Lecture 26 • Energy and Power • Resistors in Series • Real batteries • Resistors in Parallel • Example of Resistor Circuits • Grounding Energy and Power In battery: chemical energy potential...of charges:∆U = q∆V = q • bat E power (rate at which energy supplied to charges):P = dU = dq • bat dt dt E • In resistor: work done on charges qEd kinetic (accelerate) between collisions thermal energy of lattice after collisions After many collisions over length L of resistor: ∆Eth = qEL = q∆VR rate at which energy is transferred from current to resistor: dEth dq PR = dt = dt ∆VR = I∆VR For basic circuit: PR = Pbat (energy conservation) Using Ohm’s law: E U K E light... chem → → → th → Common units: P kW in ∆t hours P ∆t kilowatt hours • R → R Example on electric power • A standard 100 W (120 V ) lightbulb contains a 9.00!cm-long tungsten filament. The high-temperature resistivity of tungsten is -7 9.0 x 10 ohm-meter. What is the diameter of the filament? Resistors in Series current same in each resistor: ∆V = IR ; ∆V = IR ∆V = I (R + R ) • 1 1 2 2 ⇒ ab 1 2 equivalent resistor: ∆Vab I(R1+R2) • Rab = I = I = R1 + R2 Ammeters • measures current in circuit element (placed in series): Req = Rload + Rammeter ideal, R =0 current not changed ammeter ⇒ Real Batteries • terminal (user) voltage ∆Vba t = E − I r • for resistor connected: I = E = E Req R+r ∆V = IR = R R R+r E ∆Vbat = Ir; Short Circuit ∆V = ∆EV− = R bat " E replace high by E • I s ho r t = r zero resistance ( ∞ for ideal, r = 0) • maximum possible current battery can produce Parallel Resistors • potential differences same: ∆V1 = ∆V2 = = Vcd • Kirchhoff’s junction law: I = I 1 + I 2 • Ohm’s law: I = ∆V1 + ∆V2 = ∆V 1 + 1 R1 R2 cd R1 R2 • Replace by equivalent resistance:! " − 1 R = ∆ V c d = 1 + 1 cd I R 1 R 2 ! " (R = R ) R • Identical resistors 1 2 : Rse r i es eq =2R; Rp a r a l lel eq = 2 • In general, R e q < R 1 or R 2 ...in parallel • measure potential difference across circuit Voltmeters element (place in parallel; don’t need to break connections, unlike for ammeter) − 1 1 1 total resistance: R eq = + • R R v o l t m e t e r • in order not to change voltage: R R (ideally, ) voltmeter ! ∞ Strategy for Resistor Circuits • assume ideal wires, batteries • use Kirchhoff’s laws and rules for series and parallel resistors • reduce to equivalent resistor (basic circuit) • rebuild using current same for series and potential difference same for parallel Example of Resistor Circuit • Find I a n d ∆ V for each resistor • so far, potential difference, ∆ V (physically Grounding meaningful), enters Ohm’s law (no need to establish zero point) • need common reference point for connecting 2 circuits • grounding: connect 1 point in circuit to earth ( ) by ideal wire: no return Ve a r t h = 0 wire (no current); wire is equipotential point in circuit has earth’s potential specific value of potential at each point • does not change how circuit functions... unless malfunction/breaking of circuit (enclose circuit in grounded case insulated from circuit): case comes in contact with circuit ( touching would have caused electrocution) current thru’ wire (fuse blows...).