Download Physics 202, Lecture 9 Charge Motion in a Conductor

Physics 202, Lecture 9

Current and Resistance (Ch 25)
 DC currents
 Ohm’s Law: Resistors and Resistance
 Conductivity and Resistivity
Next lecture: DC circuits
Reminder: HW #4 due tomorrow, 11 PM
Charge Motion in a Conductor
Electrons in a conductor have random motion (vave=0)
In an external electric field (e.g. as supplied by a source of
potential difference such as a battery),
electrons accelerate, produces current:
Average current:
I=
!Q
!t
Instantaneous current:
I=
dQ
dt
direct current (DC): I constant
Text: 25.1
1
Current: Macroscopic View
Current: rate at which charge
flows through surface:
Q
+
+
+
+
v
+
+
Q
+
+
A
Unit:1 Ampere = 1 C/s
Current is directional: Follows positive charge (convention)
+q moving in +x direction  –q in moving –x direction
+
+
+
+
+
+
+
+
+
I
Charge conservation  Current conservation
Iout
Iin
Iin = Iout
Current: Microscopic View
Current: motion of charged particles
I
vd: average
drift velocity
n: number density
I average =
Current density:
(vector!)
!Q
= nqvd A = I
!t
J=
I
= nqvd
A
! !
J
! idA = I
2
Drift velocity of conduction electrons
In external electric field:
acceleration a=qE/m
velocity v = vo + qEt / m
vo= velocity after
last collision
t = time since
last collision
Lots of particles: average
over all particle velocities
vave = (vo)ave + qEtave / m = qEτ / m
vd= Drift velocity= qEτ / m
x
x
x
x
τ=ave. time since last collision
x
x
Ohm’s Law: Resistance
I = current = nqvdA
vd= Drift velocity= qEτ / m
" nq 2! A %
" qE! %
I = nq$
'A = $
'V ( V = IR
# m &
# m L&
" nq 2! A %(1
R =$
'
# m L&
Voltage proportional to current! This is Ohm’s law.
Text: 25.6
!
! 1 !
! " nq ! % !
J = nqvd = $
E = (E = E
'
)
# m &
2
resistivity
conductivity
3
Conductivity, Resistivity, Resistance
Ohm’s Law (microscopic): J=σE
Ohm’s Law (macroscopic): ΔV=IR
Resistance R (unit: Ohm Ω = Volt/Ampere)
Exercise: relate R to ρ
Resistance
!
R=!
A
Length &
Cross-section
(shape)
Resistivity (intrinsic)
Resistors
Resistivity For Various Materials
R=!
Text: 25.11, 25.14
Resistors
!
A
4
Ohmic and non-Ohmic Materials
Ohmic:
Linear I-V relationship
(constant resistance over
wide range of voltages)
non-Ohmic:
Nonlinear I-V relationship
Resistance And Temperature
Resistivity is usually temperature dependent.
ρ
T
Semiconductor
Superconductor
Normal Metal
5
Superconductivity
Superconductors: temperature T<TC, resistivity ρ=0
(a quantum phenomenon!)
Electrical Power
R
I
I
Electrical Power:
ΔV
dU d(Q!V )
P=
=
= I !V
dt
dt
+
Ohmic:
-
ΔV
(!V )2
P=I R=
R
2
(power delivered to resistor)
Unit: watts (W)
energy unit: kWh
1 KWH = 3.6 MJ
Text example: 25.35
6
Example: Battery Connected To A Resistor
Energy flow of this battery-resistor set-up
eChemical Process  ΔV =1.5V
ΔV on Resistor  Current I= ΔV/R
I
+
+++++
e-------
R
Resistor
1.5V
++++
-----
e-
motion due to
motion due to
E. field
chemical process
Charge flow through the resistor in Δt:
Q=IΔt = ΔV/RΔt
Electrical potential energy released:
U=QΔV = ΔV/RΔt ΔV = (ΔV)2/RΔt
Power: P=U/dt = (ΔV)2/R
collisions
Energy Flow: Chemical  Electrical U  KE thermal/light
Quiz: Consumption of Electric Power On Resistors
A voltage is applied to a wire of length L . If L increases,
does power consumed increase or decrease?
Increases
 Decreases
Same
ΔV
Ni
7
Quiz: Consumption of Electric Power On Resistors
When a current passes through serially connected wire
segments made of copper and nichrome, which
metal: copper or nichrome, consumes more energy?
(ρCu ~ 10-8 Ωm, ρNi ~ 10-6 Ωm, All segments have about the same
length and diameter.)
Copper
 Nichrome
Same
I
Cu
Ni
Cu
Ni
Resistors in Circuits
R
I
• Resistors:
I
ΔV
Purpose is to limit the
current in a circuit.
+
Basic rule: voltage “drops”
as current flows through resistor
-
ΔV
Next lecture: DC circuits with resistors, capacitors
But first: a preview of resistors in series, parallel
8
Resistors in Series: Preview
Intuitively: voltage “drops”
I
Va ! Vb = IR1
a
Current same in both!
Vb ! Vc = IR2
R1
b
R2
a
Va ! Vc = I(R1 + R2 )
Reffective
c
c
Hence:
Reffective = (R1 + R2 )
Another (intuitive) way…
Consider two cylindrical resistors with lengths L1 and L2 :
R1
L
R1 = ! 1
A
L
R2 = ! 2
A
L1
V
L2
R
2
Put them together, end to end to make a longer one...
Reffective = !
L1 + L2
= R1 + R2
A
R = R1 + R2
What about parallel?
9