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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 18: Electric Current and Circuits Electric current EMF Current & Drift Velocity Resistance & Resistivity Kirchhoff’s Rules Series & Parallel Circuit Elements Applications of Kichhoff’s Rules Power & Energy Ammeters & Voltmeters RC Circuits

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 §18.1 Electric Current e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- A metal wire. Assume electrons flow to the right. Current is a measure of the amount of charge that passes though an area perpendicular to the flow of charge. Current units: 1C/sec = 1 amp

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 A current will flow until there is no potential difference. The direction of current flow in a wire is opposite the flow of the electrons. (In the previous drawing the current is to the left.)

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 Example: If a current of 80.0 mA exists in a metal wire, how many electrons flow past a given cross-section of the wire in 10.0 minutes?

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 §18.2 EMF and Circuits An ideal battery maintains a constant potential difference. This potential difference is called the battery’s EMF( ). The work done by an ideal battery in pumping a charge q is W=q .

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 The circuit symbol for a battery (EMF source) is + - At high potential At low potential Batteries do work by converting chemical energy into electrical energy. A battery dies when it can no longer sustain its chemical reactions and so can do no more work to move charges.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 §18.3 Microscopic View of Current in a Metal Electrons in a metal might have a speed of ~10 6 m/s, but since the direction of travel is random, an electron has v drift = 0.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 Only when the ends of a wire are at different potentials (E 0) will there be a net flow of electrons along the wire (v drift 0). Typically, v drift < 1 mm/sec.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 Calculate the number of charges (N e ) that pass through the shaded region in a time t: The current in the wire is: l

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 Example (text problem 18.19): A copper wire of cross- sectional area 1.00 mm 2 has a constant current of 2.0 A flowing along its length. What is the drift speed of the conduction electrons? Copper has 1.10 10 29 electrons/m 3.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 §18.4 Resistance and Resistivity A material is considered ohmic if V I, where The proportionality constant R is called resistance and is measured in ohms ( ; and 1 = 1 V/A).

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 The resistance of a conductor is: where is the resistivity of the material, L is the length of the conductor, and A is its cross sectional area. With R a material is considered a conductor if is “small” and an insulator if is “large”.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 The resistivity of a material depends on its temperature: where 0 is the resistivity at the temperature T 0, and is the temperature coefficient of resistivity. A material is called a superconductor if =0.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 Example (text problem 18.28): The resistance of a conductor is 19.8 at 15.0 C and 25.0 at 85.0 C. What is the temperature coefficient of resistivity? Values of R are given at different temperatures, not values of . But the two quantities are related. Multiply both sides of equation (2) by L/A and use equation (1) to get:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 Example continued: Solve equation (3) for and evaluate using the given quantities:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 §18.5 Kirchhoff’s Rules Junction rule: The current that flows into a junction is the same as the current that flows out. (Charge is conserved) A junction is a place where two or more wires (or other components) meet. Loop rule: The sum of the voltage dropped around a closed loop is zero. (Energy is conserved.)

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 For a resistor: If you cross a resistor in the direction of the current flow, the voltage drops by an amount IR (write as – IR). There is a voltage rise if you cross the other way (write as +IR).

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 For batteries (or other sources of EMF): If you move from the positive to the negative terminal the potential drops by (write as – ). The potential rises if you cross in the other direction (write as + ).

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 A current will only flow around a closed loop. B V AB is the terminal voltage. A Applying the loop rule:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 In a circuit, if the current always flows in the same direction it is called a direct current (DC) circuit.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 § 18.6 Series and Parallel Circuits The current through the two resistors is the same. It is not “used up” as it flows around the circuit! These resistors are in series. Apply Kirchhoff’s loop rule: Resistors:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 The pair of resistors R 1 and R 2 can be replaced with a single equivalent resistor provided that R eq =R 1 + R 2. In general, for resistors in series

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 Current only flows around closed loops. When the current reaches point A it splits into two currents. R 1 and R 2 do not have the same current through them, they are in parallel. Apply Kirchhoff’s loop rule: The potential drop across each resistor is the same.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 Applying the junction rule at A: I =I 1 +I 2. From the loop rules: Substituting for I 1 and I 2 in the junction rule:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 In general, for resistors in parallel The pair of resistors R 1 and R 2 can be replaced with a single equivalent resistor provided that

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 Example (text problem 18.40): In the given circuit, what is the total resistance between points A and B? R 2 and R 3 are in parallel. Replace with an equivalent resistor R 23. R 3 = 24 R 2 = 12 R 1 = 15 A B

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 R 23 =8 R 1 = 15 A B The resistors R 23 and R 1 are in series: The circuit can now be redrawn: B R 123 =23 A Is the equivalent circuit and the total resistance is 23 . Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 Capacitors: C2C2 C1C1 For capacitors in series the charge on the plates is the same. Apply Kirchhoff’s loop rule:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 In general, for capacitors in series The pair of capacitors C 1 and C 2 can be replaced with a single equivalent capacitor provided that

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 C2C2 C1C1 For capacitors in parallel the charge on the plates may be different. Here Apply Kirchhoff’s loop rule:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 In general, for capacitors in parallel The pair of capacitors C 1 and C 2 can be replaced with a single equivalent capacitor provided that C cq = C 1 + C 2.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 32 Example (text problem 18.49): Find the value of a single capacitor that replaces the three in the circuit below if C 1 = C 2 = C 3 = 12 F. C2C2 C1C1 C3C3 B A C 2 and C 3 are in parallel

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 The circuit can be redrawn: C1C1 C 23 B A The remaining two capacitors are in series. C 123 B A Is the final, equivalent circuit. Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 § 18.7 Circuit Analysis Using Kirchhoff’s Rules To solve multiloop circuit problems: 1.Assign polarity (+/-) to all EMF sources. 2.Assign currents to each branch of the circuit. 3.Apply Kirchhoff’s rules. 4.Solve for the unknowns.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 35 Example (text problem 18.53): Find the three unknown currents (the current in each resistor). I3I3 I2I2 I1I1 - + + - R2R2 R3R3 R1R1 22 11

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 36 Loop EDCFE: Loop AFCBA: Note: Could also use Loop AFEDCBA Junction C: Note: could also use junction F Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 37 The point is to write down three equations for the three unknown currents. (1) (2) (3) Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 38 Substitute (3) into (1): Multiply the top equation by –R 3 and the bottom equation by +R 1, add the equations together, then solve for I 2. (4) (2) Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 39 Substitute I 2 = -0.123 amps in to (2): Now substitute the known values of I 2 and I 3 into (3): Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 40 Example continued: The negative sign on I 2 means that instead of the current going from right to left (from point C to point F) in the branch with resistor 2, it really goes from left to right. It is essential to keep the negative sign when evaluating your equations numerically. Make the correction only when the problem is finished.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 41 § 18.8 Power and Energy in Circuits The energy dissipation rate is: For an EMF source: For a resistor:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 42 § 18.9 Measuring Currents and Voltages Current is measured with an ammeter. An ammeter is placed in series with a circuit component. R2R2 R1R1 A1A1 A3A3 A2A2 A 1 measures the current through R 1. A 2 measures the current through R 2. An ammeter has a low internal resistance. A 3 measures the current drawn from the EMF.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 43 A voltmeter is used to measure the potential drop across a circuit element. It is placed in parallel with the component. A voltmeter has a large internal resistance. R2R2 R1R1 V The voltmeter measures the voltage drop across R 1.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 44 § 18.10 RC Circuits C R Switch Close the switch at t=0 to start the flow of current. The capacitor is being charged. Apply Kirchhoff’s loop rule: + - + -

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 45 The current I(t) that satisfies Kirchhoff’s loop rule is: where is the RC time constant and is a measure of the charge (and discharge) rate of a capacitor.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 46 The voltage drop across the capacitor is: The voltage drop across the resistor is: The charge on the capacitor is: Note: Kirchhoff’s loop rule must be satisfied for all times.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 47 Plots of the voltage drop across the (charging) capacitor and current in the circuit.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 48 C R S1S1 S2S2 + - I While the capacitor is charging S 2 is open. After the capacitor is fully charged S 1 is opened at the same time S 2 is closed: this removes the battery from the circuit. Current will now flow in the right hand loop only, discharging the capacitor. Apply Kirchhoff’s loop rule: The current in the circuit is But the voltage drop across the capacitor is now

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 49 The voltage drop across the discharging capacitor:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 50 Example (text problem 18.83): A capacitor is charged to an initial voltage of V 0 =9.0 volts. The capacitor is then discharged through a resistor. The current is measured and is shown in the figure.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 51 (a) Find C, R, and the total energy dissipated in the resistor. Use the graph to determine . I 0 =100 mA; the current is I 0 /e = 36.8 mA at t= 13 msec. Since = V 0 = 9.0 volts, R = 90 and C = 144 F. All of the energy stored in the capacitor is eventually dissipated by the resistor. Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 52 (b) At what time is the energy in the capacitor half of the initial value? Want: Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 53 Solve for t: Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 54 Summary Current & Drift Velocity Resistance & Resistivity Ohm’s Law Kirchhoff’s Rules Series/Parallel Resistors/Capacitors Power Voltmeters & Ammeters RC Circuits

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