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M. Electricity 1. Electric Charge • Electric charge is a fundamental attribute of particles. • Electrostatics are defined as the interactions between electric charges that are at rest (or nearly so). • The figure shows some experiments used to demonstrate electrostatics. 2005 Pearson Education South Asia Pte Ltd M. Electricity 1. Electric Charge Fig. 21.1 2005 Pearson Education South Asia Pte Ltd M. Electricity 1. Electric Charge • Electrostatics experiments show that there are exactly two kinds of electric charge, negative and positive. • Two positive charges or two negative charges repel each other. A positive charge and a negative charge attract each other. 2005 Pearson Education South Asia Pte Ltd M. Electricity 1. Electric Charge • CAUTION – “Like charges” only mean that two charges have the same algebraic sign (both positive or both negative). – “Opposite charges” means that the electric charges on both objects have different signs (one positive and the other negative). • A technological application is in a laser printer; the figure shows a schematic diagram of such a printer in operation. 2005 Pearson Education South Asia Pte Ltd M. Electricity 2005 Pearson Education South Asia Pte Ltd Fig. 21.2 M. Electricity 1. Electric Charge Electric Charge and the Structure of Matter • The atomic structure consists of three particles: the negatively charged electron, the positively charged proton, and the uncharged neutron. • Protons and neutrons make up the nucleus while electrons orbit it from a distance. • The figure shows how changes in the atomic structure of lithium determines its net electric charge. 2005 Pearson Education South Asia Pte Ltd M. Electricity 1. Electric Charge Electric Charge and the Structure of Matter 2005 Pearson Education South Asia Pte Ltd Fig. 21.4 M. Electricity 1. Electric Charge Electric Charge and the Structure of Matter • Atomic number is defined as the number of protons or electrons in a neutral atom of an element. • A positive ion is formed by removing one or more electrons from an atom; a negative ion is one that has gained one or more electrons. This process is called ionization. 2005 Pearson Education South Asia Pte Ltd M. Electricity 1. Electric Charge Electric Charge and the Structure of Matter • When the total number of protons equals the total number of electrons in a macroscopic body, its total charge is zero and the body as a whole is electrically neutral. • When we speak of the charge of a body, we always mean its net charge. 2005 Pearson Education South Asia Pte Ltd M. Electricity 1. Electric Charge Electric Charge and the Structure of Matter • The 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 but merely transferred from one body to another. 2005 Pearson Education South Asia Pte Ltd M. Electricity 1. Electric Charge Electric Charge and the Structure of Matter • The magnitude of charge of the electron or proton is a natural unit of charge. • Every observable amount of electric charge on any macroscopic body is always either zero or an integer multiple (positive or negative) of this basic unit, the electron charge – quantization of charge. 2005 Pearson Education South Asia Pte Ltd M. Electricity 2. Conductors, Insulators, and Induced Charges • Conductors of electricity are materials that permit electric charge to move easily through them; Insulators do not. • Most metals are good conductors while most nonmetals are insulators. Semiconductors are intermediate in their properties between good conductors and good insulators. • The figure shows the use of copper as a good conductor, and glass and nylon as good insulators. 2005 Pearson Education South Asia Pte Ltd M. Electricity 2. Conductors, Insulators, and Induced Charges • In a metallic conductor, the mobile charges are always negative electrons. • In ionic solutions and ionized gases, both positive and negative charges are mobile. 2005 Pearson Education South Asia Pte Ltd M. Electricity 3. Coulomb’s Law • Coulomb’s Law states that: 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. • The directions of the forces the two charges exert on each other are always along the line joining them, as shown in the figure. 2005 Pearson Education South Asia Pte Ltd M. Electricity 3. Coulomb’s Law 2005 Pearson Education South Asia Pte Ltd Fig. 21.9 M. Electricity 3. Coulomb’s Law • Coulomb’s Law is usually written as: where ∈ 0 = 8 . 854 × 10 − 12 C 2 / N • m2 1 = 9 . 0 × 10 9 N • m 2 / C 4π ∈0 2005 Pearson Education South Asia Pte Ltd 2 M. Electricity 3. Coulomb’s Law • The most fundamental unit of charge is the magnitude of the charge of an electron or proton, denoted by e, where e = 1.602176462(63) x 10-19 C 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 21.4 Vector addition of electric forces in a plane In Fig. 21.12, two equal positive point charges q1 = q2 = 2.0 µC interact with a third point charge Q = 4.0 µC. Find the magnitude and direction of the total (net) force on Q. Fig. 21.12 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.4 Electric Field and Electric Forces r E at a point is defined as the • The electric field r electric force F0 experienced by a test charge q0 at the point, divided by the charge q0. • That is, the electric field at a certain point is equal to the electric force per unit charge experienced by a charge at that point: • SI unit is 1 newton per coulomb (1 N/C). 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.4 Electric Field and Electric Forces • The figure shows how a charged body creates an electric field in the space around it. Fig. 21.13 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.4 Electric Field and Electric Forces • The electric field that a charged body produces exists at all points in the region around itself. • The electric force is an “action-at-a-distance” force that acts across empty space without needing any matter to transmit it through the intervening space. • The electric force on a charged body is exerted by the electric field created by other charged bodies. 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.4 Electric Field and Electric Forces • The figure shows the electric force exerted by an electric field on a point charge. r r F0 = q0 E (21.4 ) Fig. 21.14 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.4 Electric Field and Electric Forces r • The electric field E, or electric force per unit charge, is useful because it does not depend on the charge of the body on which the electric force is exerted. • CAUTION The electric force experienced by a test charge q0 can vary from point to point, so the electric field can also be different at different points. For this reason, Eq. (21.4) can be used only to find the electric force on a point charge, not on large bodies where the electric field may vary with different locations on the body. 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.4 Electric Field and Electric Forces • The source point S is the location of the point charge that is producing the electric field we are investigating. • The field point P is the location where we are determining the field. • The unit vector r̂ denotes the direction along the line from source point to field point. • The figure illustrates these terms. 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.4 Electric Field and Electric Forces Fig. 21.15 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.4 Electric Field and Electric Forces r • The magnitude and direction of the electric field E can be given by the vector equation: • The vector field is thus the infinite set of vectors associated with every point in space in an electric field. • The figure shows how a point charge produces an electric field at all points in space. 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.4 Electric Field and Electric Forces Fig. 21.16 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.4 Electric Field and Electric Forces • The field vectors point outwards from the charge for positive charges, and inward for negative charges. • When the magnitude and direction of the field (and hence its vector components) are the same everywhere throughout an electric field, the field is uniform. 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 21.7 Electron in a uniform field Fig. 21.18 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 21.8 An electron trajectory If we launch an electron into the electric field of Example 21.7 with an initial horizontal velocity v0 (Fig. 21.19), what is the equation of its trajectory? Fig. 21.19 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.5 Electric-Field Calculations • In most realistic situations that involve electric fields and forces, charge is distributed over space. • Thus, the principle of superposition of electric fields is used to find the total electric field at a given point P, set up by the individual electric fields produced by each point charge in the field: r r F0 r r r E = = E1 + E 2 + E 3 + L q0 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 21.9 Field of an electric dipole Fig. 21.20 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 21.10 Field of a ring of charge A ring-shaped conductor with a radius a carries a total charge Q uniformly distributed around it (Fig. 21.21). Find the electric field at a point P that lies on the axis of the ring at a distance x from its center. 2005 Pearson Education South Asia Pte Ltd Fig. 21.21 M. Electricity Example 21.11 Field of a line of charge Positive electric charge Q is distributed uniformly along a line with length 2a, lying along the y-axis between y = -a and y = +a. (This might represent one of the charged rods in Fig. 21.1.) Find the electric field at point P on the x-axis at a distance x from the origin. 2005 Pearson Education South Asia Pte Ltd Fig. 21.22 M. Electricity Example 21.12 Field of a uniformly charged disk Find the electric field caused by a disk of radius R with a uniform positive surface charge density (charge per unit area) σ, at a point along the axis of the disk a distance x from its center. Assume that x is positive. 2005 Pearson Education South Asia Pte Ltd Fig. 21.23 M. Electricity 21.6 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. • This is illustrated by the following figure. 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.6 Electric Field Lines Fig. 21.25 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.6 Electric Field Lines r • They show the direction of E at each point, and their spacing indicates the corresponding magnitude. • At any particular point, the electric field has a unique direction, so field lines never intersect. • The figure shows the field maps of the electric field lines for three different charge distributions. 2005 Pearson Education South Asia Pte Ltd M. Electricity 21.6 Electric Field Lines Fig. 21.26 2005 Pearson Education South Asia Pte Ltd M. Electricity 23.1 Electric Potential Energy • Potential Energy U is the work done when a conservative force acts on a particle to move it from one point to another. 2005 Pearson Education South Asia Pte Ltd M. Electricity 23.1 Electric Potential Energy Electric Potential Energy in a Uniform Field • The force on a test charge that moves from one point to another is constant and independent of its location. • The work done by this force is also independent of the particle’s path, as shown in Fig. 23.1. 2005 Pearson Education South Asia Pte Ltd M. Electricity 23.1 Electric Potential Energy 2005 Pearson Education South Asia Pte Ltd Fig. 23.1 M. Electricity 23.1 Electric Potential Energy 2005 Pearson Education South Asia Pte Ltd Fig. 23.2 M. Electricity 23.1 Electric Potential Energy 2005 Pearson Education South Asia Pte Ltd Fig. 23.3 M. Electricity 23.1 Electric Potential Energy Electric Potential Energy of Two Point Charges • The concept of electric potential energy can also be applied to a point charge in any electric field caused by a static charge distribution. • This is shown in Fig. 23.4 where displacement is along a radial line in the electric field set up by a single, stationary charge. 2005 Pearson Education South Asia Pte Ltd M. Electricity 23.1 Electric Potential Energy 2005 Pearson Education South Asia Pte Ltd Fig. 23.4 M. Electricity 23.1 Electric Potential Energy • The potential energy U when the test charge q0 is at any distance r from charge q is thus 2005 Pearson Education South Asia Pte Ltd M. Electricity 23.1 Electric Potential Energy • Potential energy is always defined relative to some reference point where U = 0. • It is a shared property of the two charges q and q0 because it is a consequence of the interactions (attractive or repulsive) between these two bodies. • Fig. 23.6 is the graphical illustration of how U varies with these interactions. 2005 Pearson Education South Asia Pte Ltd M. Electricity 23.1 Electric Potential Energy 2005 Pearson Education South Asia Pte Ltd Fig. 23.6 M. Electricity 23.1 Electric Potential Energy The total electric field at each point is the vector sum of the fields due to the individual charges, and the total work done on the charge q0 during any displacement is the sum of the contributions from the individual charges. 2005 Pearson Education South Asia Pte Ltd Fig. 23.7 M. Electricity 23.1 Electric Potential Energy • For every electric field due to a static charge distribution, the force exerted by that field is conservative. 2005 Pearson Education South Asia Pte Ltd M. Electricity 23.2 Electric Potential • Generally, potential is potential energy per unit charge. • SI units: volt (1 V). • Vab, the potential of a with respect to b, is the work done by the electric force when a UNIT charge moves from a to b. In other words, it is also the work that must be done against the electric force to move a UNIT charge slowly from b to a. 2005 Pearson Education South Asia Pte Ltd M. Electricity 23.2 Electric Potential 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.2 Resistivity • Resistivity ρ is described by Ohm’s law: • SI units: ohm-meter, Ω·m. • Conductivity – the reciprocal of resistivity, with units as (Ω·m)-1. 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.2 Resistivity 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.2 Resistivity • Over a small temperature range (up to 100°C or so), the resistivity of a metal can be represented approximately by the equation • Fig. 25.6 illustrates this relationship graphically. 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.2 Resistivity Fig. 25.6 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.3 Resistance • In a conductor with resistivity ρ, the direction of the current is always from the higher-potential end to the lower-potential end. Fig. 25.7 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.3 Resistance • The resistance R of a particular conductor is related to the resistivity ρ of its material by • Generally, • SI units: ohm, Ω 1 Ω = 1 V/A 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.3 Resistance • A circuit device made to have a specific value of resistance between its ends is called a resistor. Fig. 25.8 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.3 Resistance 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.3 Resistance Fig. 25.9 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.4 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. • If the circuit is incomplete, the current flows for only a very short time. Fig. 25.11 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.4 Electromotive Force and Circuits • Fig. 25.13 is a schematic diagram of an ideal source of emf that maintains a potential difference between conductors a and b, called the terminals of the device. Fig. 25.13 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.4 Electromotive Force and Circuits • Fig. 25.14 is the schematic diagram of the same ideal source of emf in a complete circuit. Fig. 25.14 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.4 Electromotive Force and Circuits 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 25.5 A source on open circuit Figure 25.16 shows a source (a battery) with an emf ξ of 12 V and an internal resistance r of 2 Ω. (For comparison, the internal resistance of a commercial 12-V lead storage battery is only a few thousandths of an ohm.) The wires to the left of a and to the right of the ammeter A are not connected to anything. Determine the readings of the idealized voltmeter V and the idealized ammeter A. 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 25.5 A source on open circuit 2005 Pearson Education South Asia Pte Ltd Fig. 25.16 M. Electricity Example 25.6 A source in a complete circuit Using the battery in Conceptual Example 25.5, we add a 4-Ω resistor to form the complete circuit shown in Fig. 25.17. What are the voltmeter and ammeter readings now? Fig. 23.17 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 25.7 Using voltmeters and ammeters The voltmeter and ammeter in Example 25.6 are moved to different positions in the circuit. What are the voltmeter and ammeter readings in the situations shown in a) Fig. 25.18a and b) Fig. 25.18b? 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 25.7 Using voltmeters and ammeters 2005 Pearson Education South Asia Pte Ltd Fig. 23.18 M. Electricity Example 25.8 A source with a short circuit Using the same battery as in the preceding three examples, we now replace the 4-Ω resistor with a zero-resistance conductor, as shown in Fig. 25.19. What are the meter readings now? Fig. 25.19 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.4 Electromotive Force and Circuits • The net change in potential energy for a charge q making a round trip around a complete circuit must be zero. • Hence the net change in potential around the circuit must also be zero. • Fig. 25.20 is a graph showing how the potential varies as we go around the complete circuit of Fig. 25.17. 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.4 Electromotive Force and Circuits Fig. 25.20 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.5 Energy and Power in Electric Circuits • In electric circuits we are most often interested in the rate at which energy is either delivered to or extracted from a circuit element. • The time rate of energy transfer is power, denoted by P: • Units: watt, W 1 W = 1 J/s 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.5 Energy and Power in Electric Circuits • The power input to the circuit element between a and b is P = (Va – Vb)I = VabI. Fig. 25.21 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.5 Energy and Power in Electric Circuits • If the circuit element is a resistor, the rate of transfer of electric potential energy into it is • The internal energy of the material will increase and energy (heat) will be dissipated in the resistor at a rate I2R. • Every resistor has a power rating, the maximum power the device can dissipate without becoming overheated and damaged. 2005 Pearson Education South Asia Pte Ltd M. Electricity 25.5 Energy and Power in Electric Circuits Power Output of a Source 2005 Pearson Education South Asia Pte Ltd Fig. 25.22 M. Electricity 25.5 Energy and Power in Electric Circuits Power Input to a Source Fig. 25.23 2005 Pearson Education South Asia Pte Ltd M. Electricity 26.1 Resistors in Series and Parallel Fig. 26.1 Four different ways of connecting three resistors 2005 Pearson Education South Asia Pte Ltd M. Electricity 26.1 Resistors in Series and Parallel • When several resistors R1, R2, R3,…are connected in series, the equivalent resistance Req is the sum of the individual resistances. • The same current flows through all the resistors in a series connection. • In general: • The equivalent resistance of any number of resistors in series equals the sum of their individual resistances and is greater than any individual resistance. 2005 Pearson Education South Asia Pte Ltd M. Electricity 26.1 Resistors in Series and Parallel • When several resistors are connected in parallel, the reciprocal of the equivalent resistance Req is the sum of the reciprocals of the individual resistances. • All resistors in a parallel connection have the same potential difference between their terminals. • In general: • The equivalent resistance is always less than any individual resistance. 2005 Pearson Education South Asia Pte Ltd M. Electricity 26.1 Resistors in Series and Parallel • For the special case of two resistors in parallel: R1R2 Req = (two resistors in parallel) (26.3) R1 + R2 • Because Vab = I1R1 = I 2 R2 , it follows that I1 I 2 = (two resistors in parallel) (26.4) R1 R2 • Currents carried by two resistors in parallel are inversely proportional to their resistances. More current goes through the path of least resistance. 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 26.1 Equivalent resistance Compute the equivalent resistance of the network in Fig. 26.3a, and find the current in each resistor. The source of emf has negligible internal resistance. Fig. (26.3) 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 26.1 (SOLN) • The 6-Ω and 3-Ω resistors are in parallel with an equivalent resistance of 2Ω. The remaining series combination of the 2-Ω resistor with the 4-Ω resistor is reduced to 6Ω. • To find the current, reverse the steps used to reduce the network. (Fig. 26.3) 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 26.2 Series versus parallel combinations Two identical light bulbs are connected to a source with ε=8V and negligible internal resistance. Each light bulb has a resistance R=2Ω. Find the current through each bulb, the potential difference across each bulb and the power delivered to each bulb and to the entire network if the bulbs are connected a) in series, b) in parallel c) Suppose one of the bulbs burns out (filament breaks and no current can flow through it) What happens to the other bulb in the series case? Parallel case? 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 26.2 Series versus parallel combinations Fig. 26.4 Circuit diagrams for two light bulbs (a) in series and (b) in parallel 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 26.2 (SOLN) Identify and Set up • Simple series and parallel connections are involved. Use Eq.(25.18) to find the power 2 2 delivered to each resistor: P = I R = V / R Execute • (a) Potential difference across each of the identical bulbs is the same: Vab = Vbc = IR = (2 A)(2Ω) = 4V P = I 2 R = (2 A) 2 (2Ω) = 8W • Ptotal = 2 P = 16W 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 26.2 (SOLN) Execute • (b) Similarly, Vde 8V = = 4A I= R 2Ω 2 2 Vde (8V ) P= = = 32W R 2Ω 2 Ω • For the parallel case, the value of Req is less, thus Ptotal = V 2 / Req is greater. 2005 Pearson Education South Asia Pte Ltd M. Electricity Example 26.2 (SOLN) Execute • (c) In the series case there will be no current at all in the circuit and neither bulbs will glow. • In the parallel case the potential difference across either bulb remains equal to 8V even if one of the bulbs burns out. 2005 Pearson Education South Asia Pte Ltd M. Electricity 26.2 Kirchhoff’s Rules • Kirchhoff’s junction rule is based on conservation of charge. This states that the algebraic sum of the currents into any junction must be zero. • Kirchhoff’s loop rule is based on conservation of energy and the conservative nature of electrostatic fields. It states that the algebraic sum of potential differences around any loop must be zero 2005 Pearson Education South Asia Pte Ltd M. Electricity 26.2 Kirchhoff’s Rules • Kirchhoff’s loop rule is based on conservation of energy and the conservative nature of electrostatic fields. It states that the algebraic sum of potential differences around any loop must be zero. • Careful use of consistent sign rules is essential in applying Kirchoff’s rules. 2005 Pearson Education South Asia Pte Ltd M. Electricity 26.2 Kirchhoff’s Rules • A junction in a circuit is a point where three or more conductors meet. Junctions are also called nodes or branch points. • A loop is any closed conducting path. Fig. 26.7 Kirchhoff’s junction rule states that as much current flows into a junction as flows out of it 2005 Pearson Education South Asia Pte Ltd M. Electricity 26.2 Kirchhoff’s Rules Fig. 26.6 Two networks that cannot be reduced to simple series-parallel combinations of resistors 2005 Pearson Education South Asia Pte Ltd