b Physics 2102 a Jonathan Dowling Lecture 15 MON 16 FEB DC Circuits Ch27 1 4 EMF Devices and Single Loop Circuits b 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 etc i a The difference in potential energy that the device establishes is called the EMF and denoted by E c b d i iR E E iR 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 to follow two rules When walking through an EMF add E if you flow with the current or E against How to remember current gains potential in a battery When walking through a resistor add iR if flowing with the current or iR against How to remember resistors are passive current flows potential down Example Walking clockwise from a E iR 0 Walking counter clockwise 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 E true E ir The true EMF is a function of current the more current we want the smaller Etrue we get 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 Walking the loops results in E i1R1 0 E i2 R2 0 E i3 R3 0 The total current delivered by the battery is i i1 i2 i3 E R1 E R2 E R3 E 1 R1 1 R2 1 R3 In the circuit with the equivalent i E Req resistor 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 a What is the current in each strand Ans ip 0 05 A i I 100 b What is the applied potential difference Ans vp 0 1 V vp V isr constant c 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 d What is the potential difference through each strand Ans vs 0 001 V vs V 100 Series e What is the current in each strand Ans is 0 0005 A is vs r constant f What is the resistance of the cable Ans 200 Rs r r r 100r g Which cable gets hotter the one with strands in parallel or the one with strands in series Ans Each strand in parallel dissipates Pp ivp 5mW and the cable dissipates 100 Pp 500mW Each strand in series dissipates Ps is vs 50 W and the cable dissipates 5mW Example Bottom loop all else is irrelevant V same in parallel 12V 8W i V 12V 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 across it 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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