Chapter 27
Current
Resistance
And
Resistor
Review
The current is defined and
its unit is ampere (A), a
base unit in the SI system.
+
+
+
A
+
dQ
I=
dt
+
+
I
The charges passing through the area could be positive
or negative or both. They are called charge carriers.
Current direction is chosen to the electric field direction
or the direction as the flow of positive charges.
Review
The motion of the charge carriers
is the result of the movement
driven by the electric field plus
random collisions with other
particles in the conductor. The
average velocity of thismotion is
called the drift velocity Vd
E
The current and the drift velocity has a linear relationship.
I = nqv d ⋅ A
n is the number of charge carriers per unit volume.
q is the carrier charge
The drift velocity is very small. What is actually prorogating in
the conducting wire is the electric field. So the conducting wire
is actually the carrier or guide for the electric field, which
delivers the electric power.
Current Density J, the definition
The current density J of the
current I a conductor is defined as
the current per unit area
I = ∫ J ⋅ dA
Compare with, I = nqv d ⋅ A one has:
J = nqVd
J has SI units of A/m2
Current density is a vector and is in the direction of the
positive charge carriers, or in the direction of the electric
field.
Conductivity and Ohm’s Law
For some materials, the current density is
directly proportional to the electric field:
J = σE
The constant of proportionality, σ, is called the
conductivity of the conductor.
Another often used parameter is resistivity, ρ,
is just inverse of the conductivity: ρ = 1/σ
And this formula, J = σE is called Ohm’s Law.
Ohm’s Law
Ohm’s law states the ratio of the current density
to the electric field is a constant σ
Mathematically, J = σ E
Most metals obey Ohm’s law
Materials that obey Ohm’s law are said to be ohmic
But
Not all materials follow Ohm’s law
Materials
that do not obey Ohm’s law are said to be
nonohmic
Ohm’s
law is not a fundamental law of nature
Ohm’s law is an empirical relationship valid only
for certain materials
Resistance and a more popular form
of Ohm’s Law
l
From Ohm’s Law: J = σ E
E
One has: E =
1
σ
J = ρJ
Now exam the section of wire with length l and
cross section A, one has:
I
l
∆V = El = Jl = ρ Jl = ρ l = ρ I
σ
A
A
1
Now we define a new parameter, the resistance of
this section of the wire, R:
l
R≡ρ
A
Ohm’s Law
becomes
∆V = RI
or
R≡
∆V
I
Resistance, discussion
SI units of resistance are ohms (Ω)
1Ω=1V/A
SI units of resistivity are Ω·m
Resistance proportional to the resistivity
constant of the material, the length of the
conductor (wire), inversely proportional to
the cross section.
In a conductor, the voltage applied
across the ends of the conductor is
proportional to the current through the
conductor
The constant of proportionality is called
the resistance of the conductor
l
R≡ρ
A
∆V = RI
R≡
∆V
I
Resistivity Values and the temperature
coefficient α
Resistance and Resistors
All (almost) materials have resistance
Those that are call ohmic if
Ohm’s Law R ≡ ∆V holds.
I
In a circuit, the resistance of
connecting wires (PCB
copper traces) are often
neglected.
A device made to have
certain resistance value is
call a resistor.
Ohmic Material, Graph
For an ohmic device
The resistance is constant
over a wide range of
voltages
The relationship between
current and voltage is
linear
The slope is related to the
resistance
Nonohmic Material, Graph
Nonohmic materials
are those whose
resistance changes
with voltage or current
The current-voltage
relationship is
nonlinear
A junction diode is a
common example of a
nonohmic device
Resistance and Temperature
Over a limited temperature range, the resistivity of
a conductor varies approximately linearly with the
temperature
ρ = ρo [1 + α (T − To )]
ρo is the resistivity at some reference temperature To
o
To is usually taken to be 20 C
α is the temperature coefficient of resistivity
SI units of α are oC-1
With a resistor:
R = Ro [1+ α (T − To )]
This is often used to measure temperature with
materials that have large α
Resistivity and Temperature,
Graphical View
For some metals, the
resistivity is nearly
proportional to the
temperature
A nonlinear region always
exists at very low
temperatures
The resistivity usually
reaches some finite value
as the temperature
approaches absolute zero
Electrical Power: work down by the
electric field in the circuit
As a charge Q moves from a to b, the
electric potential energy of the system
increases by Q∆V
so
This energy comes from the chemical
energy in the battery
Q
As the charge moves through the
resistor (c to d), the system loses
this electric potential energy, turning
Q
it to heat by the resistor
The electric power (energy time rate)
the resistor consumes is
dQ
energy Q∆V
because
I
≡
p=
=
= I ⋅ ∆V
time
t
dt
p = I ⋅ ∆V
2
∆
V
With Ohm’s Law: p = I 2R =
R
PLAY
ACTIVE FIGURE
Electric Power, to summarize
The electric power is given by the equation:
p= I ∆V
and is always correct.
If Ohm’s Law is applicable, alternative expressions
are often useful:
2
p= I∆V =I
2
∆V )
(
R=
R
Units: I is in A, R is in Ω, V is in V, and p is in W
Example: Electric Power Transmission
The question:
Dallas needs 100 MW electric power which
is generated 100 miles away. The
transmission line (aluminum) has a diameter
of 2 inches. How much power must be
generated to deliver 100 MW to Dallas, (a) If
110 V is used? (b) if 600,000 V is used?
Example: two incandescent light bulbs
In the US, the standard in our grid power system is
110 V. When two incandescent light bulbs (a pure
resistive device, with light output proportional to the
power consumed), with power specifications of 60 W
and 100 W, are connected in parallel, which one is
brighter?
Example: two incandescent light bulbs
Now connect them in series, which one is brighter in
this case?