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## Chapter 18: Direct-Current Circuits

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**Chapter 18: Direct-Current Circuits**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. Source of EMF • 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.**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 source of emf • If a positive charge q is moved from b to a inside the • 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. b a - + current flow**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**R c d c d I Battery r a b b a + − Source of EMF • Real battery has internal resistance, r. • Terminal voltage,ΔVoutput= (Va−Vb) = − I r.**Potential in an ideal resistor circuit**c d a b Source of EMF b a c d b**Potential in a resistor circuit in realistic situation**c d R I Battery r b a b a - + Source of EMF r V R - + e I r IR 0 d c b a a b**Example**A ammeter a b Source of EMF voltmeter V**Resistors in series**DV DV Resistors in Series The equivalent resistance of a series combination of resistors is algebraic sum of the individual resistances.**Resistors in parallel**DV DV + - - + Resistors in Parallel**Example 1:**Resistors in Series and Parallel**Example: (cont’d)**I4 I2 R2 R4 I3 Resistors in Series and Parallel I R3 DV**Example: (cont’d)**Resistors in Series and Parallel**Loop 2**i i i2 i1 Loop 1 i i i2 • 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 Kirchhoff’s Rules junction**Kirchhoff’s junction rule**• The algebraic sum of the currents into any unction is zero: 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. Kirchhoff’s Rules**Rules for Kirchhoff’s loop rule**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**Loop 2**i i i2 i1 Loop 1 i i i2 Find all the currents including directions. • Example 2 Kirchhoff’s Rules Loop 1 Loop 2 multiply by 2 i = i1+ i2**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= 8.00x105W, 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. R-C Circuits (a) (b) From Kirchhoff’s loop rule: when I=0, q=Q at max.**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= 8.00x105W, 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. R-C Circuits (c) (d) (e)