Microscopic view of electric current in a metallic wire,
the Drude model.
–
–
–
–
When no electric or magnetic field is
applied, these electrons move along straight
lines, until they undergo a collision (with
impurity atoms, defects, …).
–
–
In a metal, the valence electrons, typically
one per atom, are free to move.
The direction of the electron’s movement
after a collision is random, it has nothing to
do with the direction before the collision.
–
The speed of an electron after a collision
depends only on the temperature.
What is the average velocity of the electrons?
The average velocity v0 is zero, since the
1. zero
electrons move equally in all directions.
2. depends
–
–
The acceleration changes the velocity
of the electrons between collisions.
G
G G G G eE
v = v0 + at = v0 −
t
me
What is the average velocity of the electrons?
1. still zero 2. depends on the electric field
The average or drift velocity vd is the acceleration
times the average time τ between collisions.
–
–
G
An applied electric field exerts a
E
force on the electrons, which
accelerates the electrons between
G
G
collisions.
G
G
F
=
−
e
E
G
F = −eE ⎫
eE
G
G
G⎬ ⇒ a = − m
F = me a ⎭
e
–
–
G
eE
G
vd = − τ
me
Typically, random speed v0 ~ 106 m/s , collision time = τ ~ 10-14 s,
mean free path l0 = v0τ ~ 10-8 m, drift speed = vd ~ 10-5 – 10-3 m/s
Relation between drift velocity and current in a wire.
Pick a time interval Δt.
In this time, the electrons travel on
average a distance l = vdΔt .
In the time interval Δt, all electrons in the cylinder of length l will
cross the (pink) cross sectional area that bounds the cylinder on
the left, and no electrons outside the cylinder will cross this area.
What is the number of conduction electrons in the cylinder, if their
number density is n = N/V ?
1. N = nAl
2. N = nA/l
3. N = nl/A
Remember, the current is the flow of
charge through the cross section:
V = Al
I=
N
ΔQ N (−e)
=
= −e
Δt
Δt
Δt
Finally: I = −e nAl = −e nAvd Δt = −enAvd
Δt
Δt
⇒
Macroscopic form of Ohm’s law
V = RI
Resistance of a cylindrical wire
R=ρ
Electric field due to a voltage V
across a straight wire of length l
E=
Electrical current density,
σ = 1/ρ is the conductivity
j=
Microscopic form of Ohm’s law
V
l
j=
I
= − nevd
A
⇒ I=
V
R
l
A
⇒ V = El
I V
El/
El
1
=
=
=
= E ≡ σE
A RA RA ρ l/ A
/ ρ
/
A
j = σE =
1
ρ
E
ne 2τ
⇒ j=
E
We found before j = -nevd and vd = -eEτ/me
me
m
ne 2τ
1
Microscopic expression
j=
E = E ⇒ ρ = 2e
for the resistivity
me
ρ
ne τ