1. Lecture 11-1 Electric Current Current = charges in motion Magnitude rate at which net positive charges move across a cross sectional surface Units: [I] = C/s = A (ampere) Current is a scalar, signed quantity, whose sign corresponds to the direction of motion of net positive charges by convention J = current density (vector) in A/m²

2. Lecture 11-2 Ohm’s Law constant R Ohm’s Law Resistance (definition) V R I Power dissipation :

3. Lecture 11-3 EMF – Electromotive Force An EMF device is a charge pump that can maintain a potential difference across two terminals by doing work on the charges when necessary. Examples: battery, fuel cell, electric generator, solar cell, fuel cell, thermopile, … Converts energy (chemical, mechanical, solar, thermal, …) into electrical energy.  Within the EMF device, positive charges are lifted from lower to higher potential.  If work dW is required to lift charge dq, EMF

4. Lecture 11-4 Resistors in Series  The current through devices in series is always the same. R1R1 R2R2 i i ε R eq i ε For multiple resistors in series: Same equation for parallel connected capacitors

5. Lecture 11-5 Real Battery = Resistors in Series  The current through devices in series is always the same. R eq i ε internal resistance terminal voltage

6. Lecture 11-6 Resistors in Parallel Devices in parallel has the same potential drop Generally, Same equation for capacitors connected in serial

7. Lecture 11-7 Kirchhoff’s Rules Kirchhoff’s Rule 1: Loop Rule  When any closed loop is traversed completely in a circuit, the algebraic sum of the changes in potential is equal to zero. Kirchhoff’s Rule 2: Junction Rule  The sum of currents entering any junction in a circuit is equal to the sum of currents leaving that junction.  Conservation of charge  In and Out branches  Assign I i to each branch  Coulomb force is conservative

8. Lecture 11-8 Circuit Analysis Tips Simplify using equivalent resistors Label currents with arbitary directions If the calculated current is negative, the real direction is opposite to the one defined by you. Apply Junction Rule to all the labeled currents. Useful when having multiple loops in a circuit. Choose independent loops and define loop direction Imagine your following the loop and it’s direction to walk around the circuit. Use Loop Rule for each single loop If current I direction across a resistor R is the same as the loop direction, potential drop across R is ∆V = −I×R, otherwise, ∆V = I×R For a device, e.g. battery or capacitor, rely on the direction of the electric field in the device and the loop direction to determine the Potential drop across the device Solve simultaneous linear equations

9. Lecture 11-9 Loop Example with Two EMF Devices  If  1 <  2, we have I<0 !? This just means the actual current flows reverse to the assumed direction. No problem!

10. Lecture 11-10 Finding Potential and Power in a Circuit Just means 0 V here But what is I? Must solve for I first! supplied by 12V battery dissipated by resistors The rest? into 4V battery (charging)

11. Lecture 11-11 Charging a Battery Positive terminal to positive terminal Charging EMF > EMF of charged device Say, R+r 1 +r 2 =0.05  (R is for jumper cables). Then, power into battery 2 battery being charged (11V) good battery (12V) If connected backward,  Large amount of gas produced  Huge power dissipation in wires

12. Lecture 11-12 Using Kirchhoff’s Laws in Multiple Loop Circuits Identify nodes and use Junction Rule: Identify independent loops and use Loop Rule: Only two are independent.

13. Lecture 11-13 Warm-up quiz What’s the current I 1 ? I 1 +I 2 I2I2 I1I1 (a). 2.0A (b). 1.0A (c). -2.0A (d). -1.0A (e). Need more information to calculate the value.

14. Lecture 11-14 Answer for the Warm-up quiz Replace by equivalent R=2  first. Sketch the diagram Simplify using equivalent resistors Label currents with directions Use Junction Rule in labeling Choose independent loops Use Loop Rule Solve simultaneous linear equations I 1 +I 2 I2I2 I1I1

15. Lecture 11-15 Ammeter and Voltmeter Ammeter: an instrument used to measure currents It must be connected in series. The internal resistance of an ammeter must be kept as small as possible. Voltmeter: an instrument used to measure potential differences It must be connected in parallel. The internal resistance of a voltmeter must be made as large as possible.

16. Lecture 11-16 Galvanometer Inside Ammeter and Voltmeter Ammeter: an instrument used to measure currents Voltmeter: an instrument used to measure potential differences galvanometer shunt resistor galvanometer Galvanometer: a device that detects small currents and indicates its magnitude. Its own resistance R g is small for not disturbing what is being measured.

17. Lecture 11-17 What is the current through R 1 ? a.0.575A b.0.5A c.0.75A d.0.33A e.1.5A PHYS241 – Quiz 11A 45V 30  45V 30  R1R1 R2R2 R3R3

18. Lecture 11-18 What is the current through R 2 ? a.0.33A b.2.5A c.0.75A d.1.5A e.0.5A PHYS241 – Quiz11B 15V 10  15V 10  R1R1 R2R2 R3R3

19. Lecture 11-19 What is the current through R 3 ? a.0.375A b.0.5A c.0.75A d.1A e.1.5A PHYS241 – Quiz 11C 30V 20  30V 20  R1R1 R2R2 R3R3