e1
R
I1
I2
e2
R
I3
e3
R
V
n
loop
0
I in   I out
Devices
• Capacitors:
Purpose is to store charge (energy).
• We have calculated the capacitance of a system
• We had to modify Gauss' Law to account for bulk
matter effects (dielectrics) … C = kC0
• We calculated effective capacitance of series or
parallel combinations of capacitors
• Batteries (Voltage sources, sources of
emf):
Purpose is to provide a constant potential
+ OR
difference between two points.
• Cannot calculate the potential difference
from first principles... chemical  electrical
energy conversion. Non-ideal batteries will
be dealt with in terms of an "internal
resistance".
+
V
-
Current is charge in motion
• Charge, e.g. free electrons, exists in conductors
with a density, ne (ne approx 1029 m-3)
• “Somehow” put that charge in motion:
– effective picture -- all charge moves with a velocity, ve
– real picture -- a lot of “random motion” of charges with
a small average equal to ve
• Current density, J, is given by J  qeneve
– unit of J is C/m2sec or A/m2 (A ≡ Ampere) and 1A  1C/s
– current, I, is J times cross sectional area, I  J pr2
– for 10 Amp in 1mm x 1mm area, J  10+7 A/m2, and ve is
about 10-3 m/s (Yes, the average velocity is only 1mm/s!)
Devices
• Resistors:
Purpose is to limit current drawn in a circuit.
Note:
dQ
I
dt
UNIT: Ampere = A = C/s
• Resistance can be calculated from knowledge of the
geometry of the resistor AND the “resistivity” of the
material out of which it is made (often “conductors”).
• The effective resistance of series and parallel
combinations of resistors will be calculated using
Kirchhoff's Laws (Notion of potential difference,
current conservation).
Ohm’s Law
• Demo:
• Vary applied voltage V.
I
R
I
• Measure current I
V
V
• Does ratio remain
I
constant?
V
R
I
V
slope = R
How to calculate the resistance?
Include “resistivity” of material
I
Include geometry of resistor
Resistance
I
R
• What about acceleration?
• V  E-field  constant force eE on
electrons  constant acceleration
I
V
 ever-increasing currents
• These very large currents and “funny” I(V) do not occur.
• Charges are ballistically accelerated, but scatter in a very
short time (t = 10-14 s) from things that get in the way
– defects, lattice vibrations (phonons), etc.
• Average velocity attained in this time is v = eEt/m
• Current density is J = env so current is proportional to E
which is proportional to Voltage
• OHM’s LAW J = (e2nt/m)E or J = s E
s = conductivity
Resistance
• Resistance
R
I
I
Resistance is defined to be the
ratio of the applied voltage to
the current passing through.
R
V
I
UNIT: OHM = W
V
• How do we calculate it?
Recall the case of capacitance: (C=Q/V) depended on the geometry
(and dielectric constant), not on Q or V individually
Similarly, for resistance
–part depends on the geometry (length L and cross-sectional area A)
–part depends on the “resistivity” ρ of the material
L
R   
 A
• Increase the length  flow of electrons impeded
• Increase the cross sectional area  flow facilitated
• What about ?
Resistivity
• Property of bulk matter related to
resistance of a sample is the
resistivity () defined as:
1
E
m
   2
s J e nt
where E = electric field and
J = current density in conductor.
I
For uniform case:
J  , V
A
V  EL   JL  
E
J
A
L
1
 EL
I
 ρL 
L I

A
 A 
V  IR where R  
L
A
The resistivity varies greatly with the sort of material:
e.g., for copper  ~ 10-8 W-m; for glass,  ~ 10+12 W-m; for
semiconductors  ~ 1 W-m; for superconductors,  = 0 [see Appendix]
cylindrical resistors, R1 and R2, are made of identical material.
1A Two
R has twice the length of R but half the radius of R .
2
1
1
– These resistors are then connected to a battery V as shown:
V
I1
I2
– What is the relation between I1, the current flowing in R1 ,
and I2 , the current flowing in R2?
(a) I1 < I2
1B
(b) I1 = I2
(c) I1 > I2
• A very thin metal wire patterned as
shown is bonded to some structure.
• As the structure is deformed slightly,
this stretches the wire (slightly).
– When this happens, the resistance of
the wire:
(a) decreases
(b) increases
(c) stays the same
• Two cylindrical resistors, R1 and R2, are made of identical material.
R2 has twice the length of R1 but half the radius of R1.
– These resistors are then connected to a battery V as shown:
I1
I2
V
– What is the relation between I1, the current flowing in R1 ,
and I2 , the current flowing in R2?
(b) I1 = I2
(a) I1 < I2
(c) I1 > I2
• The resistivity of both resistors is the same ().
• Therefore the resistances are related as:
R2  
L2
2 L1
L

 8  1  8 R1
A2
( A1 / 4)
A1
• The resistors have the same voltage across them; therefore
I2 
V
V
1

 I1
R2 8 R1 8
1B
• A very thin metal wire patterned as
shown is bonded to some structure.
• As the structure is deformed slightly,
this stretches the wire (slightly).
– When this happens, the resistance of
the wire:
(a) decreases
(b) increases
(c) stays the same
1B
• A very thin metal wire patterned as
shown is bonded to some structure.
• As the structure is deformed slightly,
this stretches the wire (slightly).
– When this happens, the resistance of
the wire:
(a) decreases
(b) increases
(c) stays the same
Because the wire is slightly longer, R ~ L A is slightly increased.
Also, because the overall volume of the wire is ~constant, increasing
the length decreases the area A, which also increases the resistance.
By carefully measuring the change in resistance, the strain in the
structure may be determined (we’ll see later how to do this optically).
Preflight 9:
Two cylindrical resistors are made
from the same material, and they
are equal in length. The first
resistor has diameter d, and the
second resistor has diameter 2d.
2) Compare the resistance of the two cylinders.
a) R1 > R2
b) R1 = R2
c) R1 < R2
3) If the same current flows through both resistors, compare the
average velocities of the electrons in the two resistors:
a) v1 > v2
b) v1 = v2
c) v1 < v2
The Voltage “drops”:
Va  Vb  IR1
Resistors
in Series
I
R1
Vb  Vc  IR2
b
Va  Vc  I ( R1  R2 )
R2
a
c
Whenever devices are in SERIES, the
current is the same through both !
This reduces the circuit to:
Hence:
a
Reffective  ( R1  R2 )
Reffective
c
Another (intuitive) way…
R1
Consider two cylindrical resistors with
lengths L1 and L2
R1  
L1
A
R2  
L2
A
L1
V
Put them together, end to end to make a longer one...
Reffective
L1  L2

 R1  R2
A
R  R1  R2
L2
R2
Preflight 9:
Two resistors are connected in series to a
battery with emf E. The resistances are
such that R1 = 2R2.
5) Compare the current through R1 with the
current through R2:
a) I1 > I2
b) I1 = I2
c) I1 < I2
6) What is the potential difference across R2?
a) V2 = E
b) V2 = 1/2 E
c) V2 = 1/3 E
The World’s Simplest (and most useful) circuit:
Voltage Divider
R1
V0
V
R2
By varying R2 we can
controllably adjust
the output voltage!
V ?
 V0 
V  IR2  
 R2
 R1  R2 
R2  R1
V=0
R2  R1
V0
V=
2
R2  R1
V=V0
Kirchhoff’s First Rule
“Loop Rule” or “Kirchhoff’s Voltage Law (KVL)”
"When any closed circuit loop is traversed, the algebraic
sum of the changes in potential must equal zero."
KVL:
V
n
0
loop
• This is just a restatement of what you already know: that the
potential difference is independent of path!
e1
I
e
 1
R1
 IR1
R2
 IR2
e2
e
 2 0
Rules of the Road
Our convention:
•
•
•
•
Voltage gains enter with a + sign, and voltage drops enter with a  sign.
We choose a direction for the current and move around the circuit in that
direction.
When a battery is traversed from the negative terminal to the positive
terminal, the voltage increases, and hence the battery voltage enters KVL
with a + sign.
When moving across a resistor, the voltage drops, and hence enters KVL
with a  sign.
e1
I
e
 1
R1
 IR1
R2
 IR2
e2
e
 2 0
Note: In the ECE convention, voltage drops enter with a + sign
and voltage gains enter with a  sign.
Loop Demo
R1 e
1
b
R4
f
a
I
I
d
c
R2
KVL:
V
n
0 
loop

e2
e
R3
 IR1  IR2  e 2  IR3  IR4  e1  0
I
e1  e 2
R1  R2  R3  R4
2
• Consider the circuit shown.
– The switch is initially open and the current
flowing through the bottom resistor is I0.
– After the switch is closed, the current
12V
flowing through the bottom resistor is I1.
– What is the relation between I0 and I1?
(a) I1 < I0
(b) I1 = I0
12V
R
a
I
R
b
(c) I1 > I0
12V
• Consider the circuit shown.
– The switch is initially open and the current
flowing through the bottom resistor is I0.
– After the switch is closed, the current
12V
flowing through the bottom resistor is I1.
– What is the relation between I0 and I1?
(a) I1 < I0
(b) I1 = I0
• Write a loop law for original loop:
12V +12V  I0R  I0R = 0
I0 = 12V/R
• Write a loop law for the new loop:
12V  I1R = 0
I1 = 12V/R
12V
R
a
I
R
b
(c) I1 > I0
12V
Summary
• When you are given a circuit, you must first
carefully analyze circuit topology
– find the nodes and distinct branches
– assign branch currents
• Use KVL for all independent loops in circuit
– sum of the voltages around these loops is zero!
Appendix: Superconductivity
• 1911: H. K. Onnes, who had figured
out how to make liquid helium, used it
to cool mercury to 4.2 K and looked at
its resistance:
• At low temperatures the resistance of
some metals0, measured to be less
than 10-16•ρconductor (i.e., ρ<10-24 Ωm)!
–Current can flow, even if E=0.
–Current in superconducting rings can flow for years with no
decrease!
• 1957: Bardeen (UIUC!), Cooper, and Schrieffer (“BCS”)
publish theoretical explanation, for which they get the
Nobel prize in 1972.
– It was Bardeen’s second Nobel prize (1956 – transistor)
Appendix: Superconductivity
• 1986: “High” temp. superconductors (77K) discovered
– Important because liquid N (77 K) is much cheaper than liquid He
– Highest critical temperature to date 138 K (-135˚ C = -211˚ F)
• Today: Superconducting loops are used to produce
“lossless” electromagnets (only need to cool them, not
fight dissipation of current) for particle physics.
[Fermilab accelerator, IL]
• The Future: Smaller motors, “lossless” power lines,
magnetic levitation trains, quantum computers?? ...
• 2003: UIUC Prof. Tony Leggett shares Nobel prize for
helping to explain the related phenomenon of superfluids.
– Superconductivity arises from quantum correlations between pairs
of electrons in the metal, resulting a total loss of “friction”
– In some materials at low temperature, a similar effect allows them
to flow with no viscosity  “superfluidity”