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WARNING:
Exam is Thursday, 7:30 – 9 pm
Room 54-100
Review Sessions:
Me:
Tuesday
Imran:
Wednesday
In class: Thursday
Office Hours:
Brian F.: Tuesday
Brian P.: Wednesday
4-6 pm 6-120
8-10 pm 4-270
12-2 pm; 2-4 pm
8-10 pm 6-402
4-6 pm 6-106
Please email me questions
P12- 1
Class 12: Outline
Hour 1:
DC Circuits
Hour 2:
Expt. 6: Simple Circuits
P12- 2
Last Time:
Capacitors & Dielectrics
P12- 3
Capacitors & Dielectrics
Capacitance
To calculate:
1) Put on arbitrary ±Q
2) Calculate E
3) Calculate V
Q
C
V
Energy
2

E
Q2 1
1
2
U
 Q V  C V   uE d 3r   o d 3r
2
2C 2
2
Dielectrics
  qin
 E  dA 
S
0
 CFilled with Dielectric   C0
P12-
This Time:
DC Circuits
P12-
Examples of Circuits
P12- 6
Current: Flow Of Charge
Average current Iav: Charge Q
flowing across area A in time t
Q
I av 
t
Instantaneous current:
differential limit of Iav
dQ
I
dt
Units of Current: Coulombs/second = Ampere
P12- 7
Direction of The Current
Direction of current is direction of flow of pos. charge
or, opposite direction of flow of negative charge
P12- 8
Current Density J
J: current/unit area
 I
J  Iˆ
A
Î points in direction of current
I   J  nˆ dA   J  d A
S
S
P12- 9
Why Does Current Flow?
If an electric field is set up in a conductor, charge
will move (making a current in direction of E)
Note that when current is flowing, the conductor is
not an equipotential surface (and Einside ≠ 0)!
P12-10
Microscopic Picture
Drift speed is velocity forced by applied electric field
in the presence of collisions.
It is typically 4x10-5 m/sec, or 0.04 mm/second!
To go one meter at this speed takes about 10 hours!
How Can This Be?
P12- 11
Conductivity and Resistivity
Ability of current to
flow depends on
density of charges &
rate of scattering
Two quantities summarize this:
s: conductivity
r: resistivity
P12-12
Microscopic Ohm’s Law
E  rJ
r
or
J sE
1
s
r and s depend only on the microscopic properties
of the material, not on its shape
P12-13
Demonstrations:
Temperature Effects on r
P12-14
PRS Questions:
Resistance?
P12-15
Why Does Current Flow?
Instead of thinking of Electric Field, think of potential
difference across the conductor
P12-16
Ohm’s Law
What is relationship between V and current?
b
V  Vb  Va   E  d s  E
a
E V /
J 
r
r
I
J
A


r 
  V  I 
  IR
 A 


P12-17
Ohm’s Law
V  IR
R
r
A
R has units of Ohms (W) = Volts/Amp
P12-18
Examples of Circuits
P12-19
Wires in Circuits
Wires in circuit diagrams are “perfect conductors.”
They are thus equipotentials, and can be stretched,
shrunk... as long as you don’t change the potentials!
Must maintain
all equipotential
surfaces!
P12-20
Symbols for Circuit Elements
Battery
Resistor
Capacitor
Switch
P12-21
Sign Conventions - Battery
Moving from the negative to positive terminal of a
battery increases your potential
 V  Vb  Va
Think:
Ski Lift
P12-22
Sign Conventions - Resistor
Moving across a resistor in the direction of current
decreases your potential
 V  Vb  Va
Think:
Ski Slope
P12-23
Sign Conventions - Capacitor
Moving across a capacitor from the negatively to
positively charged plate increases your potential
 V  Vb  Va
Think:
Ski Lodge
P12-24
Series vs. Parallel
Series
Parallel
P12-25
Resistors In Series
The same current I must flow through both resistors
V  I R1  I R2  I ( R1  R2 )  I Req
Req  R1  R2
P12-26
Resistors In Parallel
Voltage drop across the resistors must be the same
V  V1  V2  I1R1  I 2 R2  IReq
V V V
I  I1  I 2 


R1
R2
Req
1
1
1
 
Req R1 R2
P12-27
Measuring Voltage & Current
P12-28
Measuring Potential Difference
A voltmeter must be hooked in parallel across the
element you want to measure the potential difference
across
Voltmeters have a very large resistance, so that
they don’t affect the circuit too much
P12-
Measuring Current
An ammeter must be hooked in series with the
element you want to measure the current through
Ammeters have a very low resistance, so that
they don’t affect the circuit too much
P12-
Measuring Resistance
An ohmmeter must be hooked in parallel across the
element you want to measure the resistance of
Here we are measuring R1
Ohmmeters apply a voltage and measure the
current that flows. They typically won’t work if the
resistor is powered (connected to a battery)
P12-
Experiment 6:
Simple Circuits
P12-32
PRS Questions:
Light Bulbs
P12-33
Kirchhoff’s Loop Rules
P12-34
Kirchhoff’s Rules
1. Sum of currents entering any junction in a circuit
must equal sum of currents leaving that junction.
I1  I 2  I 3
P12-35
Kirchhoff’s Rules
2. Sum of potential differences across all elements
around any closed circuit loop must be zero.
 
V    E  d s  0
Closed
Path
P12-36
Internal Resistance
Real batteries have an internal resistance, r, which is
small but non-zero
Terminal voltage:
V  Vb  Va    I r
(Even if you short the leads you don’t get infinite current)
P12-
Steps of Solving Circuit Problem
1.
2.
3.
4.
5.
Straighten out circuit (make squares)
Simplify resistors in series/parallel
Assign current loops (arbitrary)
Write loop equations (1 per loop)
Solve
P12-38
Example: Simple Circuit
You can simplify
resistors in series
(but don’t need to)
What is current
through the bottom
battery?
P12-39
Example: Simple Circuit
Start at a in both loops
Walk in direction of current
2  I1R   I1  I 2  R  0
  I 2  I1  R    0

Add these: 2  I1 R    0  I1 
R
We wanted I2:
 I 2  I1  R    I 2 
I2  0

R
 I1
P12-40
Group Problem: Circuit
Find meters’ values. All resistors are R, batteries are 
HARDER
EASIER
P12-41