1. Exam 2 covers Ch. 28-33, Lecture, Discussion, HW, Lab
Exam 2 is Tue. Oct. 28, 5:30-7 pm, Ch
Chapter 28: Electric flux & Gauss’ law
Chapter 29: Electric potential & work
Chapter 30: Electric potential & field
(exclude 30.7)
Chapter 31: Current & Conductivity
Chapter 32: Circuits
(exclude 32.8)
Chapter 33: Magnetic fields & forces
(exclude 33.3, 33.6, 32.10, Hall effect)

2. Electric flux E = EA cos  Flux SI units are N·m2/C
Suppose surface make angle  surface normal
Component || surface
Component  surface
Only  component ‘goes through’ surface
Other component is parallel to the surface
E = EA cos 
E =0 if E parallel A
E = EA (max) if E  A
Flux SI units are N·m2/C

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

4. Properties of conductors
everywhere inside a conductor
Charge in conductor is only on the surface
surface of conductor - +

5. Gauss’ law example: Charges on parallel-plate capacitor
Determine fields by superposition Q -Q
Apply Gauss’ law:
E=0 inside metal
E=0 to left
No charge on outer surface
Apply Gauss’ law:
E=0 inside metal
E=Q/Ao in middle
Area A=Length X Width

6. Electric potential: general
Electric potential energy difference U proportional to charge q that work is done on
Electric potential difference
Depends only on charges that create E-fields
Electric field usually created by some charge distribution.
V(r) is electric potential of that charge distribution
V has units of Joules / Coulomb = Volts

7. Electric Potential
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
Q source of the electric potential, q ‘experiences’ it

8. Example: Electric Potentialy
Calculate the electric potential at B B x d
d2=4 m
-12 μC
+12 μC A - +
Calculate the electric potential at A
d1=3 m
3 m
3 m
Calculate the work YOU must do to move a Q=+5 mC charge from A to B.
Work done by electric fields

9. Work and electrostatic potential energy
Question: How much work would it take YOU to assemble 3 negative charges?
A. W = mJ
B. W = mJ
C. W= 0
Likes repel, so YOU will still do positive work! q3
−3μC
5 m
5 m q2 q1
−1μC
−2μC
5 m
electric potential energy of the system increases

10. 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=Vo

11. Electric Potential and Field
Uniform electric field of
What is the electric potential difference VA-VB? A x y 2m 5m B
A) -12V
B) +12V
C) -24V
D) +24V

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

13. Stored energy Isolated charged capacitor Plate separation increased
The stored energy
Increases
Decreases
Does not change A) B) C)
q unchanged because C isolated
q is the same
E is the same = q/(Aε0)
ΔV increases = Ed
C decreases
U increases

14. Spherical capacitor Charge Q moved from outer to inner sphere
Gauss’ law says E=kQ/r2 until second sphere
Potential difference
Gaussian surface to find E
Along path shown +
Path to find V

15. 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’

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

17. Resistance and resistivity
Ohm’s Law: ΔV = R I (J = σ E or E = ρ J)
ΔV = EL and E = ρ J => ρ I/A = ΔV/L
R = ρL/A Resistance in ohms (Ω)

18. Current conservation I2 Iin I1 I3 I1=I2+I3 I1 I3 Iout I2 Iout = Iin

19. Resistors in Series and parallel
I1 = I2 = I
Req = R1+R2
Parallel
V1 = V2 = V
Req = (R1-1+R2-1)-1
I1+I2 I R1 R2 R1
R1+R2 I = = I1 I2 R2 I
2 resistors in series:
R  L
Like summing lengths

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

21. Quick Quiz What is the current through resistor R1? R1=200Ω R4=100Ω 9V
5 mA
10 mA
20 mA
30 mA
60 mA 6V
Req=100Ω
Req=50Ω 9V 3V

22. 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

23. Capacitors in parallel and series
ΔV1 = ΔV2 = ΔV
Qtotal = Q1 + Q2
Ceq = C1 + C2
Q1=Q2 =Q
ΔV = ΔV1+ΔV2
1/Ceq = 1/C1 + 1/C2
Parallel
Series

24. Example: Equivalent Capacitance
C1 = 30 μF
C2 = 15 μF
C3 = 15 μF
C4 = 30 μF
in series C2 V C3 C1 C4
Parallel combination Ceq=C1||C2

25. RC Circuits Start w/uncharged C Close switch at t=0
Discharge R C ε R C
RC Circuits
Time constant
Start w/uncharged C Close switch at t=0
Start w/charged C Close switch at t=0

26. Question
What is the current through R1 Immediately after the switch is closed?
10V
R1=100Ω
R2=100Ω
C=1µF
A. 10A
B. 1 A
C. 0.1A
D. 0.05A
E. 0.01A

27. Question
What is the charge on the capacitor a long time after the switch is closed?
10V
R1=100Ω
R2=100Ω
C=1µF
A. 0.05µC
B. 0.1µC
C. 1µC
D. 5µC
E. 10µC

28. 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

29. FB on a Charge Moving in a Magnetic Field, Formula
FB = q v x B
FB is the magnetic force
q is the charge
v is the velocity of the moving charge
B is the magnetic field
SI unit of magnetic field: tesla (T)
CGS unit: gauss (G): 1 T = 104 G (Earth surface 0.5 G)

30. Magnetic Force on a Current
Force on each charge
Force on length of wire
Force on straight section of wire, length L
Current N
Magnetic force S
Magnetic field

31. Magnetic field from long straight wire: Directiony
What direction is the magnetic field from an infinitely-long straight wire? x I
= permeability of free space
r = distance from wire

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