1. Physics 102: Lecture 4, Slide 1 Capacitors (& batteries) Physics 102: Lecture 04

2. Physics 102: Lecture 4, Slide 2 Phys 102 so far Lecture 1 – electric charge & electric force Lecture 2 – electric field Lecture 3 – electric potential energy and electric potential Lecture 4 – capacitance Lecture 5 – resistance Lecture 6 – Kirchhoff’s rules Lecture 7 – RC circuits Lecture 12 & 13 – AC circuits Basic principles of electricity Applications of electricity – circuits

3. Physics 102: Lecture 4, Slide 3 Recall from last lecture….. Electric Fields, Electric Potential

4. Physics 102: Lecture 4, Slide 4 Comparison: Electric Potential Energy vs. Electric Potential  V AB : the difference in electric potential between points B and A  U AB : the change in electric potential energy of a charge q when moved from A to B  U AB = q  V AB q AB

5. Physics 102: Lecture 4, Slide 5 Electric Potential: Summary E field lines point from higher to lower potential For positive charges, going from higher to lower potential is “downhill” Positive charges tend to go “downhill”, from + to  Negative charges go in the opposite direction, from  to +  U AB = q  V AB +– E-field Equipotential lines

6. Physics 102: Lecture 4, Slide 6 Important Special Case Uniform Electric Field Two large parallel conducting plates of area A +Q on one plate  Q on other plate Then E is uniform between the two plates: E=4  kQ/A zero everywhere else This result is independent of plate separation This is called a parallel plate capacitor d A E +Q –Q A ++++++++++++++++ ––––––––––––––––

7. Physics 102: Lecture 4, Slide 7 Parallel Plate Capacitor: Potential Difference A B E=E 0 d V =V A – V B = +E 0 d ++++++++++ ---------- A B d ++++++++++ ---------- ++++++++++ ---------- E=2 E 0 V =V A – V B = +2E 0 d Potential difference is proportional to charge: Double Q  Double V Charge Q on platesCharge 2Q on plates E 0 =4  kQ/A

8. Physics 102: Lecture 4, Slide 8 Capacitance: The ability to store separated charge C  Q/V Any pair conductors separated by a small distance. (e.g. two metal plates) Capacitor stores separated charge –Positive Q on one conductor, negative Q on other –Net charge is zero E d ++++++++++ –––––––––– Q=CV U =(½) Q V Stores Energy Units: 1 Coulomb/Volt = 1 Farad (F)

9. Physics 102: Lecture 4, Slide 9 Why Separate Charge? Camera Flash Defibrillator—see example in textbook AC → DC Tuners –Radio –Cell phones Cell membranes

10. Physics 102: Lecture 4, Slide 10 Capacitance of Parallel Plate Capacitor V = Ed E=4  kQ/A (Between two large plates) So: V = 4  kQd/A Recall: C  Q/V So: C = A/(4  kd) Recall:  0 =1  k)=8.85x10 -12 C 2 /Nm 2 C =  0 A/d Parallel plate capacitor d A E +Q –Q A V

11. Physics 102: Lecture 4, Slide 11 Dielectric Placing a dielectric between the plates increases the capacitance. C =  C 0 Capacitance with dielectric Dielectric constant (  > 1) Capacitance without dielectric d ++++++++++ ---------- ++++++++++ ---------- - + +- +- +- +- For same charge Q, E (and V) is reduced so C = Q/V increases

12. Physics 102: Lecture 4, Slide 12 A parallel plate capacitor given a charge q. The plates are then pulled a small distance further apart. What happens to the charge q on each plate of the capacitor? ACT: Parallel Plates 1) Increases2) Constant3) Decreases +q-q ++++++++ -------- d pull

13. Physics 102: Lecture 4, Slide 13 A parallel plate capacitor given a charge q. The plates are then pulled a small distance further apart. Which of the following apply to the situation after the plates have been moved? 1)The capacitance increases True False 2)The electric field increases True False 3)The voltage between the plates increases True False Preflight 4.1 +q-q ++++++++ -------- d pull

14. Physics 102: Lecture 4, Slide 14 A parallel plate capacitor given a charge q. The plates are then pulled a small distance further apart. Which of the following apply to the situation after the plates have been moved? ACT/Preflight 4.1 The energy stored in the capacitor A) increasesB) constantC) decreases +q-q ++++++++ -------- d pull

15. Physics 102: Lecture 4, Slide 15 Circuits Elements are connected by wires. Any connected region of wire has the same potential. V wire 1 = 0 VV wire 2 = 5 VV wire 3 = 12 VV wire 4 = 15 V V C 1 = 5-0 V= 5 VV C 3 = 15-12 V= 3 VV C 2 = 12-5 V= 7 V C1C1 C2C2 C3C3 The potential difference across an element is the element’s “voltage.”

16. Physics 102: Lecture 4, Slide 16 Capacitors in Parallel Both ends connected together by wire C1C1 C2C2 C eq 15 V 10 V 15 V 10 V 15 V 10 V Share Charge: Q eq = Q 1 +Q 2 Equivalent C:C eq = C 1 +C 2 = V eq Same voltage: V 1 = V 2 Add areas – remember C=  0 A/d

17. Physics 102: Lecture 4, Slide 17 Parallel Practice A 4  F capacitor and 6  F capacitor are connected in parallel and charged to 5 volts. Calculate C eq, and the charge on each capacitor. C4C4 C6C6 C eq 5 V 0 V 5 V 0 V 5 V 0 V C eq = C 4 +C 6 Q 4 = C 4 V 4 Q 6 = C 6 V 6 Q eq = C eq V eq = 4  F+6  F = 10  F = (4  F)(5 V) = 20  C = (6  F)(5 V) = 30  C = (10  F)(5 V) = 50  C = Q 4 +Q 6 V = 5 V

18. Physics 102: Lecture 4, Slide 18 Equivalent C: Capacitors in Series Connected end-to-end with NO other exits Same Charge: Q 1 = Q 2 = Q eq C eq C1C1 C2C2 -- - + - + - +Q -Q +Q -Q +Q -Q Share Voltage:V 1 +V 2 =V eq + Add d – remember C=  0 A/d

19. Physics 102: Lecture 4, Slide 19 Series Practice C eq C4C4 C6C6 + - + - + - +Q -Q +Q -Q +Q -Q A 4  F capacitor and 6  F capacitor are connected in series and charged to 5 volts. Calculate C eq, and the charge on the 4  F capacitor. 5 V 0 V 5 V 0 V Q = CV

20. Physics 102: Lecture 4, Slide 20 Comparison: Series vs. Parallel Series Can follow a wire from one element to the other with no branches in between. Parallel Can find a loop of wire containing both elements but no others (may have branches). C1C1 C2C2 C2C2 C1C1

21. Physics 102: Lecture 4, Slide 21 Electromotive Force Battery –Maintains constant potential difference V (electromotive force – emf ε) –Does NOT produce or supply charges, just “pushes” them. Analogy to pump… + -

22. Physics 102: Lecture 4, Slide 22 A circuit consists of three initially uncharged capacitors C 1, C 2, and C 3, which are then connected to a battery of emf ε. The capacitors obtain charges q 1, q 2,q 3, and have voltages across their plates V 1, V 2, and V 3. Which of these are true? C 2 C 3 C 1 ε + - -q 2 +q 1 -q 1 +q 3 -q 3 +q 2 V1V1 V2V2 V3V3 1)q 1 = q 2 2)q 2 = q 3 3)V 2 = V 3 4)ε = V 1 5)V 1 < V 2 6)C eq > C 1 Preflight 4.4

23. Physics 102: Lecture 4, Slide 23 C 2 C 3 C 1 ε + - -q 2 +q 1 -q 1 +q 3 -q 3 +q 2 V1V1 V2V2 V3V3 A circuit consists of three initially uncharged capacitors C 1, C 2, and C 3, which are then connected to a battery of emf ε. The capacitors obtain charges q 1, q 2, q 3, and have voltages across their plates V 1, V 2, and V 3. 1) q 1 = q 2 2) q 2 = q 3 ACT/Preflight 4.4: Which is true?

24. Physics 102: Lecture 4, Slide 24 C 2 C 3 C 1 ε + - -q 2 +q 1 -q 1 +q 3 -q 3 +q 2 V1V1 V2V2 V3V3 1) V 2 = V 3 2) ε = V 1 A circuit consists of three initially uncharged capacitors C 1, C 2, and C 3, which are then connected to a battery of emf ε. The capacitors obtain charges q 1, q 2, q 3, and have voltages across their plates V 1, V 2, and V 3. ACT/Preflight 4.4: Which is true?

25. Physics 102: Lecture 4, Slide 25 C 2 C 3 C 1 ε + - -q 2 +q 1 -q 1 +q 3 -q 3 +q 2 V1V1 V2V2 V3V3 1) V 1 < V 2 2) C eq > C 1 A circuit consists of three initially uncharged capacitors C 1, C 2, and C 3, which are then connected to a battery of emf ε. The capacitors obtain charges q 1, q 2, q 3, and have voltages across their plates V 1, V 2, and V 3. ACT/Preflight 4.4: Which is true?

26. Physics 102: Lecture 4, Slide 26 Capacitance C = Q/V Parallel Plate: C =  0 A/d Capacitors in parallel: C eq = C 1 +C 2 Capacitors in series: 1/C eq = 1/C 1 +1/C 2 Batteries provide fixed potential difference Recap of Today’s Lecture