1. Lecture 22: WED 14 OCT Exam 2: Review Session CH 24–27.3 & HW 05–07 Physics2113 Jonathan Dowling Some links on exam stress: http://appl003.lsu.edu/slas/cas.nsf/$Content/Stress+Management+Tip+1 http://wso.williams.edu/orgs/peerh/stress/exams.html http://www.thecalmzone.net/Home/ExamStress.php http://www.staithes.demon.co.uk/exams.html

2. Exam 2 (Ch24) Sec.11 (Electric Potential Energy of a System of Point Charges); Sec.12 (Potential of Charged Isolated Conductor) (Ch 25) Capacitors: capacitance and capacitors; caps in parallel and in series, dielectrics; energy, field and potential in capacitors. (Ch 26) Current and Resistance. Current Density. Ohm’s Law. Power in a Resistor. (Ch 27) Single-Loop Circuits, Series & Parallel Circuits, Multi-loop Circuits. NO RC Circuits.

3. Electric potential: –What is the potential produced by a system of charges? (Several point charges, or a continuous distribution) Electric field lines, equipotential surfaces: lines go from +ve to –ve charges; lines are perpendicular to equipotentials; lines (and equipotentials) never cross each other… Electric potential, work and potential energy: work to bring a charge somewhere is W = –qV (signs!). Potential energy of a system = negative work done to build it. Conductors: field and potential inside conductors, and on the surface. Shell theorem: systems with spherical symmetry can be thought of as a single point charge (but how much charge?) Symmetry, and “ infinite ” systems.

4. Electric potential, electric potential energy, work In Fig. 25-39, point P is at the center of the rectangle. With V = 0 at infinity, what is the net electric potential in terms of q/d at P due to the six charged particles?

5. Derive an expression in terms of q 2 /a for the work required to set up the four-charge configuration of Fig. 25-50, assuming the charges are initially infinitely far apart. The electric potential at points in an xy plane is given by V = (2.0 V/m 2 )x 2 - (4.0 V/m 2 )y 2. What are the magnitude and direction of the electric field at point (3.0 m, 3.0 m)?

6. Potential Energy of A System of Charges 4 point charges (each +Q) are connected by strings, forming a square of side L If all four strings suddenly snap, what is the kinetic energy of each charge when they are very far apart? Use conservation of energy: –Final kinetic energy of all four charges = initial potential energy stored = energy required to assemble the system of charges +Q Do this from scratch! Don ’ t memorize the formula in the book! We will change the numbers!!!

7. Potential Energy of A System of Charges: Solution No energy needed to bring in first charge: U 1 =0 Energy needed to bring in 2nd charge: Energy needed to bring in 3rd charge = Energy needed to bring in 4th charge = +Q Total potential energy is sum of all the individual terms shown on left hand side = So, final kinetic energy of each charge =

8. Potential V of Continuous Charge Distributions Straight Line Charge: dq=  dx  =Q/L Curved Line Charge: dq=  ds  =Q/2  R Surface Charge: dq=  dA  =Q/  R 2 dA=2  R’dR’ Straight Line Charge: dq=  dx  =Q/L Curved Line Charge: dq=  ds  =Q/2  R Surface Charge: dq=  dA  =Q/  R 2 dA=2  R’dR’

9. Potential V of Continuous Charge Distributions Straight Line Charge: dq=  dx  =Q/L Straight Line Charge: dq=  dx  =Q/L Curved Line Charge: dq=  ds  =Q/2  R Curved Line Charge: dq=  ds  =Q/2  R

10. Potential V of Continuous Charge Distributions Surface Charge: dq=  dA  =Q/  R 2 dA=2πR’dR‘ Straight Line Charge: dq=  dx  =bx is given to you. Straight Line Charge: dq=  dx  =bx is given to you.

11. ✔ ✔ ✔ ✔

12. ICPP: Consider a positive and a negative charge, freely moving in a uniform electric field. True or false? (a) Positive charge moves to points with lower potential voltage. (b) Negative charge moves to points with lower potential voltage. (c)Positive charge moves to a lower potential energy. (d)Negative charge moves to a lower potential energy. –Q+Q0 +V –V (a) True (b) False (c) True (d) True + + + + + + + + + – – – –

13. ✔ ✔ ✔ ✔ electron

14. downhill

15. No vectors! Just add with sign. One over distance. Since all charges same and all distances same all potentials same.

17. ΔxΔx (a) Since Δx is the same, only |ΔV| matters! |ΔV 1 | =200, |ΔV 2 | =220, |ΔV 3 | =200 |E 2 | > |E 3 | = |E 1 | The bigger the voltage drop the stronger the field. (b) = 3 (c) F = qE = ma accelerate leftward

18. Capacitors E =  /  0 = q/A  0 E = V d q = C V C plate =   0 A/d C plate =  0 A/d C sphere =  0 ab/(b-a) Connected to Battery: V=Constant Disconnected: Q=Constant Connected to Battery: V=Constant Disconnected: Q=Constant

19. Isolated Parallel Plate Capacitor: ICPP A parallel plate capacitor of capacitance C is charged using a battery. Charge = Q, potential voltage difference = V. Battery is then disconnected. If the plate separation is INCREASED, does the capacitance C: (a) Increase? (b) Remain the same? (c) Decrease? If the plate separation is INCREASED, does the Voltage V: (a) Increase? (b) Remain the same? (c) Decrease? +Q –Q Q is fixed! d increases! C decreases (=  0 A/d) V=Q/C; V increases.

20. Parallel Plate Capacitor & Battery: ICPP A parallel plate capacitor of capacitance C is charged using a battery. Charge = Q, potential difference = V. Plate separation is INCREASED while battery remains connected. +Q –Q V is fixed constant by battery! C decreases (=  0 A/d) Q=CV; Q decreases E = σ/  0 = Q/  0 A decreases Does the Electric Field Inside: (a) Increase? (b) Remain the Same? (c) Decrease? Battery does work on capacitor to maintain constant V!

21. Capacitors Capacitors Q=CV In series: same charge 1/C eq = ∑1/C j In parallel: same voltage C eq =∑C j

22. Capacitors in Series and in Parallel What ’ s the equivalent capacitance? What ’ s the charge in each capacitor? What ’ s the potential across each capacitor? What ’ s the charge delivered by the battery?

25. Capacitors: Checkpoints, Questions

26. Parallel plates: C =  0 A/d Spherical: Cylindrical: C = 2  0 L/ln(b/a)] Hooked to battery V is constant. Q=VC (a) d increases -> C decreases -> Q decreases (b) a inc -> separation d=b-a dec. -> C inc. -> Q increases (c) b increases -> separation d=b-a inc.-> C dec. -> Q decreases

27. PARALLEL: V is same for all capacitors Total charge = sum of Q (a)Parallel: Voltage is same V on each but charge is q/2 on each since q/2+q/2=q. (b) Series: Charge is same q on each but voltage is V/2 on each since V/2+V/2=V. SERIES: Q is same for all capacitors Total potential difference = sum of V

28. Initial values: capacitance = C; charge = Q; potential difference = V; electric field = E; Battery remains connected V is FIXED; V new = V (same) C new =  C (increases) Q new = (  C)V =  Q (increases). Since V new = V, E new = V/d=E (same) dielectric slab Energy stored? u=  0 E 2 /2 => u=  0 E 2 /2 =  E 2 /2 increases Example: Battery Connected — Voltage V is Constant but Charge Q Changes

29. Initial values: capacitance = C; charge = Q; potential difference = V; electric field = E; Battery remains disconnected Q is FIXED; Q new = Q (same) C new =  C (increases) V new = Q/ C new = Q/ (  C) (decreases). Since V new < V, E new = V new /d = E/  (decreases) dielectric slab Energy stored? Example: Battery Disconnected — Voltage V Changes but Charge Q is Constant

30. Current and Resistance i = dq/dt V = i R E = J   =  0 (1+  (T  T 0 )) R =  L/A Junction rule

31. Current and Resistance

32. A cylindrical resistor of radius 5.0mm and length 2.0 cm is made of a material that has a resistivity of 3.5 x 10 -5 m. What are the (a) current density and (b) the potential difference when the energy dissipation rate in the resistor is 1.0W?

34. Current and Resistance: Checkpoints, Questions

35. 26.2: Electric Current, Conservation of Charge, and Direction of Current: Fill in the blanks. Think water in hose!

36. All quantities defined in terms of + charge movement! (a) right (c) right(b) right (d) right

37. DC Circuits Single loop Multiloop V = iR P = iV Loop rule

38. Resistors vs Capacitors Resistors Capacitors Key formula: V=iR Q=CV In series: same current same charge R eq =∑R j 1/C eq = ∑1/C j In parallel: same voltage same voltage 1/R eq = ∑1/R j C eq =∑C j

39. Resistors in Series and in Parallel What ’ s the equivalent resistance? What ’ s the current in each resistor? What ’ s the potential across each resistor? What ’ s the current delivered by the battery? What’s the power dissipated by each resisitor?

40. Series and Parallel

43. Problem: 27.P.018. [406649] Figure 27-33 shows five 5.00 resistors. (Hint: For each pair of points, imagine that a battery is connected across the pair.) Fig. 27-33 (a) Find the equivalent resistance between points F and H. (b) Find the equivalent resistance between points F and G. Slide Rules: You may bend the wires but not break them. You may slide any circuit element along a wire so long as you don’t slide it past a three (or more) point junction or another circuit element.

46. Too Many Batteries!

47. Circuits: Checkpoints, Questions

48. (a) Rightward (EMF is in direction of current) (b) All tie (no junctions so current is conserved) (c) b, then a and c tie (Voltage is highest near battery +) (d) b, then a and c tie (U=qV and assume q is +) +E+E -iR

49. (a) all tie (current is the same in series) The voltage drop is –iR proportional to R since i is same.

50. (a) V batt < 12V (walking with current voltage drop –ir (b) V batt > 12V (walking against current voltage increase +ir (c) V batt = 12V (no current and so ir=0)