Current, Resistance and
Electromotive Force
Young and Freedman
Chapter 25
Electric Current: Analogy, water flowing in a pipe
H20
gallons/minute
Individual molecules are
bouncing around with
speeds of km/s!
“Flow Rate” is the NET amount
of water passing through a
surface per unit time
Net water velocity is m/s
I
Coulombs/s
“Electric Current” is the
NET amount of charge
passing through a surface
per unit time
- - - - - -
Individual electrons are
bouncing around with
very high speed
Electron “drift velocity
may be mm/s
Electric Current
In a Conductor, Charges are free to move.
The charges may be positive;
This is usually relevant only for
“special cases” like ions in a
solution. (Holes in semiconductors
act like positive charges)
The charges may be negative;
This is the normal case for
metallic conductors.
dQ
I=
dt
Inside a conductor there are LOTS of charges
There could be 1024 electrons /cm2+
Area A
Current I
vd is “drift velocity”
r
vd
n = # of charges q per m3
Total Current through area A is given by
I = nqvd A
Current per unit area is given by
I
J = = nqvd
A
J can vary in magnitude and direction in Space
r
r
J = nqvd
Vector Current Density
Conductors, in general, follow Ohm’s law
For many materials, the local current density is
proportional to the local electric field
v
r E
J=
!
E
or
!=
J
ρ is known as the Resistivity of a material
A material with a linear relationship between J and E is said to follow
“Ohm’s Law”
Important note: Not all material follow Ohm’s Law. Most metals do follow Ohm’s Law
so when we speak of a metallic conductor we are implicitly assume that the material
follows Ohm’s Law. This is not to be confused with a “perfect” conductor which has
zero resistivity. There are real materials called “superconductors”
There are many important examples of “Non-Ohmic” materials. Many extremely
important semi-conductor devices are non-ohmic.
Current
V
E=
L
r
v
E = !J
Ohm’s Law
Uniform E Field
What is the total Current through this object?
I = JA
E
I= A
!
V
I=
A
L!
L!
V=
I
A
Collect all the terms that
describe the object and
call them “R” the:
RESISTANCE
V = IR
Usual Statement
of Ohm’s Law
Resistivity and Resistance
IMPORTANT:
Do not confuse “Resistivity” with
Resisitance
Resistivity is a property of a type of
Material (copper, steel, water,…)
Resistance is a property of a particular,
specific object (a car key, a piece of wire…)
Circuits
Direct Current – “DC”
• In a DC Circuit ALL quantities (Voltage, Current, …) are constant
• Consider that the circuit has been running for a long time and will
continue to run longer.
In a steady state system – Charge can only flow in a “Loop”
E
I
-
+
+
+
E=0 I=0
V
Current can flow in continuous loop
BUT
If Resistance is NOT ZERO,
We require something to keep current flowing,
“ELECTRO MOTIVE FORCE”
ε
Continuing with “flowing water” analogy: EMF
In a closed water “circuit” because of
viscosity (“fluid friction”), there must be
some “motive force” to maintain a steady
state flow of water.
In a closed electrical “circuit” because of
resistivity (“electrical friction”), there must
be some “electro-motive force” to maintain
a steady state current.
ε
An Ideal “Electromotive Force” ε
provides a constant voltage between two
“terminals” –
No Matter How Much Current Flows!
Inside the “Ideal EMF”
r
A Non Electrostatic Force Fn acts on the the
charges inside the EMF. This cause the chargesrto
be displaces and leads to a electrostatic force Fe
which “balances” the non-electrostatic force.
A “resistive” path
Potential difference between ends of
resistive path:
V =!
V = IR }
! = IR
Symbols for circuit elements
Ideal conductor - generally assume that that R=0
Ideal EMF
NOTE – device is asymmetric
Ideal Resistor
EMF with internal resistance
Ideal Voltmeter - generally assume that that R=∞
- No current flows through an ideal voltmeter A
Ideal Ammeter - generally assume that that R=0
Electrically, an ideal ammeter is a perfect conductor
Open Circuit EMF Ex 25.2
Question: What do the meters read?
First simplify circuit by replacing the meters by equivalent resistors:
No complete circuit means No current
c
Vab = Vac + Vcb
Vab = IR + Vcb
Vab = 0 + Vcb
Vab = ! = 12V
Voltmeter reads V=12 volt
Ammeter reads A= 0 amperes
Open Circuit EMF Ex 25.2
=
c
Electrically
First Determine the Current:
V = IRtotal
V = I (r + R)
I
12V
I=
=
= 2A
( r + R ) 6!
Next Determine the Voltage:
Vab = Vcb " Vac
Vab = # " Ir
Vab = 12v " ( 2 A)( 2!)
Vab = 8V
Important Suggestion
for doing problems:
First completely solve
the problem
algebraically…
Then substitute
numerical quantities to
determine the
numerical answer
Electric potential through a complete circuit
FIGURE 25.20
If I go around the circuit and come back to the same point,
THE VOLTAGE MUST BE THE SAME!
Power in electric circuits
Power is defined as Energy (Work) per Unit Time
dW = VabdQ
dW
dQ
= Vab
dt
dt
dW
= Vab I
dt
For Pure Resistance
dW
P=
= IV
dt
V2
2
P=I R=
R
but V = IR
The sign of the power is important
dW > 0
Power added to system
Changes chemical energy to
electrical energy and adds it to
the energy in the circuit
dW < 0
Power removed from system
Changes electrical energy
to heat and removes it
from the circuit
Chapter 25 Summary
Chapter 25 Summary cont.
End of Chapter 25
You are responsible for the material covered in T&F Sections 25.1-25.5
You are expected to:
•
Understand the following terms:
Current, Resistivity, Resistance, EMF, Internal Resistance, Open
Circuit, Complete Circuit, Ammeter, Voltmeter, Short Circuit, Power
•
Determine Current and Voltage in a simple circuit.
•
Understand how voltmeters and ammeter’s are used and how they
respond.
•
Determine power dissipation in a simple circuit
Recommended Y&F Exercises chapter 25:
1, 10, 11, 31, 32, 35, 36, 44, 49