2. Microscopic Model of Electric Current in a Metal
- We’ll picture the metal as a regular array of atoms plus “free” or “conduction”
electrons that move around in random directions very rapidly ( with velocities of
106 m/s, close to 1% of the speed of light!!). These electrons make a series of
collisions with the atoms and ricochet in random directions – just like gas
molecules in air – and , in fact, we often use the term electron gas.
- In the absence of an external electric field in the wire, all the electrons move
about randomly and there is no net I flowing in the wire (see Figure 27-12 on
page 623 in your text).
- Now, suppose an external electric field is produced in the wire – this requires an
outside agent – usually a battery or power supply. Let’s imagine that the electric
field is uniform in the wire. What happens?
- Each electron with a charge q ( = -e) and mass me will experience a force given by
G
G
G
F = me a = qE , so that an acceleration
G q G
a=
E will be produced. According to our kinematics equations, then
me
G G G
v = vi + at
But the electrons will make collisions with the metal atoms and the initial
velocity after each collision will be random in direction. Each time an
electron makes a collision with an atom, it gives up the extra kinetic
energy it gained (so that the heavier atoms vibrate a bit more – we will see
that this causes a temperature increase of the metal) and then moves off in
a random direction.
- If τ is the average time between collisions with an atom, then (where < > means the
time average value
q G
G
G
<v> = <vi > +
E <t >
me
and since the average initial velocity is zero (see just above), we have that
q G
G G
<v>= vd =
Eτ , where vd is the drift velocity.
me
We’ll see that vd is about 1 mm/s – much !! slower (9 order of magnitude)
than the random velocity of the electron.
- Now let’s see how to relate vd to the current flowing along the wire:
With n = the number of electrons per unit volume, or density of electrons, (see sketch
of wire)
∆Q (in hatched region) = nA(vd ∆t) q
vd ∆t
so I ave =
∆Q
= qnAv d
∆t
-
Next, we introduce the current density, J = I/A, where A is an area perpendicular to
the velocity of the charges. Then we can write that
nq 2
J = nqv d =
Eτ
me
or
nq 2
J = σ E where σ =
τ
me
σ is called the electrical conductivity and is a constant independent of E
Any material that obeys this relation is said to obey Ohm’s Law.