Lecture 10. Resistor Circuits, Batteries and EMF

Transcription

1 Lecture 10. Resistor Circuits, Batteries and EMF Outline: Connection of Resistors: In Parallel and In Series. Batteries. Non-ideal batteries: internal resistance. Potential distribution around a complete circuit Lecture 9: DC current: flow of charge carriers, requires E 0 in a conductor. To keep current running, we need to maintain a non-zero potential difference across a conductor. Microscopic picture: electron mosquito cloud slowly drifting in the field. Linear regime: Ohm s Law = drift velocity E Resistance: the coefficient of proportionality between V and I, depends on materials parameters. 1

2 Connections: In Series vs. In Parallel + - In Series: current is the same through all elements voltage across them can be different + - In Parallel: voltage is the same across all elements current through them can be different (compare with capacitors: replace current with charge) 4

3 Resistors: Connection in Series and in Parallel V aa In Series Common quantity: current R ee V aa I - voltage difference across R 1, R 2, R 3 - common current V aa = V 1 + V 2 + V 3 = I R 1 + R 2 + R 3 R ee V aa I R ee = R i i In Parallel Common quantity: voltage difference - common voltage difference - total current I I = I 1 + I 2 + I 3 = V aa R 1 + V aa R 2 + V aa R 3 R ee = 1 R R R 3 1 = 1 R i i R 1 R 2 : R ee = R 1R 2 R 1 + R 2 1 9

4 More Complex Circuits I 1Ω 1Ω R 4 R 5 1Ω 1Ω R 1 1Ω 2Ω 1Ω 2Ω 3Ω 3Ω 3Ω 1/3Ω 3Ω 3Ω 3Ω R 2 1/3Ω 1/3Ω R 1/3Ω 3 R 1 = = 1Ω R 2 = = 1Ω R 2 R 5 = = 2Ω R 4 R 2 R 5 = = 2 3 Ω R 3 R 4 R 2 R 5 = = 1Ω R ee = R 1 R 3 R 4 R 2 R 5 = = 0.5Ω I = 10V 0.5Ω = 20A P = II = 200W 12

5 Ideal Batteries (no energy dissipation inside battery) R + V=V - V=V V=0 R We assume that the wires have resistance much smaller than the load R (all voltage difference provided by the battery is applied to the load R). Sketch of a (conservative) electric field + - E dl = 0 V battery wire resistor wire x Conclusion: an external agent inside the battery (not the electrostatic E!!) forces the charge carriers to climb the potential hill. 13

6 External Forces (not Electrostatic Field) separate Charges + E NN E - R E Demonstrations: lemon battery, dynamo Mechanical analogy: van de Graaff generator Fictitious field E NN inside the battery: a source of an additional force on charge carriers. This field is non-conservative: E NN dl 0 (E NN is zero outside the battery). Inside the battery, electrons are driven by the total electric field E nnn = E + E NN. 14

7 Electromotive Force (EMF) + E NN E - E nnn = E + E NN V R E + E E NN dl - the electromotive force Units: volts E V battery external circuit x Ideal battery: E = V = II V voltage difference between the battery s electrodes R net resistance of the external circuit 15

8 Non-ideal Battery I What is the output voltage? R ee = R + r I = E r + R I V = E II = E R r + R r the internal resistance of the battery V = E R r + R Demonstration of the internal resistance. 16

9 Battery Discharge The internal resistance of a battery increases in the process of work: it is small (in comparison with a typical load resistor) for a fresh battery, and becomes large for an old (discharged) one. V E E/2 V = E R r + R = E 1 r/r r/r 17

10 Battery Tester Can we use a voltmeter (very high r) to test the freshness of a battery? The voltmeter will measure V E provided r R ii. But R s is usually as high as Ω, and even if R ii ~ Ω, we won t notice the battery aging. typical R load 18

11 Voltage Source The goal: to provide output voltage that is independent of the load resistance. An ideal voltage source: V = ℇ R r + R A. has zero internal resistance ( zero means that r << all possible values of load R). B. has infinite internal resistance ( infinite means that r >> all possible values of load R). 19

12 Voltage Source The goal: to provide output voltage that is independent of the load resistance. V = ℇ R r + R An ideal voltage source: A. has zero internal resistance ( zero means that r << all possible values of load R). B. has infinite internal resistance ( infinite means that r >> all possible values of load R). 20

13 Current Source I The goal: to provide output current that is independent of the load resistance. I = ℇ r + R An ideal current source: A. has zero internal resistance ( zero means that r << all possible values of load R). B. has infinite internal resistance ( infinite means that r >> all possible values of load R). 21

14 Current Source I The goal: to provide output current that is independent of the load resistance. I = ℇ r + R An ideal current source: A. has zero internal resistance ( zero means that r << all possible values of load R). B. has infinite internal resistance ( infinite means that r >> all possible values of load R). 22

15 Conclusion Batteries: the potential energy of charge carriers is increased by non-electrostatic (non-conservative) forces. Non-ideal batteries: internal resistance. Potential distribution around a complete circuit. Energy and power in electric circuits. Next time: Lecture 11: Connection of Resistors, Kirchhoff s Rules

16 Appendix 1: More Complicated Resistor Circuits R T R T R T = 2R 1 + R 2R T R 2 + R T R T 2 2R 1 R T 2R 1 R 2 = 0 Symmetry helps! Equipotential points can be connected (all a s and b s). 24