1. Physics for Scientists and Engineers II, Summer Semester 2009 1 Lecture 9: June 10 th 2009 Physics for Scientists and Engineers II

2. Physics for Scientists and Engineers II, Summer Semester 2009 2 Direct Current (DC) Circuits Batteries: Have constant electric potential difference between terminals.  Current’s direction is constant / Direct Current (DC) Batteries are a source of “electromotive force (emf)” (is not really a force…but an electric potential difference.)  is the maximum possible voltage the battery can provide between it’s terminals. Batteries have an “internal resistance” r  The terminal voltage for a real battery is LESS than  as soon as a current flows.

3. Physics for Scientists and Engineers II, Summer Semester 2009 3 Internal Resistance of Batteries Batteries have an “internal resistance” r  The terminal voltage for a real battery is LESS than  as soon as a current flows. +- a b c d e f R Delivered by battery Delivered to R Delivered to r

4. Physics for Scientists and Engineers II, Summer Semester 2009 4 Resistors in Series The same current passes through all resistors that are connected in series. + - a R1R1 R2R2 b c

5. Physics for Scientists and Engineers II, Summer Semester 2009 5 Resistors in Parallel The potential difference across resistors that are connected in parallel is the same. + - a R1R1 b c R2R2 d Points c and d are called “junctions”: The current I splits up into I 1 and I 2 at point c. The currents I 1 and I 2 combine at point d into I again.

6. Physics for Scientists and Engineers II, Summer Semester 2009 6 Example + - 4.00  1.00  48.0  3.0  10.0  6.0  34.0  1)What is R eq of this circuit? 2)Determine the current through each resistor. 3)Determine the voltage drop across each resistor. 4)Determine the power dissipated in each resistor. 5)What happens to the current through the battery if a resistor is removed? (Discuss qualitatively for each R).

7. Physics for Scientists and Engineers II, Summer Semester 2009 7 Example : What is R eq of this circuit? + - 4.00  1.00  48.0  3.0  10.0  6.0  34.0 

8. Physics for Scientists and Engineers II, Summer Semester 2009 8 Example : What is R eq of this circuit? + - 4.00  48.0  34.0 

9. Physics for Scientists and Engineers II, Summer Semester 2009 9 Example : What is R eq of this circuit? + - 34.0 

10. Physics for Scientists and Engineers II, Summer Semester 2009 10 Example : What is R eq of this circuit? + -

11. Physics for Scientists and Engineers II, Summer Semester 2009 11 Example : What is the current through each resistor? + - + - 4.00  1.00  48.0  3.0  10.0  6.0  34.0 

12. Physics for Scientists and Engineers II, Summer Semester 2009 12 Example : Get voltage drop across R C and R d. + - 34.0 

13. Physics for Scientists and Engineers II, Summer Semester 2009 13 Example : Get the branch currents. + - 4.00  48.0  34.0  a b c

14. Physics for Scientists and Engineers II, Summer Semester 2009 14 Example: Get all the voltage drops. + - 4.00  1.00  48.0  3.00  10.0  6.00  34.0 

15. Physics for Scientists and Engineers II, Summer Semester 2009 15 Example: Get all the powers dissipated. + - 4.00  1.00  48.0  3.00  10.0  6.00  34.0 

16. Physics for Scientists and Engineers II, Summer Semester 2009 16 Kirchhoff’s Rules

17. Physics for Scientists and Engineers II, Summer Semester 2009 17 Kirchhoff’s Rules: More on Junction Rule Count incoming currents as positive and outgoing currents as negative.

18. Physics for Scientists and Engineers II, Summer Semester 2009 18 Kirchhoff’s Rules: More on Loop Rule + - Imagine traveling along the indicated loop in the given direction: Sum up all the electric potential changes along the closed loop. 1)The emf is traversed from the negative to the positive side:  V increases (count positive) 2)The resistor R 1 is traversed in the same direction as the current:  V drops (count negative) 3 The resistor R 2 is traversed in the same direction as the current:  V drops (count negative)

19. Physics for Scientists and Engineers II, Summer Semester 2009 19 Kirchhoff’s Rules: More on Loop Rule + - Imagine traveling along the indicated loop in the other direction: Sum up all the electric potential changes along the closed loop. 1)The emf is traversed from the positive to the negative side:  V decreases (count negative) 2)The resistor R 1 is traversed in the opposite direction as the current:  V drops (count positive) 3 The resistor R 2 is traversed in the opposite direction as the current:  V drops (count positive) This is the same equation as before, just multiplied by -1.

20. Physics for Scientists and Engineers II, Summer Semester 2009 20 Example with two batteries + - + - Notice: I indicated a direction of current, even though I do not know in which direction the current really flows here.

21. Physics for Scientists and Engineers II, Summer Semester 2009 21 Example with two batteries + - + -

22. Physics for Scientists and Engineers II, Summer Semester 2009 22 Example: Multiple branches and batteries + - + - + - Find the currents through the resistors.

23. Physics for Scientists and Engineers II, Summer Semester 2009 23 Step1: Identify junctions and indicate currents + - + - + - You can choose the current direction as you like. A B

24. Physics for Scientists and Engineers II, Summer Semester 2009 24 Step2: Write down junction rules + - + - + - A B

25. Physics for Scientists and Engineers II, Summer Semester 2009 25 Step3: Indicate loops and write loop rules + - + - + - A B Loop 1 Loop 2 Loop 3

26. Physics for Scientists and Engineers II, Summer Semester 2009 26 Step4: Look at all the independent equations:

27. Physics for Scientists and Engineers II, Summer Semester 2009 27 Step5: Solve for the unknowns

28. Physics for Scientists and Engineers II, Summer Semester 2009 28 ….and solve….

29. Physics for Scientists and Engineers II, Summer Semester 2009 29 Step6: Indicate actual current direction + - + - + - A B