# Chapter 20 Electric Circuits

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1 Chapter 0 Electric Circuits Chevy olt --- Electric vehicle of the future Goals for Chapter 9 To understand the concept of current. To study resistance and Ohm s Law. To observe examples of electromotive force and circuits to learn Ohm s Law s application. To calculate the energy and power in electric circuits. To study the similarity and differences in the combination of resistors in parallel and those connected in series. To apply Kirchhoff s ules to combinations of resistors. To observe and understand devices which measure electricity in circuits. To combine resistors and capacitors then calculate examples of the results. 0. Electromotive force and current n an electric circuit, energy is transferred from a source (battery) to a device (MP3 player) by charges that move through a conducting wires 0. Electromotive force and current Within a battery, a chemical reaction occurs that transfers from positive terminal to the negative terminal. The maximum potential difference between two terminals of a battery is called the electromotive force (emf) of the battery. The symbol ε is used. 0. Electromotive force and current The flow of charges is known as an electric current, Δq Δt The current is defined as the amount of charges per unit time that cross an imaginary surface as shown below. Movement of Charge Carriers : Between plates of a parallel plate capacitor (vacuum) L q m a The charge accelerates with a (q/m) E (q/m) (/L) f the particle (with charge q) starts at rest, and the potential difference between the plates is, then the kinetic energy upon reaching the second plate will be: K q ( in electron volts) ½ m v

2 Movement of Charge Carriers : nside a semi-conductor Direction of the Current Direction of current is direction of flow of positive charge nside a semi-conductor positive holes and negative electrons are the charge carriers f an electric field is present, the holes / electrons will start moving (in a direction of / opposite to the field). However, the motion of the charge carriers will be disrupted by frequent collisions with the ions. The net result is that the charge carriers ( holes and electrons) acquire a slow average speed. or, opposite direction of flow of negative charge Why does current flow? f an electric field is set up in a conductor, charge will move (making a current in the direction of E Note that when the current is flowing, the conductor is not an equipotential surface (and E inside is not equal 0) E Current Flow in a Conductor Conductors are made of materials (usually metals) in which some of the electrons are free to move (not bound to the ions). These are called conduction electrons. n normal state t (E0) these free electrons have random, Brownian motion in the material. Electrons move under the influence of an E field. The individual motion of electrons is still quasirandom. However, a net average flow of charge is set up when an E field is applied. nfluence of electric field on flow of electrons nfluence of electric field on flow of electrons E0 E0 E E0 An electric field modifies the trajectories of electrons between collisions. When E is nonzero, the electrons move almost randomly after each bounce, but gradually they drift in the direction opposite to the electric field.

3 Microscopic Picture Drift speed is velocity forced by electric field in the presence of collisions! t is typically 4x0-5 m / s, or 0.04 mm / second Current: Flow of Charge Average Current av : Charge ΔQ flowing across area A in time Δt ΔQ Δt To go one meter at this speed takes about 0 hours! How Can This Be? Changes in the electric field that drive the charge carriers (free electrons in a conductor) travel with the speed of light! For instance, when flipping a light switch electron already in the filament will experience the change in electric force withing nanoseconds. Units of current: Coulombs / second Ampere 0. Ohm s Law esistor alues (Ohm) has units of Ohms (Ω) olts / Ampere alues of resistors are commonly marked by colored bands Ohm s Law 0.3 esistance and esistivity The macroscopic form is the most commonly used form of Ohm s Law. is the esistance t depends d on the material type and shape: ρ L / A Units: ohms, (Ω). As ρ A / L, common units for the resistivity ρ are Ohm-meters. Similarly, common units for the conductivity σ / ρ are (Ohm m) - ρ L A

4 Examples of Circuits 0.3 Temperature dependence of resistance and resistivity The resistivity of a material depends on its temperatures. ρ ρo[ α( T T o)] where α is the temperature coefficient of resistivity. And similarly we also have the following equation for resis tance. o[ α( T To )] Symbols for Circuits Elements Sign Conventions - Battery Moving from the negative to positive terminal of a battery increases your potential Δ b a Electrical description of a battery symbol for battery E - symbol for resistance 0.4 Electrical Power The rate at which the energy is changed is the power ΔQ P Δt Δ - A battery does work on positive charges in moving them to higher potential. The EMF E is the work per unit charge exerted to move the charges uphill but you can just think of it as an applied voltage. From Ohm s Law, alternate fo of power are P

5 Units of Power 0.5 Alternating Current The S unit of power is Watt (W) must be in Amperes, in ohms and Δ in olts P P / The unit of energy used by electric companies is the kilowatt-hour This is defined in te of the unit of power and the amount of time it is supplied kwh 3.60 x 0 6 J n an AC circuit, the charge flow reverses direction periodically. 0.5 Alternating Current 0.5 Alternating Current n circuits that contain only resistance, the current reverses direction each time the polarity of the generator reverses. ( ft) o sin π o sin peak current ( π ft) sin( π ft) o 0.5 Alternating Current 0.5 Alternating Current o sin( π ft) o sin( π ft) P sin o o ( π ft) P o o o o o o,

6 0.5 Alternating Current 0.6 Series wiring P P P Series Parallel esistors in Series a b When two or more resistors are connected end-to-end, they are said to be in series The current is the same in all resistors because any charge that flows through one resistor flows through the other The sum of the potential differences across the resistors is equal to that across the combination Example: esistors in Series - Equivalent esistance eq 3 We want to write this as eq hence eq The equivalent resistance of a series combination of resistors is always greater than any of the individual resistors 0.7 Parallel wiring The potential difference a across each resistor is the b same because each is connected directly across the battery terminals The current,, that enters a point must be equal to the total current leaving that point Consequence of Conservation of Charge / / (/ / ) We want to write this as: / eq Hence / eq / / Example: esistors in Parallel - Equivalent esistance Equivalent resistance replaces the two original resistances Household circuits are wired so the electrical devices are connected in parallel Circuit breakers may be used in series with other circuit elements for safety purposes

7 Example: esistors in Parallel - Equivalent esistance Equivalent esistance eq 3 K Household electric wiring is parallel wiring so one device will not affect the others. eq eq [ ] 8 8 eq Ω The equivalent is always less than the smallest resistor in the group esistors in Series a b Summary esistors esistors in Parallel a eq / eq / / b 0.8 Series and parallel wiring Often you can replace sets of resistors step by step. eplace any resistors in series or in 6 Ω parallel using their equivalent E resistances 6 Ω - Sketch the new circuit after these changes have been made Ω Continue to replace any series or parallel combinations /6/6/(3) Continue until one equivalent resistance is found E 3 Ω 3Ω - Ω 5Ω Ω step step E - 5Ω Combinations of esistors Combine the resistors in series and parallel edraw the circuit with the equivalents of each set The 8.0-Ω and 4.0-Ω resistors are in series and can be replaced with their equivalent,.0 Ω Example : A four-resistor circuit The 6.0-Ω and 3.0-Ω resistors are in parallel and can be replaced with their equivalent,.0 Ω Determine the final equivalent resistance These equivalent resistances are in series and can be replaced with their equivalent resistance, 4.0 Ω

8 0.9 nternal esistance A real battery has some internal resistance Therefore, the terminal voltage is not equal to the emf The schematic shows the internal resistance, r The terminal voltage is Δ ε r For the entire circuit, ε r ε 0.9 nternal esistance ε is equal to the terminal voltage when the current is zero Also called the open-circuit voltage is called the load resistance The current depends on both the resistance external to the battery and the internal resistance When >> r, r can be ignored Generally assumed in problems Power relationship ε r When >> r, most of the power delivered by the battery is transferred to the load resistor Example: A dim flashlight As a flashlight battery ages, its emf stays approximately constant, but its internal resistance increases. When the battery needs replacement, its emf is still.5, but its internal resistance has increased to 000 Ω. f this old battery is supplying.0 ma to a lightbulb, what is its terminal voltage? 0.0 Kirchhoff s ules Terminal voltage ab ε - r ab A 000 Ω 0.5 There are ways in which resistors can be connected so that the circuits cannot be reduced to a single equivalent resistor, in this case, we need to use Kirchhoff s rules. Kirchhoff s Loop ule Loop ule The sum of the potential differences across all elements around any closed circuit loop must be zero A statement of Conservation of Energy Mathematically, closed loop Δ 0 r E - Loop ule To apply Kirchhoff s ules, Assign symbols and directions to the currents in all branches of the circuit f the direction of a current is incorrect, the answer will be negative, but have the correct magnitude Choose a direction to transverse the loops ecord voltage rises and drops Traveling around the loop from a to b n (a), the resistor is traversed in the direction of the current, the potential across the resistor is

9 Loop ule n (b) (current reversed), the resistor is traversed in the direction opposite of the current, the potential across the resistor is n (c), the source of emf is traversed in the direction of the emf (from to ), and dthe change in the electric potential is ε n (d) (battery reversed), the source of emf is traversed in the direction opposite of the emf (from to -), and the change in the electric potential is -ε Example: Simple Circuit What is current through the bottom battery? You can simplify resistors in series (but don t need to) Example: Simple Circuit Kirchhoff s Junction ule 3 From Conservation of Charge Diagram (b) shows a mechanical analog 0 in out Steps of Solving Circuit Problems. Straighten out circuit (make squares). Simplify resistors in series / parallel 3. Assign directions to the currents. 4. Apply the junction rule to any junction in the circuit 5. Apply the loop rule to as many loops as are needed to solve for the unknowns 6. Solve the equations simultaneously for the unknown quantities 7. Check your answers Example: Kirchhoff s rules Taking.00 kω and ε 50 in the circuit shown, determine the currents through the and resistors a) label the currents in the branches b) reduce the circuit by combining the two parallel resistors

10 000 Ω 000 Ω Kirchhoff s loop rule ( ) ( ) simultaneous solution of these equations 0 ma 30 ma 0. The measurement of current and voltage A voltmeter (ammeter) is used to measure voltage (current). The essential feature of these devices is a dc galvanometer. A galvanometer consists of a magnet, a coil of wire, a spring, a pointer, and a calibrated scale. When a current is 0.0 ma, the pointer deflects full scale. 0. Ammeter with a full scale of 60 ma To measure the voltage between two points in a circuit, a voltmeter is connected between the points. An ammeter must be inserted into a circuit so that the current passes directly through it. f a galvanometer with a full-scale limit of 0.00 ma is to be used to measure the current of 60.0 ma, a shunt resistance must be used so that the excess current of 59.9 ma can detour around the galvanometer coil. 0. Capacitors in series and in parallel Capacitors in Parallel q q q C C C C ) C C p C C ( C 3... P 0. Capacitors in series and in parallel Capacitors in parallel q C q C q q C C C S... C C C C s 3

11 0.3 C Circuits capacitor uncharged 0.3 C Circuits Capacitor charging charging the capacitor C [ ] t q q o e When a direct current circuit contains capacitors and resistors, the current will vary with time When the switch is closed, the capacitor starts to charge The capacitor continues to charge until it reaches its maximum charge (Q C) (ε) Once the capacitor is fully charged, the current in the circuit is zero time constant τ C Discharging Capacitor in a C Circuits Electrical Safety and Effects of arious Currents When a charged capacitor is placed in closed circuit, it can be discharged q Q e t C Electric shock can result in fatal burns Electric shock can cause the muscles of vital organs (such as the heart) to malfunction time constant, τ C The charge decreases exponentially At t τ C, the charge decreases to Q max n other words, in one time constant, the capacitor loses 63.% of its initial charge 5 ma or less Can cause a sensation of shock Generally little or no damage 0 ma Hand muscles contract May be unable to let go of a live wire 00 ma f passes through the body for just a few seconds, can be fatal The importance of proper grounding

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