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# PHY204H Study Guide for Exam 2 PHY 204H

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Physics Study Guide for Exam 2: Chapters 23, 24 and 25 Chapter 23: Electric Potential • Electric Potential is a scalar field Potential Difference • The electrostatic force exerted by a point charge on another point charge is directed along the line joining the charges and varies inversely with the square of their separation distance ▯▯ • Definition-Potential Difference: ???????? = ▯▯= − ????∙ ???????? • For a finite displacement from point a to point b the change in potential is: ∆▯ ▯ Definition-Finite Potential Difference: ∆???? = ???????? − ???????? = ▯▯ = ▯ ???? ∙ ???????? • The potential difference V b V ia the negative of the work per unit charge done by the electric field on a test charge when the test charge moves from point a to point b (along any path) • The function V is called the electric potential; it is often referred to as the potential • The potential V is a function of position. Unlike the electric field, V is a scalar function. • As with potential energy U, only differences in the potential V are physically significant. We are free to choose the potential to be zero at any convenient point • Relation between Potential Energy and Potential: ???? = ???????????? Continuity of V: • The potential function is continuous everywhere, except at points where the electric field is infinite (points occupied by a point charge or a line charge). Units: • The SI unit for potential and potential difference is the joule per coulomb, called the volt (V): 1 V=1 J/C • The potential difference between two points (measured in volts) is commonly referred to as the voltage between two points • The unit of the electric field is equal to one volt per meter: 1 N/C = 1 V/m • We may think of the magnitude of the electric field E as either a force per unit charge or as a rate of change of potential (V) with respect to distance in a given direction • A unit of energy is defined as the product of the fundamental charge unit e and a volt. This particularly useful unit is called an electron volt (eV) • The Electron Volt: 1 eV = 1.60X10 CV = 1.60X10 J -19 Potential and Electric Fields: • The charge accelerates toward a region where its electric potential energy is less • The electric potential energy U is related to the electric potential V by U = qV, so for a positive charge a region where the charge has lower potential energy U is also a region of lower electric potential V • The electric field ???? points in the direction in which the potential V decreases most rapidly • The point where the value of potential is specified is called the reference point for the potential function V • The potential at a field point b is obtained by calculation ???? − 0 = ▯ − ▯ ???? ∙ ????????, where the potential at a is taken to be zero. The integral is to be evaluated along any path from a to b. Potential Due to a System of Point Charges ▯▯ • Coulomb Potential: ???? = ▯ • Electrostatic Potential Energy of a Two-Charge System: The potential energy U of a point charge q’ placed a distance r from the ▯ ▯▯▯ ▯▯▯▯ point q is ???? = ???? ???? = ???? ▯ = ▯ • At a very great distance from q, the potential energy of the particle that has charge q’ approaches zero so its kinetic energy approaches kqq’/r • The work an external agent must to do move a test charge qo from rest at point P, a distance r from q, is kqoq/r. • The work per unit charge is kq/r, which is the electric potential V at point P relative to the potential an infinite distance from P. • The potential at the field point due to the presence of several point charges is the sum of the potentials due to each of these charges separately. • The Potential Due to a System of Point Charges is ???? = ▯▯▯▯ ▯▯ • PROBLEM-SOLVING STRATEGY o Calculating V Using Equation of the Potential Due to s System of Point Charges o PICTURE We can use Equation 23-10 to calculate the potential at a field point due to any collection of point charges if each point charge is a finite distance from every other point charge. o SOLVE § 1. Sketch the charge configuration and include suitable coordinate axes. Label each point charge with a distinct symbol, such as q . Draw a straight line from each point 1 charge q to the field point P and label it with a suitable i symbol, such as r .iP careful drawing can be very helpful in relating the distances of interest to the distances given in the problem statement. ▯▯▯ § 2. Use the formula ???? = ▯ ▯▯to calculate the potential at P due to the presence of the point charges o CHECK If the field point is arbitrarily chosen, take the limit as the field point goes to infinity. In that limit, the potential must approach zero. Computing the Electric Field From the Potential • A vector that points in the direction of the greatest change in a scalar function and that has a magnitude equal to the derivative of that function with respect to the distance in that direction is called the gradient of the function • ???????? = − ▯▯(▯) ▯▯ • Potential Due to a Continuous Charge Distribution: ???? = ▯▯▯ ▯ o Assumes that V=0 at an infinite distance from the charges, so we cannot use it for any charge distributions of infinite extent V on the Axis of a Charged Ring: ▯▯ • Potential on the Axis of a Charged Ring: ???? = ▯ ▯ ▯ ▯▯ o When z (the distance from the ring) is much greater than a (the radius of the ring), then the potential approaches kQ/IzI the same as the potential due to a point charge Q at the origin V on the Axis of a Uniformly Charged Disk: • Potential on the Axis of a Uniformly Charged Disk: ▯▯ ???? = 2???????????? ???? 1 + − 1 ▯▯ V Due to an Infinite Plane of Charge: • Potential Near an Infinite Plane of Charge: ???? = ???????? − 2???????????? ???? V Inside and Outside a Spherical Shell of Charge: ▯▯ (???? ≥ ????) ▯ • Potential Due to a Think Spherical Shell: ???? = ▯▯ ▯ (???? ≥ ????) • A region of zero electric field implies that the potential field is uniform throughout the region V Due to an Infinite Line Charge: • Potential Due to a Uniform Line Charge of Infinite Length: ▯▯▯▯ ???? = 2????????ln ( ▯ ) Equipotential Surfaces • Because there is no electric field inside the material of a conductor that is in static equilibrium, the value of the potential is the same throughout the region occupied by the conducting material. That is, the conductor is a three-dimensional equipotential region and the surface of a conductor is an equipotential surface. • The potential V has the same value everywhere on an equipotential surface • We conclude that electric field lines are normal to any equipotential surfaces they intersect. • Two conductors that are separated in space will typically not be at the same potential. The potential difference between such conductors depends on their geometrical shapes, their separation in space, and the net charge on each. When two conductors touch, the charge on the conductors redistributes itself so that electrostatic equilibrium is established and the electric field is zero inside both conductors. While touching, the two conductors can be considered to be a single conductor with a single potential • When a charge is placed on a conductor of nonspherical shape, the surface of the conductor will be an equipotential surface, but the surface charge density and the electric field just outside the conductor will vary from point to point. Near a point where the radius of curvature is small, such as point A in the figure, the surface charge density and electric field will be large, whereas near a point where the radius of curvature is large, such as point B in the figure, the field and surface charge density will be small. ▯▯ ▯ ▯ ▯ ▯▯▯ ▯ ▯▯ • ???? = = = = ▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯ ▯ ▯▯ • ???? = ▯ Electrostatic Potential Energy • Objects that repel each other have more potential energy if they are close together, and objects that attract each other have more potential energy if they are far apart • The total work required to assemble the three charges is the electrostatic potential energy U of the system of three point charges: ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ???? = ▯▯▯ + ▯▯▯ + ▯▯▯ • Electrostatic Potential Energy of a System: The electrostatic potential energy of a system of point charges is the work needed to bring the charges from an infinite separation to their final positions • Electrostatic Potential Energy U of a System of n Point Charges ▯ ▯ ???? = ▯ ▯▯▯ ???????????????? o Vi is the potential at the location of the ith charge due to the presence of all other charges in the system • Electrostatic Potential Energy of a System of Conductors: ▯ ▯ ???? = ▯ ▯▯▯ ???????????????? Chapter 24: Capacitance • The potential V of a single isolated conductor due to the charge Q on it is proportional to Q and depends on the size and shape of the conductor. Typically, the larger the surface area of a conductor, the more charge it can carry for a given potential. • ???? = ▯▯ ▯ • The ratio Q/V of the charge to the potential of an isolated conductor is called its self-capacitance C. • A capacitor is a device consisting of two conductors, one that has a charge Q and the other that has a charge ▯Q. • The ratio of charge Q to the potential difference V between the two conductors is called the capacitance of the capacitor. • Definition-Capacitance: ???? = ▯ ▯ • Capacitance is a measure of the capacity to store charge for a given potential difference. Because the potential difference is proportional to the charge, this ratio does not depend on either Q or V, but only on the sizes, shapes, and relative positions of the conductors. • The SI unit of capacitance is the coulomb per volt, which is called a farad (F) • 1 F = 1 C/V • The farad is a rather -6rge unit, so submultiples suc-12s the microfarad (1 mFX10 F) or the picofarad (1 pFX10 F) are more commonly used ▯▯▯ • Electric Constant: ???????? = 8.85????10 ????/???? = 8.85 pF/m Capacitors: • A capacitor is usually charged by transferring a charge Q from one conductor to the other conductor, which leaves one of the conductors having a charge ︎ Q and the other conductor having a charge -Q. The capacitance of the device is defined to be Q/V, where Q is the magnitude of the charge on either conductor and V is the magnitude of the potential difference between the conductors. To calculate the capacitance, we place equal and opposite charges on the conductors and then find the potential difference V by first finding the electric field E due to the charges and then calculating V from E. Parallel-Plate Capacitors: • A common capacitor is the parallel-plate capacitor, which uses two parallel conducting plates. • Let A be the area of the surface (the area of that side of each plate that faces the other plate), and let d be the separation distance, which is very small compared to the length and width of the plates. We place a charge +Q on one plate and -Q on the other plate. These charges attract each other and become uniformly distributed on the inside surfaces of the plates. Because the plates are very close together, the electric field between them is uniform and has a magnitude of E=????/????????. (That the electric field strength just outside the surface of the conductor is given by E=????/????????. ▯ ▯ ▯▯▯ • Capacitance of a Parallel-Plate Capacitor: ???? = ▯ = ▯▯/(▯▯▯) = ▯ • PROBLEM-SOLVING STRATEGY o Calculating Capacitance o PICTURE Make a sketch of the capacitor that has a charge of +Q on one conductor and a charge of -Q on the other conductor. o SOLVE § 1. Determine the electric field E, usually by using Gauss’s law. ▯ § 2. Determine the magnitude of the potential difference V between the two ▯conductors by integrating ???????? = −???? ∙ ???????? § 3. The capacitance is equal to C = Q/V § CHECK: Check that the result depends only on the electric constant* and on geometrical factors such as lengths and areas. Cylindrical Capacitors: • A cylindrical capacitor consists of a long conducting cylinder of radius R 1nd a larger, concentric cylindrical conducting shell of radius R 2 The cylinders have the same length ▯▯▯▯▯ • Capacitance of a Cylindrical Capacitor: ???? = ▯▯ ) ▯▯ The Storage of Electrical Energy • When a capacitor is being charged, electrons are transferred from the positively charged conductor to the negatively charged conductor. This leaves the positively charged conductor with an electron deficit and the negatively charged conductor with an electron surplus. Alternatively, transferring positive charges from the negatively charged conductor to the positively charged conductor could also charge a capacitor. Either way, work must be done to charge a capacitor, and at least some of this work is stored as electrostatic potential energy. ▯ ▯▯ ▯ ▯ • Energy Stored in a Capacitor: ???? = = ???????? = ???????? ▯ ▯ ▯ ▯ ▯ • Suppose we charge a capacitor by connecting it to a battery. The potential difference V when the capacitor is fully energized with charge ︎ Q on one conductor and charge ︎ Q on the other is just the potential difference between the terminals of the battery before they were connected to the capacitor. The total work done by the battery in charging the capacitor is QV, which is twice the amount of energy stored in the capacitor Electrostatic Field Energy: • During the process of charging a capacitor, an electric field is produced between the plates. The work required to charge the capacitor can be thought of as the work required to establish the electric field. That is, we can think of the energy stored in a capacitor as energy stored in the electric field, called electrostatic field energy. • The energy per unit volume is called the energy density u e • Energy Density of an Electrostatic Field: ???????? = ▯▯▯▯▯▯= ???????????? ▯ ▯▯▯▯▯▯ ▯ o Works with any electric field Capacitors, Batteries, and Circuits • The potential difference between the two terminals of a battery is called its terminal voltage. The terminals of a battery are connected to dissimilar conductors called electrodes, and within the battery the electrodes are separated by a conducting liquid or paste called an electrolyte. • Because of chemical reactions in the battery, charge is transferred from one electrode to the other. This leaves one electrode of the battery (the anode) positively charged and the other electrode (the cathode) negatively charged; this charge separation is maintained by chemical reactions within the battery. Within the battery, there is an electric field directed away from the positive electrode and toward the negative electrode. • Open-circuit terminal voltage: the terminal voltage at its initial level • Devices connected in parallel share a common potential difference across each device due solely to the way they are connected. • A combination of capacitors in a circuit can sometimes be substituted with a single capacitor that is operationally equivalent to the combination. The substitute capacitor is said to have an equivalent capacitance. • Equivalent Capacitance for Capacitors in Parallel: ???????????? = ????1 + ????2 + ????3 + ⋯ • Kirchhoff’s Loop Rule: The changes in potential around any closed path always sum to zero • A junction is a point in a wire where the wire divides into two or more wires. • If two capacitors are connected so that a plate of one capacitor is connected to a plate of a second capacitor by a wire containing no junctions, they are connected in series. • Equivalent Capacitance for Equally Charged Capacitors in ▯ ▯ ▯ ▯ Series:▯▯▯= ▯▯+ ▯▯ + ▯▯+ ⋯ o The equivalence capacitance of two capacitors in series is less than the capacitance of either capacitor. Adding a capacitor in series increases 1/Ceq, which means the equivalent capacitance Ceq decreases. Dielectrics • A nonconducting material (for example, air, glass, paper, or wood) is called a dielectric. When the space between the two conductors of a capacitor is occupied by a dielectric, the capacitance is increased by a factor that is characteristic of the dielectric ▯▯ • Electric Field Inside a Dielectric: ???? = ▯ • k (Kappa) is called the dielectric constant of the inserted material • Effect of a Dielectric on Capacitance: ???? = ???????????? ▯ o ???????? = is the capacitance without the dielectric ▯▯ • The capacitance of a parallel-plate capacitor filled with a dielectric of constant k is ???? = ▯▯▯▯= ▯▯ ▯ ▯ o ???? = ???????????? • The parameter ???? is called the permittivity of the dielectric • PROBLEM-SOLVING STRATEGY o Calculating Capacitance II o PICTURE To calculate the capacitance of a capacitor that has a gap containing two or more dielectric slabs, first calculate the electric field strength E using charge Q and with no 0 dielectrics in the gap. o SOLVE § 1. When the dielectric is in the gap, the electric field strength within a dielectric slab is E ︎ E k, where k is the 0 dielectric constant. § 2. Use E within a dielectric slab to calculate the voltage V slabcross the slab. The voltage V across the entire gap is the sum of the voltages across the individual slabs in the gap plus the sum of the voltages across any empty regions of the gap. § 3. Then, calculate C using C=Q/V o CHECK Evaluate your expression for C by setting k equal to 1. Then compare your result with the expression for C (the 0 capacitance without a dielectric present). Molecular View of a Dielectric • When a dielectric is placed in the field of a charged capacitor, its molecules are polarized in such a way that there is a net dipole moment parallel to the field. If the molecules are polar, their dipole moments, originally oriented at random, tend to become aligned due to the torque exerted by the field. • The surface charge on the dielectric is called a bound charge, because the surface charge is bound to the surface molecules of the dielectric and can- not move about like the free charge on the conducting capacitor plates. This bound charge produces an electric field opposite in direction to the electric field produced by the free charge on the conductors. Thus, the net electric field between the plates is reduced Magnitude of the Bound Charge: • The bound charge density s on the surfaces of the dielectric is b related to the di- electric constant k and to the free charge density s f on the surfaces of the plates. • The bound charge density s is blways less than or equal to the free charge density s of the capacitor plates, and it is zero if k = 1, which is the case when there is no dielectric. For a conducting slab, k =∞︎ and ???? b ???? .f The Piezoelectric and Pyroelectric Effects: • In certain crystals that have polar molecules (for example, quartz, tourmaline, and topaz), a mechanical stress applied to the crystal produces polarization of the molecules. This is known as the piezoelectric effect. The polarization of the stressed crystal causes a potential difference across the crystal, which can be used to pro- duce an electric current. • The converse piezoelectric effect, in which a voltage applied to such a crystal induces mechanical strain (deformation), is used in headphones and many other devices. • Many crystals that exhibit the piezoelectric effect also exhibit the pyroelectric effect, which is the generation of a large electric field within the crystal when the temperature of the crystal is increased. Chapter 25: Electric Current and Direct-Current Circuits • The flow of charge constitutes an electric current. Current and the Motion of Charges: • When a switch is thrown to turn on a circuit, a very small amount of charge accumulates along the surfaces of the wires and other conducting elements of the circuit, and these surface charges produce electric fields that drive the motion of charges through the conducting materials of the circuit. • In steady state, charge no longer continues to accumulate at points along the circuit and the current is steady. • Electric current is the rate of flow of charge through a surface— typically a cross-sectional surface of a conducting wire. • If ∆Q is the charge that flow∆▯through the cross-sectional area A in time ∆t, the current I is ???? = ∆▯ • The SI unit of current is the ampere (A) o 1 A = 1 C/s • Mobile charges can be negatively charged or positively charged. In addition, a direction along the wire is designated as the positive direction. By convention, the sign of the current is positive if the current is due either to positive charges moving in the positive direction or to negative charges moving in the negative direction. However, the current is negative if it is due either to positive charges moving in the negative direction or to negative charges moving in the positive direction. • Free electrons move in the negative direction when the current is positive, and vice versa • In a metal wire, the motion of negatively charged free electrons is quite complex. When there is no electric field in the wire, the free electrons move in random directions with relatively large speeds of the order of 10 m/s. In addition, the electrons collide repeatedly with the lattice ions in the wire. Because the velocity vectors of the electrons are randomly oriented, the average velocity is zero. When an electric field is applied, the field exerts a force -eE on each free electron, giving it a change in velocity in the direction opposite the field. However, any additional kinetic energy acquired is quickly dissipated by collisions with the lattice ions in the wire. During the time between collisions with the lattice ions, the free electrons, on average, acquire an additional velocity in the direction opposite to the field. The net result of this repeated acceleration and dissipation of energy is that the electrons drift along the wire with a small average velocity, directed opposite to the electric-field direction, called the drift velocity. The drift speed is the magnitude of the drift velocity. • When there is no applied electric field, the average velocity of all the free electrons in a metal is zero, but when there is an applied electric field, the average velocity is not zero due to the small drift velocities of the free electrons. • Relation Between Current and Drift Speed: ???? = ∆▯ = ???????????????? ∆▯ ▯ o Let n be the number of mobile charged particles (charge carriers) per unit volume in a conducting wire of cross- sectional area A. We call n the number density of the charge carriers. Assume that each particle carries a charge q and moves in the positive direction with a drift speed v . d o Volume Av ∆t d • If the current is the result of the motion of more than one species of mobile charge, as it sometimes is in ionic solutions such as salt water, then the total current is the sum of the currents for each of the individual species of mobile charges. • The current density vector, ????, is specified by ???? = ???????????????? • The current through a surface S is defined as the flux of the current density vector J through the surface. That is, ???? = ???? ???? = ???? ∙ ???? = ???? ∙ ???????? = ???????????????????????? ▯ o ???? is the area of the surface and ???? is the angle between ???? and ????. The sign of the current I is the same as the sign of cos????. If ???? < 90° I is positive, and if ???? > 90° then I is negative • There are always a very large number of conduction electrons throughout the metal wire. Thus, electrons start moving along the entire length of the wire (including the wire inside the light-bulb) almost immediately after the light switch is turned on. The transport of a significant amount of electrons in a wire is accomplished not by a few electrons moving rapidly down the wire, but by a very large number of electrons slowly drifting down the wire. Surface charges are established on the wires, and these surface charges produce an electric field. It is the electric field produced by these surface charges that drives the conduction electrons through the wire. Resistance and Ohm’s Law • Current in a conductor is driven by an electric field Einside the conductor that exerts a force qEon the free charges. (In electrostatic equilibrium, the electric field must be zero inside a conductor, but when there is a current in a conductor, the conductor is no longer in electrostatic equilibrium.) • In a metal, the free charges are negatively charged, so the free charges are driven in a direction opposite to the direction of the electric field E. • If we model the current as the flow of positive charge carriers, these positive charge carriers drift in the direction of decreasing potential. • The ratio of the potential drop in the direction of the current* to the current is called the resistance of the segment, ???? = ▯ ▯ o The definition of the current refers to the direction of the current density vector • The SI unit of resistance, the volt per ampere, is called an ohm (Ω) o 1 Ω = 1 ????/???? • For many materials, the resistance of a sample of the material does not depend on either the potential drop or the current. Such materials, which include most metals, are called ohmic materials. • Ohm’s Law: ???? = ???????? • Ohm’s law is not a fundamental law of nature, like Newton’s laws or the laws of thermodynamics, but rather is an empirical description of a property shared by many materials under specified conditions. o The resistance of a conductor does vary with the temperature of the conductor. • The resistance R of a conducting wire is found to be proportional to the length L of the wire and inversely proportional to its cross- ▯ sectional area A: ???? = ???? ▯ o The proportionality constant p is called the resistivity of the conducting material. The unit of resistivity is the ohm-meter (Ω ∙ ????) • For a segment of wire that has a length L, a cross-sectional area A, a current I, and a resistance R, the voltage drop V across the length of the segment is related to the current I in the segment by ???? = ???????? = ▯ ???????? ▯ • ???? = ???????? o It states that the current density vector J at a point in a current-carrying conductor is equal to the reciprocal of the resistivity multiplied by the electric field vector Eat the same point. • The resistivity of any given metal depends on the temperature. • In tables, the resistivity is usually given in terms of its value at 20°C, p20 along with the temperature coefficient of resistivity,????, which is the ratio of the fractional change in resistivity to the change in (▯▯▯▯)/▯▯ temperature: ???? = ▯▯▯▯ o po is the resistivity at temperature To and p is the resistivity at temperature T Energy in Electric Circuits • When there is an electric field in a conductor, the free electrons gain kinetic energy due to the work done on the free electrons by the field. However, steady state is soon achieved as the kinetic energy gain is continuously dissipated into the thermal energy of the conductor by interactions between the free electrons and the lattice ions of the conductor. This mechanism for increasing the thermal energy of a conductor is called Joule heating. • The rate of potential energy loss is the power P delivered to the conducting segment, and it is equal to the rate of dissipation of electrical potential energy in the segment: ???? = ???????? o If V is in volts and I is in amperes, the power is in watts. The power loss is the product IV, where V is the decrease in potential energy per unit charge, and I is the rate at which the charge flows past a cross section of the segment. This equation applies to any device in a circuit. • In a conductor (a resistor is a conductor), the potential energy is dissipated as thermal energy. ▯ • Power Delivered to a Resistor: ???? = ???????? = ???? ???? = ▯ ▯ EMF and Batteries: • A device that supplies electrical energy to a circuit is called a source of emf. (The letters emf stand for electromotive force a term that is now rarely used.) • Examples of emf sources are a battery, which converts chemical energy into electrical energy, and a generator, which converts mechanical energy into electrical energy. A source of emf does non- conservative work on the charge passing through it, increasing or decreasing the potential energy of the charge • The work per unit charge is called the emf E of the source. The unit of emf is the volt, the same unit as potential difference. An ideal battery is a source of emf that maintains a constant potential difference between its two terminals, independent of the current through the battery. The potential difference between the terminals of an ideal battery is equal in magnitude to the emf of the battery. • The rate at which energy is sup- plied by the source of emf is the ∆▯ ▯ power output of the source: ???? = ∆▯ =I???? • In a real battery, the potential difference across the battery terminals, called the terminal voltage, is not simply equal to the emf of the battery. • We can consider a real battery to consist of an ideal source of emf E and a resistor with resistance r, called the internal resistance of the battery. • The terminal voltage is less than the emf of the battery because of the decrease in potential due to the internal resistance of the battery. Combinations of Resistors Resistors in Series: • ???? = ????????1 + ????????2 = ????(????1 + ????2) • Equivalent Resistance for Resistors in Series: ???????????? = ????1 + ????2 + ????3+... Resistors in Parallel: • ???? = ????1 + ????2 • ???? = ???????????????? ▯ ▯ ▯ ▯ • Equivalent Resistance for Resistors in Parallel: ▯▯▯= ▯▯+ ▯▯+ ▯▯+ ⋯ • The equivalent resistance of a parallel combination of resistors is less than the resistance of any single resistor in the combination. • Adding more resistors in parallel means adding more conducting paths for charges to flow along. The creation of additional parallel paths lowers the equivalent resistance of the combination. • PROBLEM-SOLVING STRATEGY o Problems Involving Series and/or Parallel Combinations of Resistors o PICTURE If no circuit diagram is provided, draw one. o SOLVE § Identify each series and/or parallel combination of resistors and calculate its equivalent resistance. ▯ § Redraw the circuit so that each series or parallel combination of resistors is replaced by a single resistor of equivalent resistance. ▯ § Repeat steps 2 and 3 until there are no more series or parallel combinations. (At this point the circuit should contain only a single resistor.) Apply V ▯ IR and calculate the current. ▯ § Return to the previous drawing and calculate the voltage across and/or the current in each resistor in that drawing. ▯ § Repeat step 4 until you calculate all currents and/or voltages of interest. o CHECK Calculate the power delivered to each resistor (using P = IV or its equivalent) and calculate the power supplied by the chemical reactions in each battery using P = IE. Then check to see that the total power being delivered equals the total power being supplied. Kirchoff’s Rules: • When any closed loop is traversed, the algebraic sum of the changes in potential around the loop must equal zero. ▯ • At any junction (branch point) in a circuit where the current can divide, the sum of the currents into the junction must equal the sum of the currents out of the junction. ▯ • Kirchoff’s first rule, the loop rule:???? ???? ∙ ???????? = 0 o where the integral is taken around any closed curve C. Changes in potential ∆???? and E are related by ∆???? = ???????? − ???????? = ▯ ▯ ???? ∙ ???????? o Implies that the sum of the changes in potential (the sum of the ∆Vs) around any closed path equals zero. • Kirchoss’s second rule, called the junction rule, follows from the conservation of charge o Because charge does not originate or accumulate at this point, the conservation of charge implies the junction rule, which for this case gives ????1 = ????2 + ????3 Single-Loop Circuits: • The analysis of a circuit is usually simplified if we define the potential to equal zero at a convenient point in the circuit. Then we calculate the potential at the other points relative to it. Because only potential differences are important, any point in a circuit can be chosen to have zero potential. In many circuits, however, one point is connected to a rod that is driven into the ground. Such a point is said to be grounded or put to earth, and the potential is defined to be zero at that point. Multi-loop Circuits: • In multi-loop circuits, often the direction of the current in one or more branches of the circuit are not obvious. • For each branch of the circuit we arbitrarily assign a positive direction along the branch, and we indicate this assignment by placing a corresponding arrow on the circuit diagram. If the current density in the branch is in this positive direction, then when we solve for this current we will get a positive value. However, if the current density is opposite to the assigned positive direction, when we solve for the cur- rent we will get a negative value. • Sign Rule for the Change in Potential Across a Resistor: For each branch of a circuit, we draw an arrow to indicate the positive direction for that branch. Then, if we traverse a resistor in the direction of the arrow, the change in potential∆V is equal to -IR (and if we traverse a resistor in the direction opposite the direction of the arrow, ∆V is equal to +IR). • If we traverse a resistor in the positive direction, and if I is positive, then -IR is negative. This is as expected, because in a resistor, the current is always in the direction of decreasing potential. However, if we traverse a resistor in the positive direction and if I is negative, then -IR is positive. Similarly, if we traverse a resistor in the negative direction, and if I is positive, then +IR is positive. And if we traverse a resistor in the negative direction and if I is negative, then +IR is negative. • PROBLEM-SOLVING STRATEGY o Method for Analyzing Multiloop Circuits o PICTURE Draw a sketch of the circuit. o SOLVE § 1. Replace any series or parallel resistor combinations or capacitor combinations with their equivalent values. ▯ § 2. Repeat step 1 as many times as possible. ▯ § 3. Next, assign a positive direction for each branch of the circuit and indicate this direction with an arrow. Label the current in each branch. Add a plus sign and a minus sign to indicate the high-potential terminal and low-potential terminal of each source of emf. ▯ § 4. Apply the junction rule to all but one of the junctions. ▯ § 5. Apply the loop rule to the different loops until the total number of independent equations equals the total number of unknowns. When traversing a resistor in the positive direction, the change in potential equals ▯IR. When traversing a battery from the negative terminal to the positive terminal, the change in potential equals E ▯ Ir. ▯ § 6. Solve the equations to obtain the values of the unknowns. o CHECK Check your results by assigning a potential of zero to one point in the circuit and use the values of the currents found to determine the potentials at other points in the circuit. Ammeteres, Voltmeters, and Ohmmeteres: • The devices that measure current, potential difference, and resistance are called ammeters, voltmeters, and ohmmeters, respectively. Often, all three of these meters are included in a single multimeter that can be switched from one use to another. • A galvanometer, is a device that detects small currents passing through it. RC Circuits: • A circuit containing a resistor and a capacitor is called an RC circuit. The current in an RC circuit is in a single direction, as in all dc circuits, but the magnitude of the current varies with time. Discharging a Capacitor: • Definition- Time Constant: ???? = ???????? o ???? = the time it takes for the charge to decrease by a factor of ????▯▯ • This type of decrease, which is called an exponential decrease, is very common in nature. It occurs whenever the rate at which a quantity decreases is proportional to the quantity itself. Charging a Capacitor: • The capacitor is initially uncharged • The switch, originally open is closed at time t=0 • Charge immediately begins to flow through the battery Works Cited Tipler, Paul Allen, and Gene Mosca. Physics for Scientists and Engineers. New York: W.H. Freeman, 2008. Print.

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