b Physics 2102 a Jonathan Dowling Lecture 15 MON 16 FEB DC Circuits Ch27 1 4 QuickTime and a decompressor are needed to see this picture b EMF Devices and Single Loop Circuits The battery operates as a pump that moves positive charges from lower to higher electric potential A battery is an example of an electromotive force EMF device a These come in various kinds and all transform one source of energy into electrical energy A battery uses chemical energy a generator mechanical energy a solar cell energy from light i etc a The difference in potential energy that the device establishes is called the EMF and denoted by E E iR c b d i iR E Va a b c d a Circuit Problems Given the EMF devices and resistors in a circuit we want to calculate the circulating currents Circuit solving consists in taking a walk along the wires As one walks through the circuit in any direction one needs toEMF follow When walking through an add two E ifrules you flow with the current or E against How to remember current gains potential lking through a resistor add iR if flowing with the current or i in a battery How to remember resistors are passive current flows potential d xample alking clockwise from a alking counter clockwise E iR 0 from a E iR 0 Ideal vs Real Batteries If one connects resistors of lower and lower value of R to get higher and higher currents eventually a real battery fails to establish the potential difference E and settles for a lower value One can represent a real EMF device as an ideal one attached to a resistor called internal resistance of the EMF device E ir iR 0 i E r R Etrue E ir The true current EMF the is a more function of current we want Resistances in Series i is Constant Two resistors are in series if they are connected such that the same current i flows in both The equivalent resistance is a single imaginary resistor that can replace the resistances in series In the circuit with the equivalent resistance Walking the loop results in E iR1 iR2 iR3 0 i E R1 R2 R3 E iReq 0 i E Req Thus n Req R j j 1 Resistors in Parallel V is Constant Two resistors are in parallel if they are connected such that there is the same potential V drop through both The equivalent resistance is a single imaginary resistor that can replace the resistances in parallel E i1R1 0 E i2 R2 0 E i3 R3 0 Walking the loops results in i i1 i2 i3 E R1 E R2 E R3 E 1 R1 1 R2 1 R3 The total current delivered i E Reqis by the battery n 1 1 Req j 1 R j Resistors Capacitors V iR Q CV Series I dQ dt Same Series Q Same Rser R1 R2 R3 1 Cser 1 C1 1 C2 1 C3 Parallel V Same Parallel V Same 1 Rpar 1 R1 1 R2 1 R3 Cpar C1 C2 C3 Resistors in Series and Parallel An electrical cable consists of 100 strands of fine wire each having r 2 resistance The same potential difference is applied between the ends of all the strands and results in a total current of I 5 A What is the current in each strand Ans ip 0 05 A i I 100 What is the applied potential difference Ans vp 0 1 V vp V isr constant What is the resistance of the cable Ans Rp r 0 02 1 Rp 1 r 1 r 100 r R r 100 Parallel Assume now that the same 2 strands in the cable are tied in series one after the other and the 100 times longer cable connected to the same V 0 1 Volts potential difference as before a What is the potential difference through each strand Ans vs 0 001 V vs V 100 b What is the current in each strand Ans is 0 0005 A is vs r constant c What is the resistance of the cable Ans 200 Rs r r r 100r d Which cable gets hotter the one with strands in parallel or the one with strands in series Series Ans Each strand in parallel dissipates P p ivp 5mW and the cable dissipates 100 P p 500mW Each strand in series dissipates P s is vs 50 W and the cable dissipates 5mW Example Bottom loop all else is irrelevan V same in parallel 12V 8W V 12V i 1 5 A R 8 Which resistor 3 or 5 gets hotter P i2R Example a Which circuit has the largest equivalent resistance b Assuming that all resistors are the same which one dissipates more power c Which resistor has the smallest potential difference Example Find the equivalent resistance between points a F and H and b F and G Hint For each pair of points imagine that a battery is connected across the pair Monster Mazes If all resistors have a resistance of 4 and all batteries are ideal and have an emf of 4V what is the current through R If all capacitors have a capacitance of 6 F and all batteries are ideal and have an emf of 10V what is the charge on capacitor C
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