Download Quantum ElectroDynamics

TOPIC 6
Electric current
and resistance
1
Electrons in Conductors
Conductors have free electrons, which
• Are in continuous rapid motion – thermal and
quantum effects
• Undergo frequent scattering from the crystal lattice
(positive ions)
• Random motion does not constitute a current
• An applied electric field results in a small drift velocity
superimposed on the random motion
• This drift gives a net movement of charge – an electric
current, I
dQ
I
dt
2
Electric Current
Unit of current is Amp (Ampère), 1 A = 1 C s–1
Continuous current through conductor  potential
difference between ends – eg due to battery
Battery raises positive charges from low potential
(negative terminal) to high potential (positive terminal)
As much charge enters one end of the conductor
(eg wire) as leaves at the other end – it does not
charge up!
Current only flows in a closed loop or circuit
Current density J = current flow per unit area
(perpendicular to current) J = I/A
3
Electrical Resistance
The current I flowing through a component depends on
the potential difference between its ends, V.
We can define the resistance R of the component from
V
R
I
Unit of resistance : ohm  (1  = 1 volt per amp)
Conductance G = 1 / R (units –1 or mho)
4

Drude Model
vd
A
q
Assume n electrons (of charge q = –e) per unit volume
Applied electric field E gives acceleration a  qE
m
Mean time between collisions with lattice = 
Drift velocity (superimposed on random motion)
qE
vd 

m
Charge Q in dotted cylinder = n  A q, passes end
plane in time t = /vd, so I = n A q vd
Hence
nq 2 
J  nqvd 
E
m
5
Resistivity & Conductivity
2
Drude model 
We can write this
nq 
J
E
m
1
J  E  E

 is the resistivity, with units  m
 = 1 /  = conductivity

vd
q
Note J = I / A
V=E
A
V E

R 

I JA A
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Example 1 – Resistance, drift velocity
Copper has a resistivity of 1.710–8  m, and
8.5 1028 m–3 free electrons. What is the mean time
between collisions between a conduction electron
and the lattice? What will the drift velocity be when
2 V is applied across a 5 m sample of copper?
What resistance will a copper coil have if it is formed
of 1000 turns of wire, of diameter 1 mm, wrapped
around a tube of radius 3 cm?
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Temperature Coefficient of Resistivity
Increased temperature  increased lattice vibrations
 increased electron scattering
 increased resistivity in most materials
Approximation for modest temperature changes:
 T   0 1   T  T0  
Here  is the temperature coefficient of resistivity.
Example 2: A sample of platinum has a resistance of
30.00 at 20C, and 39.41 at 100C. What is the
coefficient of resistivity for platinum? What would the
resistance be at 0C?
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Electrical Power
The energy loss of a charge Q falling through potential
difference V is Q V. The power dissipated (rate of
energy loss) is therefore P = dQ/dt V = I V.
Using V = I R, this can be expressed in a variety of
useful ways:
2
V
2
P  IV  I R 
R
Example 3
A resistance of 3 is connected across a potential
difference of 2 V. What is the current which flows, and
how much power is dissipated?
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