2. Microscopic Model of Electric Current in a Metal

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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…
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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 G G q G v = vi + E t me and since the average initial velocity is zero (see just above), we have that G G q 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 ∆Q so I ave = = 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.
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