Current and Resistance

```Current and Resistance
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Putting electrons in motion
Electron movement through conductors
Resistivity and Resistance – Ohm’s Law
Electrical Power
Current
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Current : Flow
rate of charges
through a surface
area.
The direction of
the current is the
direction of the
flow of positive
charges.
ΔQ
I av =
Δt
dQ
I (t ) =
dt
(C/s=Amperes)
Microscopic Model
ΔQ = ( nV ) q
ΔQ = ( nAΔx ) q
ΔQ = ( nAvd Δt ) q
ΔQ
I av =
= nqvd A
Δt
vd : Drift Velocity
Drift Speed In Copper Wire
A = 3.31 x 10-6 m2
I = 10 A
ρCu = 8.92 g/cm3
ΔQ
I av =
= nqvd A
Δt
10 A = nvd 1.6 &times;10 −19 3.31&times; 10 −6
(
V=
m = 63.5 g/mol
NA = 6.02 x 1023 at/mol
1 electron per Cu atom
m
ρ
=
)(
)
63.5
= 7.12cm3 / mol
8.92
N A 6.02 &times; 10 23
=
= 8.46 &times; 10 22 el / cm3 = 8.46 &times; 10 28 el / m 3
n=
V
7.12
10
−4
vd =
=
2
.
23
&times;
10
m/s
28
−19
−6
1.6 &times; 10
3.31&times; 10 8.46 &times;10
(
)(
)(
)
Ohm’s Law
Current Density:
I
J = = nqvd
A
J = nqv d
(for a uniform current
perpendicular to the cross-section)
General vector definition
In some materials, the field applied and the current
density are proportional.
Ohm’s Law:
J = σE
Most metals obey Ohm’s Law
σ: conductivity
Resistance
⎛ l ⎞
ΔV = El = l = ⎜ ⎟ I
σ
⎝ σA ⎠
J
l
ΔV
R=
≡
σA
I
Metal
Semiconductor diode
(V/A=Ω)
Resistivity
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Resistivity is the
inverse of
conductivity.
Both are intrinsic
properties of the
material.
Resistance is a also
a function of the
shape and size of the
device.
1
l
ρ=
R=ρ
σ
A
Material
ρ (Ω.m)
Copper
1.7x10-8
Gold
2.44x10-8
Aluminum
2.82x10-8
Silicon
640
Rubber
~1013
Resistance of a Conductor
A=2x10-4 m2
ρAl = 2.82x10-8 Ω.m
10 cm
RAl = ρ Al
ρGlass = 3x1010 Ω.m
l
= 2.82 &times;10 −8
A
)
(
)
RGlass = ρ Glass
(
⎛ 0.1 ⎞
−5
1
.
41
10
=
&times;
Ω
⎜
⎟
−4
⎝ 2 &times;10 ⎠
l
⎛ 0.1 ⎞
13
=
1
.
5
&times;
10
Ω
= 3 &times;1010 ⎜
⎟
−4
A
⎝ 2 &times;10 ⎠
Resistance of a Coaxial Cable
ρ ⎛b⎞
R=
ln⎜ ⎟
2πL ⎝ a ⎠
A Model for Electrical Conduction:
The Drude Model
Atoms
Electrons
A regular array of
atoms surrounded
by a “cloud” of free
electrons
Random
movement under
zero field
Random
movement
modified by a field
field
Drude Model
F = qE = mea
qE
a=
me
J = σE
qE
v f = v i + at = v i +
t
me
Now take the average over all
times. Then vi=0 (random
movement),
v f = vd =
nq 2 E
τ
J = nqvd =
me
qE
qE
τ
t=
me
me
, where t is the mean
time between collisions
nq 2τ
σ=
me
me
ρ= 2
nq τ
l = τ ⋅ vd
is the mean free path
Resistivity and Temperature
Resistivity in metals is linear with temperature over a limited range
ρ = ρ 0 [1 + α (T − T0 )]
R = R0 [1 + α (T − T0 )]
α : temperature coefficient of resistivity
1 Δρ
α=
ρ 0 ΔT
(C-1)
Electrical Power
ΔU battery = ΔQΔV
I
ΔU resistor (heat ) = ΔU battery
+
-
ΔV
R
ΔU ΔQΔV
=
= IΔV
Electrical Power
Δt
Δt
V.A = Watts(W)
P = IΔV
(
ΔV )
R=
2
Symbol of a Resistor
P =I
2
R
Power Dissipated
on a Resistor
Rank the Currents
Ia=Ib
Ic=Id
Ie=If
Ia=Ic+Ie
P60W &gt; P30W
Ic&gt;Ie
Ia=Ib&gt; Ic=Id&gt;Ie=If
Summary
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Current is the net rate of charge flow.
Electrons move at the drift velocity.
Resistance is the ratio of voltage applied to
current. The ratio is linear for most metals.
Resistivity is a material property.
Electrical energy will be converted to
thermal energy on a resistor. The rate of
conversion is the power.
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