Electric Current
„
„
Electric current is the rate of flow of
charge through some region of space
The SI unit of current is the ampere (A)
„
„
1A=1C/s
The symbol for electric current is I
Average Electric Current
„
„
Assume charges are
moving perpendicular
to a surface of area A
If ΔQ is the amount of
charge that passes
through A in time Δt,
then the average
current is
I av
ΔQ
=
Δt
Instantaneous Electric Current
„
If the rate at which the charge flows
varies with time, the instantaneous
current, I, can be found
dQ
I=
dt
Direction of Current
„
„
„
„
The charges passing through the area could
be positive or negative or both
It is conventional to assign to the current the
same direction as the flow of positive charges
The direction of current flow is opposite the
direction of the flow of electrons
It is common to refer to any moving charge as
a charge carrier
Current and Drift Speed
„
„
„
Charged particles
move through a
conductor of crosssectional area A
n is the number of
charge carriers per
unit volume
nA Δx is the total
number of charge
carriers
Current and Drift Speed, cont
„
The total charge is the number of
carriers times the charge per carrier, q
„
„
The drift speed, vd, is the speed at
which the carriers move
„
„
„
ΔQ = (nA Δx)q
vd = Δx / Δt
Rewritten: ΔQ = (nAvd Δt)q
Finally, current, Iav = ΔQ/Δt = nqvdA
Charge Carrier Motion in a
Conductor
„
The zigzag black line
represents the motion
of a charge carrier in a
conductor
„
„
„
The net drift speed is
small
The sharp changes in
direction are due to
collisions
The net motion of
electrons is opposite
the direction of the
electric field
Motion of Charge Carriers,
cont.
„
„
In spite of all the collisions, the charge
carriers slowly move along the
conductor with a drift velocity, vd
Changes in the electric field that drives
the free electrons travel through the
conductor with a speed near that of light
„
This is why the effect of flipping a switch is
effectively instantaneous
Electric current
I=
dq
dt
dq = Idt ,
(C = A) ( Amp )
s
∫
Q
0
t
dq = ∫ Idt ⇒ Q = It
0
Direction of currentGmotion direction of positive charge
I
• Current density
I
A
I
L
⊕
E
I
J=
A
r r
A
(
) I = ∫ J ⋅ dA
m2
I
J=
L
(A )
m
+
r
dA = dAnˆ
−
Consider the conductor shown in the figure. It is connected to a
battery (not shown) and thus charges move through the conductor.
Consider one of the cross sections through the conductor
( aa′ or bb′ or cc′ ).
The electric current i is defined as: i =
dq
dt
Current = rate at which charge flows
Current SI Unit:
C/s known as the "Ampere"
dq
i=
dt
(26 - 3)
Current direction :
conductor
r
v
+q
i
conductor
r
v
-q
i
An electric current is represented by an arrow
which has the same direction as the charge velocity.
The sense of the current arrow is defined as follows:
1. If the current is due to the motion of positive
charges the current arrow is parallel to the
r
charge velocity v
2. If the current is due to the motion of negative
charges the current arrow is antiparallel to
r
the charge velocity v .
(26 - 3)
conductor
r A
v
+q
Current density
r
J
i
conductor
i
J=
i
A
2
-q
i
follows: Its magnitude J =
A
m
/
A
:
J
r
o
f
t
i
n
u
I
S
r A
v
Current density is a vector that is defined as
r
J
r
The direction of J is the same as that of the
current. The current through a conductor of
cross sectional area A is given by the equation:
i = JA , if the current density is constant.
r
r r
If J is not constant then: i = ∫ J ⋅ dA .
(26 - 4)
conductor
r
v A
+q
r
J
i
conductor
An incoming current io branches at
r
v A
-q
i
i
J=
A
We note that even though the current density
is a vector the electric current is not. This is
illustrated in the figure to the left.
point a into two currents, i1 , and i2 .
r
J
Current io = i1 + i2
This equation expresses the conservation of
charge at point a. Please note that we have
not used vector addition.
(26 - 4)
Drift speed
When a current flows through a conductor the electric field causes
the charges to move with a constant drift speed vd . This drift speed
is superimposed on the random motion of the charges.
J = nvd e
r
r
J = nevd
(26 - 5)
• Drift speed
L
⎧ - : electron
Carrier ⎨
⎩ +: electric hole
q
A
Steady current : I =
L
I =η A q
t
v
random motion
Q
number of carriers
, Q = η ( AL)q, η :
,
t
volume
⇒ vd = vav =
L I 1
J
=
= .
t A ηq ηq
v
v
v
J = η qvd = ρ vd .
ρ: charge density (
charge
)
volume
EX.
Cu:atomic mass 63.5 g
I = 1A
density = 8.9 g
Cu
A = 1mm2
1 conducting
⇒
η = N umber
cm3
=
η =?
mole
cm3
carrier
atom
vd = ?
1
N mole g
⋅
⋅ 3 = 6.02 × 1023 ⋅
⋅ 8.9 ≈ 8.5 × 1022
mole g cm
63.5
I 1
1
1
=
⋅
≅ 0.01cm
vd =
2
−19
s
A η g (0.1) η ⋅1.6 × 10
• Resistance
E
vd can be viewed as the result of the
q
A
carrier under an acceleration in an
average time interval tav .
L
(average time between two collisions)
V
vd = vav
slpoe = a
tav
Fe = qE = ma
ma m vd m J
E=
=
= 2⋅
q
q tav q η tav
m
1
resistivity
σ
=
q 2η tav
ρ
r
J r
⇒ E = ρ J = , J = σ E.
Let ρ =
conductivity
σ
EL = ρ JL ⇒ V = ρ
Let R = ρ
L
A
I
L
A
resistance ( ohm,Ω ) ⇒
V = IR.
ρ = ρ0 (1 + αΔT )
T : temperature, α : temperature coefficient of ρ .
R=ρ
L
A
⇒
R = R0 (1 + α T ).
Resistivity
r
E
i
-
+
r
r
E = ρJ
Unlike the electrostatic case, the electric field
in the conductor of the figure is not zero.
V
r
r
J =σE
r
E
L
R=ρ
A
We define as resistivity ρ of the conductor
r
r
E
the ratio ρ = . In vector form: E = ρ J
J
V/m V
SI unit for ρ :
= m = Ω⋅m
2
A/m
A
1
The conductivity σ is defined as: σ =
ρ
Using ρ the previous equation takes the form:
r
r
J =σE
(26 -7)
Consider the conductor shown in the
figure above. The electric field inside the
r
E
i
+
r
r
E = ρJ
V
r
r
J =σE
i
The current density J =
A
We substitute E and J into equation
E
ρ = and get:
J
V /L V A
A
L
ρ=
=
=R →R=ρ
i/ A
i L
L
A
r
E
R=ρ
V
.
L
conductor E =
L
A
(26 -7)
Variation of resistivity with temperature
In the figure we plot the resistivity ρ of copper as function
of temperature T . The dependence of ρ on T is almost
linear. Similar dependence is observed in many conductors.
ρ − ρo = υoα (T − To )
(26 -8)
(26 - 9)
Ohm's Law : A resistor was defined as a conductor whose resistance
does not change with the voltage V applied across it. In fig.b we plot
the current i through a resistor as function of V . The plot (known as
"i - V curve" ) is a straight line that passes through the origin. Such a
conductor is said to be "Ohmic" and it obeys Ohm's law that states:
The current i through a conductor is proportional to the voltage V
applied across it.
(26 - 9)
Not all conductors abey Ohm's law (these are known as "non - Ohmic" )
An example is given in fig.c where we plot i versus V for a
semiconductor diode.
The ratio V / i (and thus the resistance R ) is not constant. As a matter
of fact the diode does not conduct for negative voltage values.
Note : Ohm's "law" is in reality a definition of Ohmic conductors
(defined as the conductors that obey Ohm's law)
A Microspopic view of Ohm's law :
In order to understand why some materials such as metals obey
Ohm's law we must look into the details of the conduction
process at the atomic level. A schematic of an Ohmic conductor
such as copper is shown in the figure. We assume that there are
free electrons that move around in random directions with an
effective speed veff = 1.6 × 106 m/s.
The free electrons suffer collisions
with the stationary copper atoms.
r
vd
r
F
(26 - 10)
A schematic of a free electron path is shown in the figure in the
figure using the dashed gray line. The electron starts at point A
r
and ends at point B. We now assume that an electric field E is
applied. The new electron path is indicated by the dashed green line.
Under the action of the electric force the electron acquires a small
drift speed vd . The electron drifts
to the right and ends at point B′.
r
vd
r
F
(26 - 10)
Consider the motion of one of the free electrons. We assume that
the average time between collisions with the copper atoms is equal
to τ . The electic field exerts a force F = eE on the electron,
F eE
resulting in an acceleration a = = .
m m
r
vd
The drift speed is given by the equation:
eEτ
vd = aτ =
(eqs.1)
m
r
F
(26 - 11)
We can also get vd from the equation: J = nevd → vd =
J
(eqs.2)
ne
If we compare equations 1 and 2 we get:
J eEτ
⎛ m
vd =
=
→E =⎜ 2
ne
m
⎝ ne τ
⎞
⎟ J . If we compare the last equation with:
⎠
m
E = ρ J we conclude that: ρ = 2
This is a statement of Ohm's law
ne τ
(the resistance of the conductor does not
depend on voltage and thus E ) This is
r
vd
r
F
because m, n, and e are constants.
The time τ can also be considered be
independent of E since the drift spees vd
is so much smaller than veff .
(26 - 11)
• Resistor
dU = Vdq = VIdt
V
R
R
dU
V2
2
P=
= VI = I R =
dt
R
( power : J = Watt = V ⋅ A)
s
• Kirchhoffs voltage rule ( loop rule ) :
The sum of the charge in potential encounted in a complete
traversal of any loop of a circuit must be zero.
( conservation of energy )
Resistivity
Values
Resistors
„
„
„
Most circuits use
elements called
resistors
Resistors are used
to control the current
level in parts of the
circuit
Resistors can be
composite or wirewound
Resistor Values
„
Values of resistors
are commonly
marked by colored
bands
Resistance of a Cable,
Example
„
„
Assume the silicon
between the
conductors to be
concentric elements
of thickness dr
The resistance of
the hollow cylinder
of silicon is
ρ
dR =
dr
2πrL
Resistance of a Cable,
Example, cont.
„
The total resistance across the entire
thickness is
ρ
⎛b⎞
R = ∫ dR =
ln ⎜ ⎟
a
2πL ⎝ a ⎠
b
„
„
This is the radial resistance of the cable
This is fairly high, which is desirable
since you want the current to flow along
the cable and not radially out of it
Conduction Model, final
„
Using Ohm’s Law, expressions for the
conductivity and resistivity of a conductor can
be found:
nq 2τ
1
me
σ=
ρ= = 2
me
σ nq τ
„
Note, the conductivity and the resistivity do
not depend on the strength of the field
The average time is also related to the free
mean path: τ = ℓ/vav
„
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
„
„
To is usually taken to be 20° C
α is the temperature coefficient of
resistivity
„
SI units of α are oC-1
Temperature Variation of
Resistance
„
Since the resistance of a conductor with
uniform cross sectional area is proportional to
the resistivity, you can find the effect of
temperature on resistance
R = Ro[1 + α(T - To)]
Resistivity and Temperature,
Graphical View
„
„
„
For 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
Residual Resistivity
„
„
The residual resistivity near absolute
zero is caused primarily by the collisions
of electrons with impurities and
imperfections in the metal
High temperature resistivity is
predominantly characterized by
collisions between the electrons and the
metal atoms
„
This is the linear range on the graph
Semiconductors
„
„
„
Semiconductors are
materials that exhibit a
decrease in resistivity
with an increase in
temperature
α is negative
There is an increase in
the density of charge
carriers at higher
temperatures
Superconductors
„
A class of materials
and compounds whose
resistances fall to
virtually zero below a
certain temperature, TC
„
„
TC is called the critical
temperature
The graph is the same
as a normal metal
above TC, but suddenly
drops to zero at TC
Superconductors, cont
„
The value of TC is sensitive to:
„
„
„
„
chemical composition
pressure
molecular structure
Once a current is set up in a
superconductor, it persists without any
applied voltage
„
Since R = 0
Superconductor Application
„
„
„
An important
application of
superconductors is a
superconducting
magnet
The magnitude of the
magnetic field is
about 10 times
greater than a normal
electromagnet
Used in MRI units
Electrical Power
„
„
Assume a circuit as
shown
As a charge moves
from a to b, the
electric potential
energy of the system
increases by QΔV
„
The chemical energy
in the battery must
decrease by this
same amount
Electric Power, final
„
The power is given by the equation:
℘= I ΔV
„
Applying Ohm’s Law, alternative expressions
can be found:
2
V
℘= I Δ V = I2 R =
R
„ Units: I is in A, R is in Ω, V is in V, and
℘ is in W
Electric Power Transmission
„
„
Real power lines
have resistance
Power companies
transmit electricity at
high voltages and
low currents to
minimize power
losses