric Field The location of the charge is called the source point, and the point P where we are determining the field is called the field point. 𝐸= 1 𝑞 4𝜋𝜖 0 𝑟2 Using the unit vector from point charge toward where the field is measured , we can write a vector equation that gives both the magnitude and direction of the electric field 𝐸 16 18  Electric Field The electric field of a point charge always points away from a positive charge but toward a negative charge. The force on a positive charge is in the direction of the field. The force on a negative charge is opposite the direction of the field. Since 𝐸 can vary from point to point, it is not a single vector quantity but rather an infinite set of vector quantities, one associated with each point in space. This is an example of a vector field. 19 Superposition of Electric Fields 20  Electric Field Lines An electric field line is an imaginary line or curve drawn through a region of space so that its tangent at any point is in the direction of the electric field vector at that point. The spacing between the field lines gives a general idea of the magnitude of 𝐸 at each point. Stronger 𝐸 - lines are closer. At any particular point, the electric field has a unique direction, so only one field line can pass through each point of the field. In other words, field lines never intersect. 21  Electric Dipoles An electric dipole is a pair of point charges with equal magnitude and opposite sign (a positive charge q and a negative charge -q) separated by a distance d. Water is an excellent solvent for ionic substances such as table salt (sodium chloride, NaCl) precisely because the water molecule is an electric dipole. The net force on an electric dipole in a uniform external electric field is zero. The magnitude of the torque: 𝑀 = (𝑞𝐸)(𝑑 sin 𝜙) The product of the charge q and the separation d is the magnitude of a quantity called the electric dipole moment, denoted by p: 𝑝 = 𝑞𝑑 The units of p are C ∙ m. 20 22 Electric Dipoles 23 Electric Dipoles 24 Electric Flux If we know the electric-field pattern, what can we determine about the charge distribution in that region? A closed surface through which an electric field passes is called a Gaussian surface (named after mathematician Carl Friedrich Gauss). A closed surface must be a surface in three dimensions, but for representation two- dimensional cross sections are used. 25 Electric Flux The amount of electric field passing through a surface is called the electric flux. The first conclusions, stated in terms of electric flux, are o There is an outward/inward flux through a closed surface around a net positive/negative charge. o There is no net flux through a closed surface around a region of space in which there is no net charge. o Charges outside the surface do not give a net electric flux through the surface. o The net electric flux is directly proportional to the net amount of charge enclosed within the surface but is otherwise independent of the size of the closed surface 26 Calculating Electric Flux 27 Gauss’s Law 28 Electrical Potential Energy 29 Electrical Potential Energy 30  Electrical Potential Energy The potential energy is always defined relative to some reference point where 𝑈 = 0. o If q and q0 have the same sign, the interaction is repulsive, this work is positive, and U is positive at any finite separation. o If the charges have opposite signs, the interaction is attractive, the work done is negative, and U is negative. The potential energy U is shared property, it does not matter which charge is moving. 31 Electric Potential (Electric) Potential V is potential energy per unit charge. 𝑈 𝑉= 𝑈 = 𝑞0 𝑉 𝑞0 Potential is a scalar. The SI unit is one volt (1V = 1J/C). The work per unit charge: 𝑊𝑎→𝑏 Δ𝑈 𝑈𝑏 𝑈𝑎 =− =− − = − 𝑉𝑏 − 𝑉𝑎 = 𝑉𝑎 − 𝑉𝑏 𝑞0 𝑞0 𝑞0 𝑞0 𝑽𝒂𝒃, the potential (in V) of a with respect to b, equals the work (in J) done by the electric force when a UNIT (1 C) charge moves from a to b. 𝑉𝑎𝑏 = 𝑉𝑎 − 𝑉𝑏 is called potential difference. In electric circuits the potential difference between two points is often called voltage. 32 Electric Potential 33 Electric Potential 34 Potential Gradient 35 A Conductor in Electrostatic Equilibrium A conductor is in electrostatic equilibrium when all charges are at rest. Excess charges are on the surface. o The electric field is zero at any interior point of a conductor in electrostatic equilibrium. o Any two points inside a conductor in electrostatic equilibrium are at the same potential. o The exterior electric field 𝐸 of a charged conductor is perpendicular to the surface. 36 Sources of Electric Potential An electric potential difference is created by separating positive and negative charge. The most common source of electric potential is a battery, which uses chemical reactions to separate charge. A battery consists of chemicals, called electrolytes, sandwiched between two electrodes made of different metals. Chemical reactions in the electrolytes transport ions (i.e., charges) from one electrode to the other. The work done per charge is called the emf of the battery: ℰ = 𝑊𝑐ℎ𝑒𝑚/𝑞. 37  Capacitors A capacitor is a device that stores electric potential energy and electric charge. Any two conductors separated by an insulator (or a vacuum) form a capacitor. Charging the capacitor: each conductor initially has zero net charge and electrons are transferred from one conductor to the other. Both conductors have the same magnitude charge but opposite sign, net charge is zero. 38 Capacitors The ratio of charge to potential is called the capacitance C of the capacitor: 𝐶= 𝑄 𝑉𝑎𝑏 The SI unit of capacitance is called one farad (1 F = 1 C/V). Capacitance is a measure of the ability of a capacitor to store energy. The greater the capacitance C of a capacitor, the greater the magnitude Q of charge on either conductor for a given potential difference Vab and hence the greater the amount of stored energy. 39 Capacitance Parallel-plate capacitor consists of two parallel conducting plates, each with area A, separated by a distance d that is small in comparison with their dimensions. The electric-field magnitude E for this arrangement is 𝐸= 𝜎 = 𝑄 (9.37) 𝜖0 𝜖0 𝐴 The potential difference between two plates: 1 𝑄𝑑 𝑉𝑎𝑏 = 𝐸𝑑 = 𝜖0 𝐴 Capacitance of parallel-plate capacitor: 𝑄 𝐴 𝐶= = 𝜖0 (9.39) 𝑉𝑎𝑏 𝑑 1 F = 1C2/N ∙ m 𝜖0 = 8.85 × 10−12F/m 40  Connection of Capacitors In a series connection the magnitude of charge on all plates is the same 𝑄 𝑄 𝑉𝑎𝑐 = 𝑉1 = 𝑉𝑏𝑐 = 𝑉2 = 𝐶1 𝐶2 1 + 1 𝑉= 1 + 1 𝑉𝑎𝑏 = 𝑉 = 𝑉1 + 𝑉2 = 𝑄 𝐶1 𝐶2 𝑄 𝐶1 𝐶2 The equivalent capacitance Ceq of the series combination is defined as the capacitance of a single capacitor for which the charge Q is the same as for the combination, when the potential difference V is the same. 1 1 1 1 = + + +⋯ 41 𝐶𝑒𝑞 𝐶1 𝐶2 𝐶3 Connection of Capacitors Parallel connection: two capacitors are connected in parallel between points a and b. In a parallel connection the potential difference for all individual capacitors is the same and is equal to 𝑉𝑎𝑏 = 𝑉. 𝑄1 = 𝐶1𝑉 𝑄2 = 𝐶2𝑉 𝑄 = 𝑄1 + 𝑄2 = 𝐶1 + 𝐶2 𝑉 𝑄 = 𝐶1 + 𝐶2 𝑉 The equivalent capacitance of a parallel combination equals the sum of the individual capacitances. 𝐶𝑒𝑞 = 𝐶1 + 𝐶2 + 𝐶3 + ⋯ 42 40 Energy Stored in Capacitors 43  Dielectrics Most capacitors have a nonconducting material, or dielectric, between their conducting plates. Placing a solid dielectric between the plates of a capacitor serves three functions: o it solves the mechanical problem of maintaining two large metal sheets at a very small separation without actual contact; o using a dielectric increases the maximum possible potential difference between the capacitor plates. o the capacitance of a capacitor of given dimensions is greater when there is a dielectric material between the plates than when there is vacuum. 44 42  Dielectrics The original capacitance when air or vacuum is between conductor plates is 𝐶0 = 𝑄/𝑉0. When dielectric is present, 𝐶 = 𝑄/𝑉. When the space between plates is completely filled by the dielectric, the ratio of C to C0 (equal to the ratio of V0 to V) is called the dielectric constant of the material, K: 𝐾= 𝐶 𝐶0 When the charge is constant 𝑉0 𝑉= 𝐾 The dielectric constant K is a pure number and always greater than unity. 45 Dielectrics 46  Induced Charges and Polarization When a dielectric material is inserted between the plates while the charge is kept constant, the potential difference between the plates decreases by a factor K. The same holds for electric field 𝐸0 𝐸= 𝐾 When E is smaller, the surface charge density must be smaller as well. As Q is constant, an induced charge of the opposite sign appears on each side of dielectric. It appears due to redistribution of charges within dielectric material, a phenomenon known as polarization. The induced surface charge is directly proportional to the electric-field magnitude E in the material. 47  Induced Charges and Polarization If the magnitude of induced charge per unit area is 𝜎𝑖, then the net surface charge on each side of the capacitor has magnitude (𝜎 − 𝜎𝑖). The field between the plates is related to the net surface density by 𝜎 𝜎 − 𝜎𝑖 𝐸0 = 𝐸0 = 𝜖0 𝜖0 The induced surface charge density 1 − 1 𝜎𝑖 = 𝜎 𝐾 The product 𝐾𝜖0 is called the permittivity of the dielectric, denoted by 𝜖: 𝜖 = 𝐾𝜖0 48 Induced Charges and Polarization The electric field within the dielectric can be expressed as 𝜎 𝐸= 𝜖 Capacitance of a parallel-plate capacitor, when dielectric is between plates 𝐴 𝐴 𝐶 = 𝐾𝐶0 = 𝐾𝜖0 = 𝜖 𝑑 𝑑 Electric energy density in a dielectric 𝑢 = 1 𝐾𝜖0𝐸 2 = 1 𝜖𝐸 2 2 2 In empty space where K=1 and 𝜖 = 𝜖0, Equations for the capacitance and electric energy density reduce to equations as for the parallel-plate capacitor in vacuum. For this reason 𝜖0 is sometimes called the “permittivity of free space” or the 49 “permittivity of vacuum”. Dielectric Breakdown When a dielectric is subjected to a sufficiently strong electric field, dielectric breakdown takes place, and the dielectric becomes a conductor. Lightning is a dramatic example of dielectric breakdown in air. The maximum electric-field magnitude that a material can withstand without the occurrence of breakdown is called its dielectric strength. 50  Gauss’s Law in Dielectrics The total charge enclosed, including both the charge on the capacitor plate and the induced charge on the dielectric surface, is 𝑄𝑒𝑛𝑐𝑙 = 𝜎 − 𝜎𝑖 𝐴, so Gauss’s law gives 𝜎 − 𝜎𝑖 𝐴 𝐸𝐴 = 𝜖0 Combining Equations we get 𝐾𝐸𝐴 = 𝜎𝐴 𝜖0 Gauss’s law for any surface whenever the induced charge is proportional to the electric field in the material is: 𝑄𝑒𝑛𝑐𝑙−𝑓𝑟𝑒𝑒 is the total free charge enclosed by Gaussian surface. 51  Ferroelectricity Ferroelectricity is a property of certain materials in which they possess a spontaneous electric polarization that can be reversed by the application of an external electric field. The term is used in analogy to ferromagnetism, in which a material exhibits a permanent magnetic moment. Ferroelectricity was discovered in 1920 in Rochelle salt by Valasek. The prefix ferro, meaning iron, was used to describe the property despite the fact that most ferroelectric materials do not contain iron. 52  Piezoelectricity Crystals which acquire a charge when compressed, twisted or distorted are said to be piezoelectric. o Quartz demonstrates this property and is extremely stable. Quartz crystals are used for watch crystals and for precise frequency reference crystals for radio transmitters. o Barium titanate, lead zirconate, and lead titanate are ceramic materials which exhibit piezoelectricity and are used in ultrasonic transducers as well as microphones. o The standard piezoelectric material for medical imaging processes has been lead zirconate titanate (PZT). Piezoelectric ceramic materials have found use in producing motions on the order of nanometers in the control of scanning tunneling microscopes. 53  Piezoelectricity o Charge separation between opposite faces of a crystal occurs when it is stressed. This is due to the regular atomic structure within the crystal. The effect also works in reverse. o The nature of the piezoelectric effect is closely related to the occurrence of electric dipole moments in solids (BaTiO3 and PZTs). 54 Summary o Electric charges can be positive or negative. Two positive charges or two negative charges repel each other. A positive charge and a negative charge attract each other. o Coulomb’s law describes the relation between the electric force and the distance between two point charges. o Electric field is the agent that exerts an electric force on a charged particle. o Gauss’s law relates the electric field flux to the charge enclosed by the surface. o The electric force caused by any collection of charges at rest is a conservative force. o Potential, V, is potential energy per unit charge. o A capacitor is any pair of conductors separated by an insulating material. o If a dielectric is introduced in the capacitor, capacitance of the device increases by a factor K. 55 Outline Electric current: o Electric charges. Coulomb’s law. o Drift velocity, current, current density. o Electrostatic field. o Resistivity and resistance. o Electric dipole. o Electromotive force. o Electric flux. o Energy and power. o Gauss’s law. Direct-current circuits: o Electric potential. o Resistor connections: series, parallel. o Capacitors. o Kirchhoff’s Rules. o Dielectrics. Polarization. o RC circuits. o Summary. Other processes. o Summary. o Next lecture. 56 Introduction An electric current consists of charges in controlled motion from one region to another. If the charges follow a conducting path that forms a closed loop, the path is called an electric circuit. As charged particles move within a circuit, electric potential energy is transferred from a source (such as a battery or generator) to a device in which that energy is either stored or converted to another form. 57  Charge Carriers The charges that move in a conductor are called the charge carriers. Electrons are the charge carriers in metals. In equilibrium, there is no net motion in metals even though they have free electrons moving around. If steady electric field𝐸 is established inside a conductor, electrons start to move in the direction of the electric force with the drift velocity (10-4 m/s). Consequence – net current in the conductor. 58  Current Electron current is the number of electrons per second that pass through a cross section of a wire or other conductor. The units of electron current is s-1. The applied electric field 𝐸 does work on the moving charges. The collisions taking place in conductor, increases the temperature of material (vibrational energy). In different current-carrying materials, the charges of the moving particles may be positive or negative. We define the current, denoted by I, to be in the direction in which there is a flow of positive charge. 59  Current We define the current through the cross-sectional area A to be the net charge flowing through the area per unit time. Thus, if a net charge dQ flows through an area in a time dt, the current I through the area is 𝑑𝑄 𝐼= 𝑑𝑡 The SI unit of current is the ampere; one ampere is defined to be one coulomb per second 1 A = 1 C/s. The current in the flashlight is about 0.5–1 A; The current in the wires of a car engine’s starter motor is around 200 A. Currents in radio and television circuits are usually expressed mA (10-3 A) or A (10-6 A); Currents in computer circuits are expressed in nA (10-9 A) or 60 pA (10-12 A).  Current, Drift Velocity, Current Density Assume that the free charges in the conductor are positive; then the drift velocity is in the same direction as the field. Suppose there are n moving charged particles per unit volume (concentration of particles in m-3). If each particle has a charge q, the charge dQ that flows out of the end of the cylinder during time dt is 𝑑𝑄 = 𝑛𝑞𝐴𝑣𝑑𝑑𝑡 61 Resistivity 62  Resistivity The reciprocal of resistivity is conductivity 𝜎 (its units are Ω ∙ m −1). Good electrical conductors, such as metals, are usually also good conductors of heat. Poor electrical conductors, such as ceramic and plastic materials, are also poor thermal conductors. A material that obeys Ohm’s law reasonably well is called an ohmic conductor or a linear conductor. The resistivity of a metallic conductor nearly always increases with increasing temperature 𝜌 𝑇 = 𝜌0 1 + 𝛼 𝑇 − 𝑇0 The reference temperature T0 is often taken as 0°C or 20°C; the temperature T may be higher or lower than T0. The factor  is called the temperature coefficient of resistivity. 63  Resistance For a conductor with resistivity 𝜌, the current density and electric field is related by Let V be the potential difference between the higher potential and lower-potential ends of the conductor, so that V is positive. The direction of the current is always from the higher potential end to the lower potential end (𝐸 points in the direction of decreasing electric potential). When 𝜌 is constant, the total current is proportional to the potential difference. 𝜌𝐿 𝑉= 𝐼 𝐴 64 Resistance The ratio of V to I for a particular conductor is called its resistance R: 𝑉 𝑅= 𝐼 Resistance of a particular conductor is related to the resistivity of its material by 𝜌𝐿 𝑅= 𝐴 Ohm’s law: 𝑉 = 𝐼𝑅 The real content of Ohm’s law is the direct proportionality (for some materials) of V to I or of J to E. SI unit of resistance is ohm 1Ω = 1V/A. Kilohm (103Ω) and megaohm (106Ω) are widely used. 𝑅 𝑇 = 𝑅0 1 + 𝛼 𝑇 − 𝑇0 65 Resistor A circuit device made to have a specific value of resistance between its ends is called a resistor. For a resistor that obeys Ohm’s law, a graph of current as a function of potential difference (voltage) is a straight line 66  Electromotive Force and Circuits For a conductor to have a steady current, it must be part of a path that forms a closed loop or complete circuit. o If a charge q goes around a complete circuit and returns to its starting point, the potential energy must be the same at the end of the round trip as at the beginning. BUT there is always a decrease in potential energy. o There must be some part of the circuit in which the potential energy increases. o The influence that makes current flow from lower to higher potential is called electromotive force (emf, an energy-per- unit-charge quantity), and a circuit device that provides emf is called a source of emf. o The SI unit of emf is volt. 67 Electromotive Force and Circuits Batteries, electric generators, solar cells, thermocouples, and fuel cells are all examples of sources of emf. An ideal source of emf maintains a constant potential difference between its terminals, independent of the current through it. The emf source provides an additional influence – a nonelectrostatic force 𝐹Ԧ𝑛, that pushes charge from b to a against the electric force 𝐹Ԧ𝑒. For the ideal emf, 𝐹Ԧ𝑒 and 𝐹Ԧ𝑛 are equal in magnitude but opposite in direction. The total work done on the charge is zero. 𝑉𝑎𝑏 = ℰ 68  Electromotive Force and Circuits For a complete circuit where elements are connected with the wire, the emf will be ℰ = 𝑉𝑎𝑏 = 𝐼𝑅 When a positive charge q flows around the circuit, the potential rise ℰ as it passes through the ideal source is numerically equal to the potential drop 𝑉𝑎𝑏 = 𝐼𝑅 as it passes through the remainder of the circuit. Once ℰ and R are known, this relationship determines the current in the circuit. 69  Internal Resistance The potential difference across a real source in a circuit is not equal to the emf. The charge moving through the material encounters internal resistance of the source, r. If this resistance behaves according to Ohm’s law, then the potential difference between the terminal is 𝑉𝑎𝑏 = ℰ − 𝐼𝑟 The potential 𝑉𝑎𝑏 called the terminal voltage, is less than ℰ because of the term 𝐼𝑟 representing the potential drop across the internal resistance r. For a real source of emf, the terminal voltage equals the emf only if no current is flowing through the source. ℰ ℰ − 𝐼𝑟 = 𝐼𝑅 𝐼= 𝑅+𝑟 70 Circuit Diagrams An important part of analysing any electric circuit is drawing a schematic circuit diagram. 71  Energy and Power As charge passes through the circuit element, the electric field does work on the charge. The potential-energy change for the amount of charge dQ passing through the element is 𝑉𝑎𝑏𝑑𝑄 = 𝑉𝑎𝑏𝐼𝑑𝑡. Dividing this expression by dt, we obtain the rate at which energy is transferred either into or out of the circuit element (power). 𝑃 = 𝑉𝑎𝑏𝐼 The power delivered to a pure resistor by the circuit is 2 𝑉𝑎𝑏 𝑃 = 𝑉𝑎𝑏𝐼 = 𝐼 2 𝑅 = 𝑅 The energy is dissipated in the resistor at a rate 𝐼2 𝑅. Every resistor has a power rating, the maximum power the device can dissipate without becoming overheated and damaged. 72  Energy and Power If the external circuit is connected to emf, the current I is leaving the source at the higher-potential terminal. Energy is being delivered to the external circuit at a rate 𝑃 = 𝑉𝑎𝑏𝐼 = ℰ𝐼 − 𝐼 2 𝑟 ℰ𝐼 term represents the rate of conversion of nonelectrical energy to electrical energy within the source. The term 𝐼 2 𝑟 is the rate at which electrical energy is dissipated in the internal resistance of the source. ℰ𝐼 − 𝐼 2 𝑟 is the net electrical power output of the source - the rate at which the source delivers electrical energy to the remainder of the circuit. 73 Energy and Power In case of charging of the source from the circuit, we get that the current is in opposite direction; the lower source is pushing current backwards the upper source. 𝑃 = 𝑉𝑎𝑏𝐼 = ℰ𝐼 + 𝐼 2 𝑟 Work is being done on, rather than by, the agent that causes the nonelectrostatic force in the upper source. ℰ𝐼 term represents the rate of conversion of electrical energy to nonelectrical energy in the upper source. ℰ𝐼 + 𝐼 2 𝑟 is the total electrical power input to the upper source. 74  Direct-Current (DC) Circuits In direct-current (dc) circuits the direction of the current does not change with time. Alternating current (ac) – the current oscillates back and forth. Used in household appliances. 75  Resistors in Series and Parallel Resistors turn up in all kinds of circuits, ranging from hair dryers and space heaters to circuits that limit or divide current or reduce or divide a voltage. o When several circuit elements such as resistors, batteries, and motors are connected in sequence with only a single current path between the points, we say that they are connected in series. o When each resistor provides and alternative path between the points – the resistors are connected in parallel. For any combination of resistors we can always find a single resistor that could replace the combination and result in the same total current and potential difference. The resistance of this single resistor is called the equivalent resistance of the combination Req 𝑉𝑎𝑏 𝑉𝑎𝑏 = 𝐼𝑅𝑒𝑞 𝑅𝑒 = 76 𝐼 Resistors in Series When resistors are connected in series, the current is the same for all of them. 𝑉𝑎𝑥 = 𝐼𝑅1 𝑉𝑥𝑦 = 𝐼𝑅2 𝑉𝑦𝑏 = 𝐼𝑅3 The potential difference Vab across the entire combination is the sum of these individual potential differences: 𝑉𝑎𝑏 = 𝑉𝑎𝑥 + 𝑉𝑥𝑦 + 𝑉𝑦𝑏 = 𝐼(𝑅1 + 𝑅2 + 𝑅3) 𝑉𝑎 = 𝑅1 + 𝑅2 + 𝑅3 𝐼 The equivalent resistance of any number of resistors: 𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3 + ⋯ The equivalent resistance of a series combination equals the sum of the individual resistances. 77  Resistors in Parallel If the resistors are in parallel, the current through each resistor need not be the same. But the potential difference between the terminals of each resistor must be the same and equal to Vab 𝑉𝑎𝑏 𝑉𝑎𝑏 𝑉𝑎𝑏 1 1 1 𝐼1 = 𝐼2 = 𝐼3 = 𝐼 = 𝐼1 + 𝐼2 + 𝐼3 = 𝑉𝑎𝑏 + + 𝑅1 𝑅2 𝑅3 𝑅1 𝑅2 𝑅3 The equivalent resistance of any number of resistors: 1 1 1 1 = + + +⋯ 𝑅𝑒𝑞 𝑅1 𝑅2 𝑅3 The reciprocal of the equivalent resistance of a parallel combination equals the sum of the reciprocals of the individual resistances. 78  Kirchhoff’s Rules Many practical resistor networks cannot be reduced to simple series-parallel combinations. In such case we need to apply techniques developed by the German physicist Gustav Robert Kirchhoff. A junction in a circuit is a point where three or more conductors meet. A loop is any closed conducting path. Kirchhoff’s junction rule (valid for any junction): Σ 𝐼=0 Kirchhoff’s loop rule (valid for any closed loop): Σ 𝑉=0 79 Kirchhoff’s Rules The junction rule is based on conservation of electric charge. No charge can accumulate at a junction, so the total charge entering the junction per unit time must equal the total charge leaving per unit time. The loop rule is a statement that the electrostatic force is conservative. When going around the loop and measuring potential difference, at the starting point the algebraic sum must be zero. 80 Sign Conventions for the Loop Rule o Assume the direction for the current in each branch of the circuit. o Start at any point of the loop and add all emfs and IR terms as you come to it. o The emf is considered to be positive when we travel through a source in the direction from - to +; and negative when we travel from + to -. o The IR term is negative/positive when we travel through a resistor in the same/opposite direction as the assumed current. 81 Electrical Measuring Instruments A current-measuring instrument is usually called an ammeter. It always measures the current passing through it. An ideal ammeter would have zero resistance, so including it in a branch of a circuit would not affect the current in that branch. A voltage-measuring device is called a voltmeter. A voltmeter always measures the potential difference between two points, and its terminals must be connected to these points. A voltmeter and an ammeter can be used together to measure resistance and power. 82 Electrical Measuring Instruments An alternative method for measuring resistance is to use an ohmmeter. It consists of a meter, a resistor, and a source (often a battery) connected in series. The resistance R to be measured is connected between terminals x and y. The potentiometer is an instrument that can be used to measure the emf of a source without drawing any current from the source. 83 RC Circuits In the simple act of charging or discharging a capacitor we find a situation in which the currents, voltages, and powers do change with time. A circuit that has a resistor and a capacitor in series is called an R-C circuit. When switch is closed (at t=0), the circuit is complete and capacitor stars to charge. At this time, the initial current is given by Ohm’s law 𝐼0 = 𝑣𝑎𝑏/𝑅. As the capacitor charges, its voltage 𝑣𝑏𝑐 increases and the potential difference 𝑣𝑎𝑏 across the resistor decreases, corresponding to a decrease in current. The sum of these two voltages is constant and equal to ℰ. 84 RC Circuits After a long time the capacitor is fully charged, the current decreases to zero, and 𝑣𝑎𝑏 across the resistor becomes zero. Then the entire battery emf ℰ appears across the capacitor and 𝑣𝑏𝑐 = ℰ. 𝑞 𝑣𝑎𝑏 = 𝑖𝑅 𝑣𝑏𝑐 = 𝐶 Using Kirchhof’s loop rule 𝑞 ℰ − 𝑖𝑅 − = 0 𝐶 The potential drops by an amount 𝑖𝑅 as we travel from a to b and by 𝑞/𝐶 as we travel from b to c. ℰ 𝑞 𝑖= − 𝑅 𝑅𝐶 85  RC Circuits. Discharging a Capacitor Charged capacitor Battery removed, Capacitor discharges 𝑞 = 𝑄0. through the resistor , 𝑞 → 0. 𝑑𝑞 𝑞 𝑖= =− 𝑑𝑡 𝑅𝐶 The current i is now negative; this is because positive charge q is leaving the left-hand capacitor plate 86  RC Circuits. Discharging a Capacitor Capacitor charge q as a function of time during a discharge of capacitor: 𝑞 = 𝑄0 𝑒−𝑡/𝑅𝐶 The instantaneous current i is the derivative of this with respect to time: 𝑑𝑞 𝑄0 −𝑡 /𝑅𝐶 𝑖= =− 𝑒 = 𝐼0𝑒 𝑑𝑡 𝑅𝐶 −𝑡 /𝑅 The total energy supplied by the battery during charging of the capacitor equals the battery emf ℰ multiplied by the total charge 𝑄𝑓, or ℰ𝑄𝑓. Half of the energy supplied by the battery is stored in the capacitor, and the other half is dissipated in the resistor. 87 Thermoelectric Effect o The thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa. o A thermoelectric device creates a voltage when there is a different temperature on each side. o Conversely when a voltage is applied to it, it creates a temperature difference (known as the Peltier effect). This effect can be used to generate electricity, to measure temperature, to cool objects, or to heat them or cook them. 88  Thermoelectric Effect Thermoelectric effect or thermoelectricity encompasses three separately identified effects: o the Seebeck effect (conversion of temperature difference into electricity), o the Peltier effect (potential difference applied across a thermocouple causes a temperature difference), o and the Thomson effect (the evolution or absorption of heat when electric current passes through a circuit composed of a single material that has a temperature difference along its length). 89 Electric Discharge o Electric discharge describes any flow of electric charge through a gas, liquid or solid. o Electric discharge in gases occurs when electric current flows through a gaseous medium. Depending on several factors, the discharge may radiate visible light. 90 Electric Discharge Townsend discharge (dark discharge), below the breakdown voltage. At low voltages, current is due to cosmic rays. As the applied voltage is increased, the free electrons carrying the current gain enough energy to cause further ionization, causing an electron avalanche. The current increases from fA to A. The glow becomes visible near the breakdown voltage. 91  Electric Discharge Glow discharge occurs once the breakdown voltage is reached. The voltage across the electrodes suddenly drops and the current increases to mA range. At lower currents, the voltage across the tube is almost current-independent (normal glow). At higher currents the normal glow turns into abnormal glow, the voltage across the tube gradually increases, and the glow discharge covers more and more of the surface of the electrodes. 92 Electric Discharge Arc discharge occurs in the ampere range of the current; the voltage across the tube drops with increasing current. High-current switching tubes, e.g. triggered spark gap, ignitron, thyratron and krytron, high-power mercury-arc valves and high-power light sources, e.g. mercury-vapor lamps and metal halide lamps, operate in this range. 93 Corona Discharge A corona discharge is an electrical discharge caused by the ionization of a fluid such as air surrounding a conductor carrying a high voltage. A corona occurs at locations where the strength of the electric field (potential gradient) around a conductor exceeds the dielectric strength of the air. Applications: manufacture of ozone, sanitization of pool water, photocopying, air ionisers, etc. 94  Plasma Plasma is a gas consisting of charged ions and electrons. Over a large volume the plasma is quasi-neutral, meaning that the number of free negative charges is equal to the number of free positive charges. Ionization takes place and is maintained because the gas: o is very hot, so that collisions between atoms are sufficiently strong to remove electrons; o is very rarefied, so that electrons, once removed, will hardly encounter an ion with which to recombine; o is subjected to an external source of energy, such as strong electric fields or radiation capable to tear away the electrons. 95 Plasma Plasma is also called the "fourth state of matter", complementing solids, liquids and gases. o Plasmas possess all the dynamical properties of fluids, like turbulence. o Since they are formed of free charged particles, plasmas conduct electricity. They both generate and respond to electromagnetic fields, which leads to collective effects. o This means that the motions of all other particles influence the motion of each charged particle. Collective behaviour is a key concept in the definition of plasma. 96 Plasma Properties o The plasma approximation applies when the plasma parameter, Λ, representing the number of charge carriers within a sphere (called the Debye sphere whose radius is the Debye screening length) surrounding a given charged particle, is sufficiently high as to shield the electrostatic influence of the particle outside of the sphere 𝜖0𝑘𝑇𝑒 Λ = 4𝜋𝜆3 𝑛 𝜆𝐷 = 𝐷 𝑒 𝑛𝑒 𝑞𝑒2 𝜆𝐷 is the Debye length, 𝜖0 is the permittivity of free space, k is the Boltzmann constant, qe is the charge of an electron, Te is the temperature of the electrons, ne is the density of electrons. 97  Plasma Properties o Bulk interactions: Λ is short compared to the physical size of the plasma. This criterion means that interactions in the bulk of the plasma are more important than those at its edges, where boundary effects may take place. When this criterion is satisfied, the plasma is quasineutral. o Plasma frequency: The electron plasma frequency (measuring plasma oscillations of the electrons) is large compared to the electron-neutral collision frequency (measuring frequency of collisions between electrons and neutral particles). When this condition is valid, electrostatic interactions dominate over the processes of ordinary gas kinetics. 𝑛 𝑒 2 𝑒 𝜔𝑝2 = 𝜖0𝑚 98 Plasma Globe A plasma globe is usually a clear glass orb filled with a mixture of various gases (most commonly helium and neon) at low pressure (below 0.01 atm) and driven by high- frequency alternating current at approximately 35 kHz, 2–5 kV, generated by a high- voltage transformer. 99 Summary 100  Next Lecture Electromagnetic Induction Magnetic field in vacuum and material 101