1 Exam 2 covers Ch , Lecture, Discussion, HW, Lab Chapter 27: Electric flux & Gauss’ law Chapter 29: Electric potential & work Chapter 30: Electric potential & field Chapter 28: Current & Conductivity Chapter 31: Circuits Chapter 32: Magnetic fields & forces (exclude 32.6,32.8,32.10) Exam 2 is Wed. Mar. 26, 5:30-7 pm, 2103 Ch: Adam(301,310), Eli(302,311), Stephen(303,306), 180 Science Hall: Amanda(305,307), Mike(304,309), Ye(308)

2 Electric flux Suppose surface make angle  surface normal  E = EA cos   E =0 if E parallel A  E = EA (max) if E  A Flux SI units are N·m 2 /C Component || surface Component  surface Only  component ‘goes through’ surface

3 Gauss’ law net electric flux through closed surface = charge enclosed /  

4 Field outside uniformly-charged sphere Field direction: radially out from charge Gaussian surface: Sphere of radius r Surface area where Value of on this area: Flux thru Gaussian surface: Charge enclosed: Gauss’ law:

5 Electric potential energy since they repel! potential energy increases Work is Force x distance (taking into account cosθ between 2 vectors!) >0 If opposte charges they attract => W <0 and potential energy decreases

6 Electric Potential Q source of the electric potential, q ‘experiences’ it Electric potential energy per unit charge units of Joules/Coulomb = Volts Example: charge q interacting with charge Q Electric potential energy Electric potential of charge Q

7 Example: Electric Potential Calculate the electric potential at B Calculate the work YOU must do to move a Q=+5 mC charge from A to B. Calculate the electric potential at A x + - B A d 1 =3 m3 m d 2 =4 m 3 m y -12  C+12  C d

8 A. W = mJ B. W = mJ C. = 0 Work and electrostatic potential energy −  C −  C −  C  m Question: How much work would it take YOU to assemble 3 negative charges? Likes repel, so YOU will still do positive work! q3q3 q2q2 q1q1 electric potential energy of the system increases

9 Potential from electric field Electric field can be used to find changes in potential Potential changes largest in direction of E-field. Smallest (zero) perpendicular to E-field V=VoV=Vo

10 Electric Potential and Field Uniform electric field of E = 4i+3j N/C Points A at 2m and B at 5m on the x axis. What is the potential difference V A - V B ? B(5,0) E = 4i N/C x(m) A(2,0) 0 A) -12V B) +12V C) -24V D) +24V

Capacitors Energy stored in a capacitor: C = capacitance: depends on geometry of conductor(s) Conductor: electric potential proportional to charge: Example: parallel plate capacitor +Q-Q d Area A

12 Isolated charged capacitor Plate separation increased The stored energy 1)Increases 2)Decreases 3)Does not change A) B) C) Stored energy q unchanged because C isolated q is the same E is the same = q/(Aε 0 ) ΔV increases = Ed C decreases U increases

13 Conductors, charges, electric fields Electrostatic equilibrium No charges moving No electric fields inside conductor. Electric potential is constant everywhere Charges on surface of conductors. Not equilibrium Charges moving (electric current) Electric fields inside conductors -> forces on charges. Electric potential decreases around ‘circuit’

Electric current Current density J= I/A = nqv d (direction of + charge carriers) L SI unit: ampere 1 A = 1 C / s Average current: Instantaneous value: n = number of electrons/volume n x AL electrons travel distance L = v d Δt I av = ΔQ/ Δt = neAL v d /L

15 Resistance and resistivity Ohm’s Law: ΔV = R I (J =  E or E = ρ J  ΔV = EL and E =  J  /A = ΔV/L R = ρL/A Resistance in ohms ( )

16 Current conservation I in I out I out = I in I1I1 I2I2 I3I3 I 1 =I 2 +I 3 I2I2 I3I3 I1I1 I 1 +I 2 =I 3

17 Resistors in Series and parallel Series I 1 = I 2 = I R eq = R 1 +R 2 R1R1 R2R2 = R 1 +R 2 2 resistors in series: R  L Like summing lengths R1R1 R2R2 = I I I I1I1 I2I2 I 1 +I 2 Parallel V 1 = V 2 = V R eq = (R R 2 -1 ) -1

18 Quick Quiz How does brightness of bulb B compare to that of A? A.B brighter than A B.B dimmer than A C.Both the same Battery maintain constant potential difference Extra bulb makes extra resistance -> less current

19 Quick Quiz What happens to the brightness of bulb B when the switch is closed? A.Gets dimmer B.Gets brighter C.Stays same D.Something else Battery is constant voltage, not constant current

20 Quick Quiz What happens to the brightness of bulb A when the switch is closed? A.Gets dimmer B.Gets brighter C.Stays same D.Something else

21 Capacitors as circuit elements Voltage difference depends on charge Q=CV Current in circuit Q on capacitor changes with time Voltage across cap changes with time

22 RC Circuits R C  R C Start w/uncharged C Close switch at t=0 Start w/charged C Close switch at t=0

23 Capacitors in parallel and series ΔV 1 = ΔV 2 = ΔV Q total = Q 1 + Q 2 C eq = C 1 + C 2 Q 1 =Q 2 =Q ΔV = ΔV 1 +ΔV 2 1/C eq = 1/C 1 + 1/C 2

24 Calculate the equivalent Capacitance C 1 = 10  F C 2 = 20  F C 3 = 30  F C 4 = 40  F V = 50 Volts C2C2 V C3C3 C1C1 C4C4 parallel C 1, C 23, C 4 in series

25 RC Circuits What is the value of the time constant of this circuit? A) 6 ms B) 12 ms C) 25 ms D) 30 ms

26 Magnetic fields and forces I B Magnetic force on current- carrying wire Magnetic torque on current loop Magnetic force on moving charged particle I B

27 Effect of uniform magnetic field Effect of uniform B-field on charged particle If charged particle is not moving - no effect If particle is moving: force exerted perpendicular to both field and velocity vector ‘cross product’

12/09/2002U. Wisconsin, Physics 208, Fall Lorentz force Electron moves in plane of screen the page. B- field is in the plane of screen to the right. Direction of instantaneous magnetic force on electron is A) toward the top of the page B) into the page C) toward the right edge of the page D) out of the page v B F electron

29 Trajectory in Constant B Field F F v x x x v B q Charge enters B field with velocity shown. (v  B) Force is always  to velocity and to B. Path is a circle. Radius determined by velocity:

30 Current loops & magnetic dipoles Current loop produces magnetic dipole field. Magnetic dipole moment: current Area of loop magnitude direction Effect of uniform magnetic field Magnetic field exerts torque Torque rotates loop to align with

12/09/2002U. Wisconsin, Physics 208, Fall Which of these loop orientations has the largest magnitude torque? (A) a (B) b (C) c Question on torque Answer: (c). all loops enclose same area and carry same current  magnitude of μ is the same for all. (c) μ upwards, μ  B and τ = μB. (a),  = 0 (b)  =  Bsin  a b c μ μ τ τ