### Exam 2 Lectures Circuits

```Exam 2 Lectures
Circuits
Charges in Motion
• Now we will start to talk about charges in
motion instead of static charges
• Consider the three cases below
– A pipe of flowing water
– A wire without a potential difference
– A wire with a potential difference between the
two ends
Definitions
• Electric Current—charges in motion, or a
stream of moving charges
• Steady state—constant flow in time: for every
electron entering, an electron must leave.
• Ampere—unit of current 1 Amp = 1C/s
• Current density—vector quantity which has
the direction of E through a surface and
magnitude of current per unit area.
• Drift velocity—velocity of electrons drifting
through a conductor with a current in it
Current
• The flow of conduction electrons through a
metal wire (conductor)
dq
i
 q   dq  0t idt
dt
• In steady state
q  it
• Current I is a scalar, not a vector
• By convention the
arrow is drawn in the
direction (+) charges
would move
• Current can be from
the movement of
electrons, positive
ions, or both
Current Density
• Current density 
current
J 
area

 i   J  dA
• Current density J is a vector and we
must use vector math
• The same direction as the E field in the
wire
• Relationship between J and E

 

E  J & J  E
Drift Velocity
• Electrons move in the direction opposite
the E field with a drift speed vd
m
vd  10
s
5
• vd tiny compared to the random motion
speed of 106 m/s from Brownian motion
i
J
vd 

nAe ne

J  nevd
• Know how to find n (carrier density)
Batteries
• Almost any 2 different solid conductors immersed in
an active solution (electrolyte) functions as a battery
• The chemical energy stored in the interatomic bonds
is converted to electrical potential energy as the
solution and the conductors become involved in the
chemical reaction
• The electrolyte is a solution which dissolves the ions
formed by the leaving electrons allowing the ions to
move in the solution
• One of the conductors becomes the cathode (gains
electrons) and the other becomes the anode (loses
electrons)
• A salt bridge is necessary for letting the ions flow
Batteries cont
• emf—potential difference that can be used to
supply energy and sustain a current. Also voltage
measured across the terminals of the battery when
no current is being drawn from or delivered to it
• If the batteries are connected oppositely: +
terminal to – terminal then the voltages subtract
• For big i & low V – put battery cells in parallel.
• For small i & big V – put battery cells in series.
• For big i & big V – put rows of parallel battery cells
in series
Definitions
• Conductivity—the ability of a material to
conduct electricity.  is not necessarily a
constant, it could be a tensor or it could be a
function of E.
• Resistivity—the inverse of conductivity,
ability of a material to resist the flow of
electric charge
• Resistance—the ratio of V to i for a
particular conductor
• Ohm—the unit of resistance. 1  = 1 V/A
Definitions cont
• Resistors—devices in a circuit to control the
current level in various parts of the circuit.
Isotropic materials—materials whose
electrical properties are the same in all
directions (conductivity and resistivity)
• Ohm’s Law—usually stated V = iR or J =
E. Not all devices follow this law, some
are not directly proportional to V (R a
constant), for some R is a function of V (R =
f(V)). (isotropic materials)
• Resistivity is a property of the material, and
resistance is a property of the object
• Resistance depends on the geometry of the
conductor (resistor)
L L
R

A A
• Resistivity  depends on the properties
of the material and temperature
  o  o T  To 
Ohm’s Law
V  iR
• Ohm’s law is true for many substances, but
there are many materials and devices that
are nonohmic
• A device obeys Ohm’s law when its R is
independent of the magnitude and polarity of
V
• A material obeys Ohm’s law when  is
independent of the magnitude and direction
of E
• Most modern electronic devices are
nonohmic and their usefulness or
proper operation depends on how they
violate Ohm’s Law
b) An ohmic device – a resistor
c) A nonohmic device – a pn junction diode
– For a resistor, resistance is a constant of
proportionality between current and the voltage
difference and is independent of V and i
– For a resistor, resistance does not depend on either i
or V, but on the properties of the material making up
the resistor
No tolerance band – ±20%
Silver band – ±10%
Gold band – ±5%
Yellow = 4, Violet = 7, Orange = 103
47*103 = 4.7*104 Ohms
Microscopic View of Ohm’s Law
• Look at the motion of free conduction
electrons
eE
F  qE  ma  a 
m
eEt
v  at 
m
J eEt
vd  
ne m
m
m
E  2 J    2
ne t
ne t
Different Types of Conductors
• Conductors – materials that allow the flow of
charge
• Insulators – materials that don’t allow the
flow of charge
• Semiconductors—materials that are
intermediate between conductors and
insulators
• Doping—adding minute amounts of
impurities to semiconductors to change their
resistivity.
• Superconductors—materials with no
resistance to the movement of electric
charge through them
Types of Conductors cont
Materials
Conductors
Semiconductors
Insulators
Resistivity
< 10-5 m
10-5 <  < 105 m
> 105 m
Examples
Ag and Cu
Si and Ge
Glass,
rubber
Energy
Conduction Band
Valance Band
Conductor
Semiconductor
Insulator
Comparison Conductors to
Semiconductors
• Semiconductors have smaller n
• Semiconductors have a much higher 
• Semiconductors temperature coefficient of
resistivity  is large and negative
• In conductors n is large but nearly constant. As
T increases, v increases and t decreases  > o
• In semiconductors t still decreases but n starts
out small and increases fast with temperature.
 < o as n increases
m
 2
ne 
Power
• The power or the joule heating of the resistor
is how fast a resistor heats up
P  i2R
• The rate of energy transfer from battery to
some other device
P  iV
• This energy could be a conversion of
electrical potential energy to some other form
of energy such as mechanical work, thermal
energy, stored chemical energy, light or etc
Circuit Devices
• Resistor – device in a circuit to control the
current level in various parts of the circuit.
• Capacitor – device in a circuit which store
energy in an electric field
• Battery – device in a circuit which produces
a potential difference
• Conductor – material through which current
flows
Capacitor
Definitions
• Ideal emf device—has no internal resistance
and V = potential difference between the
terminals
• Real emf device—does have internal
resistance and V > potential difference
between the terminals (some energy lost
probably as heat)
•
•
Emf Devices
An emf device does work on (transfers
energy to) charge carriers
Energy comes from:
1.
2.
3.
4.
•
In batteries or fuel cells—chemical energy
In electric generator—mechanical forces
In thermopile—temperature differences
In solar cell—sun or solar energy
2 ways to calculate the current i in a simple
single loop circuit
1. Energy method
2. Potential method
Energy Method
• Using conservation of energy
dW
device
 Vdq  Vidt
P  i 2 R  energyresistor  i 2 Rdt
Vidt  i 2 Rdt
V
V  iR  i 
R
Potential Method
• Using the potential differences
• A battery from low to high potential
from high to low potential
• A resistor from low to high potential
• from high to low potential
V  V  iR  V
a
a
V
V  iR  0  V  iR  i 
R
V  V
V  V
V  iR
V  iR
•
•
Using the sign of the first terminal
Battery
– from low to high V
– from high to low V
•
Resistor
– from low to high V
– from high to low V
V  V
V  V
V  iR
V  iR
Va  V  iR  Va
V  iR  0  V  iR
V
i 
R
V   E
Internal Resistance of Battery
V
V  ir  iR  i 
Rr
This internal resistance comes from the
resistance of the internal components
of the battery and is irremovable
Resistors in Series and Parallel
• Resistors in Series
Requivalent  R1  R2  R3
• Resistors in Parallel
1
Rtotal
1 1 1
  
R1 R2 R3
Circuit Facts
Parallel
Series
Vtotal  V1  V2  V3
Vtotal  V1  V2  V3
qtotal  q1  q2  q3
qtotal  q1  q2  q3
itotal  i1  i2  i3
itotal  i1  i2  i3
Ctotal  C1  C2  C3
1
Rtotal
1 1 1
  
R1 R2 R3
1
Ctotal
1 1
1
  
C1 C2 C3
Rtotal  R1  R2  R3
Circuit Facts & Kirchhoff’s Laws
• If you have two batteries in a circuit, the
battery with the larger emf determines
the direction of the current
• Loop Rule: the algebraic sum of the
changes in potential encountered in a
complete transversal of any loop of a circuit
must be 0 (conservation of energy)
• Junction Rule: the sum of the currents
entering any junction must be equal to the
sum of the currents leaving that junction
(conservation of charge)
Meters
• Ammeter—an instrument used to measure
currents. In series & low resistance
• Voltmeter—an instrument used to measure
potential differences. In parallel & high
resistance
• Ohmmeter—an instrument used to measure
resistance of an element.
• Multimeter—a single meter which can
measure all of the above
RC Circuits
• RC circuit – circuit in which the current
 C  RC
varies with time
Charging
Discharging

dq V
i
 e 
dt R
q
V   V 1  e 
C

q  CV 1  e
t
C
t
C
t
C
 t  
q  qo  e



dq
qo  t  
i

e

dt
RC 

C
C
```