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Chapter 18: Direct-Current Circuits Source of EMF Homework assignment : 9,16,28,41,50 What is emf? • A current is maintained in a closed circuit by a source of emf. The term emf was originally an abbreviation for electromotive force but emf is NOT really a force, so the long term is discouraged. • Among such sources are any devices (batteries, generators etc.) that increase the potential energy of the circulating charges. • A source of emf works as “charge pump” that forces electrons to move in a direction opposite the electrostatic field inside the source. Source of EMF Maintaining a steady current and electromotive force • When a charge q goes around a complete circuit and returns to its starting point, the potential energy must be the same as at the beginning. • But the charge loses part of its potential energy due to resistance in a conductor. • There needs to be something in the circuit that increases the potential energy. • This something to increase the potential energy is called electromotive force (emf). Units: 1 V = 1 J/C • Emf (E) makes current flow from lower to higher potential. A device that produces emf is called a source of emf. source of emf -If a positive charge q is moved from b to a inside the b Fe - E a E + Fn current flow E source, the non-electrostatic force Fn does a positive amount of work Wn=qE on the charge. -This replacement is opposite to the electrostatic force Fe, so the potential energy associated with the charge increases by qVab . For an ideal source of emf Fe=Fn in magnitude but opposite in direction. -Wn=qE=qVab, so Vab=E=IR for an ideal source. Source of EMF Internal resistance • Real sources in a circuit do not behave ideally; the potential difference across a real source in a circuit is not equal to the emf. Vab=E – Ir (terminal voltage, source with internal resistance r) • So it is only true that Vab=E only when I=0. Furthermore, E –Ir = IR or I = E / (R + r) Source of EMF Real battery cc I a b r dd R Battery + b a − • Real battery has internal resistance, r. • Terminal voltage, ΔVoutput = (Va −Vb) = − I r. • Vout I r I R R I Rr Source of EMF Potential in an ideal resistor circuit c d a b a b c d b Source of EMF Potential in a resistor circuit in realistic situation c I d R Battery r a b - + V ba r R + - Ir IR 0 d c ab ab Source of EMF Example r 2 , 12 V, R 4 A a Vcd Vab b V ammeter I Rr Vab Vcd . 12 V 2 A. 42 Vcd IR (2 A)(4 ) 8 V. Vab Ir 12 V - (2 A)(2 ) 8 V. voltmeter The rate of energy conversion in the battery is I (12 V)(2 A) 24 W. The rate of dissipatio n of energy in the battery is Ir 2 (2 A) 2 (2 ) 8 W. The electrical power output is I I 2 r 16 W. The power output is also given by Vbc I (8 V)(2 A) 16 W. It is also given by IR 2 (2 A) 2 (4 ) 16 W. Resistors in Series Resistors in series V V IR1 IR2 V IReq Req R1 R2 In general you can extend this formula to : Req i Ri The equivalent resistance of a series combination of resistors is algebraic sum of the individual resistances. Resistors in Parallel Resistors in parallel V V + - + - V V V 1 1 1 I I1 I 2 Req R1 R2 Req R1 R2 I1 R2 V I1 R1 I 2 R2 I 2 R1 1 1 In general you can extend this formula to : i Req Ri Resistors in Series and Parallel Example 1: Resistors in Series and Parallel Example: (cont’d) I2 R2 I4 R4 I3 I R3 V I V / Req 12 V/2 6 A I 3 V / R3 12 V/3 4 A I 2 I 4 V /( R2 R3 ) 12 V/(2 4 ) 2 A Resistors in Series and Parallel Example: (cont’d) Kirchhoff’s Rules Introduction • Many practical resistor networks cannot be reduced to simple series-parallel combinations (see an example below). • Terminology: -A junction in a circuit is a point where three or more conductors meet. -A loop is any closed conducting path. junction Loop 2 i i i2 i1 i Loop 1 i i2 junction Kirchhoff’s Rules Kirchhoff’s junction rule • The algebraic sum of the currents into any unction is zero: I 0 at any junction Kirchhoff’s Rules Kirchhoff’s loop rule • The algebraic sum of the potential differences in any loop, including those associated with emfs and those of resistive elements, must equal zero. V 0 for any loop Kirchhoff’s Rules Rules for Kirchhoff’s loop rule I 0 at any junction V 0 for any loop Kirchhoff’s Rules Rules for Kirchhoff’s loop rule (cont’d) Kirchhoff’s Rules Solving problems using Kirchhoff’s rules Kirchhoff’s Rules Example 1 Kirchhoff’s Rules Example 1 (cont’d) Kirchhoff’s Rules Example 1 (cont’d) Kirchhoff’s Rules Find all the currents Example 2 including directions. Loop 2 i i i2 i1 i Loop 1 i Loop 1 0 8V 4V 4V 3i 2i1 0 8 3i1 3i 2 2i1 0 8 5i1 3i 2 multiply by 2 i = i1+ i2 i2 Loop 2 6i 2 4 2i1 0 6i 2 16 10i1 0 0 12 12i1 0 i1 1A 6i2 4 2(1A) 0 i 2 1A i 2A R-C Circuits Charging a capacitor R-C Circuits Charging a capacitor (cont’d) R-C Circuits Charging a capacitor (cont’d) R-C Circuits Charging a capacitor (cont’d) R-C Circuits Charging a capacitor (cont’d) R-C Circuits Discharging a capacitor R-C Circuits Discharging a capacitor (cont’d) R-C Circuits Discharging a capacitor (cont’d) R-C Circuits Example 18.6 : Charging a capacitor in an RC circuit An uncharged capacitor and a resistor are connected in series to a battery. If E=12.0 V, C=5.00 mF, and R= 8.00x105 , find (a) the time constant of the circuit, (b) the maximum charge on the capacitor, (c) the charge on the capacitor after 6.00 s, (d) the potential difference across the resistor after 6.00 s, and (e) the current in the resistor at that time. 5 -6 (a) RC (8.00 10 )(5.00 10 F) 4.00 s (b) From Kirchhoff’s loop rule: Vbat VC VR 0 E q IR 0 Q CE 60.0 mC c when I=0, q=Q at max. R-C Circuits Example 18.6 : Charging a capacitor in an RC circuit An uncharged capacitor and a resistor are connected in series to a battery. If E=12.0 V, C=5.00 mF, and R= 8.00x105 , find (a) the time constant of the circuit, (b) the maximum charge on the capacitor, (c) the charge on the capacitor after 6.00 s, (d) the potential difference across the resistor after 6.00 s, and (e) the current in the resistor at that time. t / 6.00 s/4.00s ) 46.6 mC (c) q Q(1 e ) (60.0 mC)(1 e (d) VC q / C 9.32 V VR Vbat VC 12.0 (9.32 V) -2.68 V (e) I VR 3.4 10 6 A R