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REVISED SCHEDULE
Class Test – 1
Day: 15 May 2019
Syllabus: Lectures 2-4
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Electrical Conduction
• Electrical conduction involves the motion of charges in a material under
the influence of an applied electric field.
• Metals the valence electrons from form a sea of electrons that are free
to move within the metal conduction electrons.
• Drude model describes electrical conduction in solids.
• The Drude model of electrical conduction was proposed
in 1900 by Paul Drude to explain the transport
properties of electrons in materials (especially metals).
• Drude model assumes the microscopic behavior of
electrons in a solid classically and looks much like
a pinball machine, with a sea of constantly jittering
electrons bouncing and re-bouncing off heavier,
relatively immobile positive ions.
The Drude Model
∆
• The electric current density: =
∆
q: net quantity of charge flowing through an
area A in time t.
• Conduction electrons move around randomly in the metal no net flow of
charge.
• When an electric field Ex is applied, conduction electrons acquire a net
velocity in the x direction.
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Drift Velocity
• The average velocity of electrons in the x direction or the drift velocity, vdx:
1
= + + + ⋯+
vxi : x direction velocity of the
ith electron
N : number of conduction
electrons
• Assume n = N∕V number of electrons per unit volume
• In time Δt, electrons move a distance Δx = vdx Δt, and Δq crossing A is enA Δx.
∆ ∆
• The current density in the x direction: = = =
∆ ∆
• Time-dependent current: = ( )
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Random Motion of Conduction Electrons
• The kinetic energy originates from the electrostatic interaction of these
electrons with the positive metal ions and also with each other.
• Conduction electrons move about randomly (with a mean speed u) being
frequently and randomly scattered by thermal vibrations of the atoms.
• In the absence of an applied field there is no net drift in any direction.
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3
Under Applied Electric Field
• Conduction electrons experience a force of eEx in the opposite direction of Ex.
• A net drift along the x direction is superimposed on the random motion of the
electron.
• The electron accelerates along the x direction under the action of the force eEx,
and then it suddenly collides with a vibrating atom and loses the gained
velocity there is an average velocity in the x direction
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Drift Velocity
Let uxi be the velocity of electron
i in the x direction just after the
collision (initial velocity). Since
eEx/me is the acceleration of the
electron, the velocity vxi in the x
direction at time t will be
= + ( − )
1
= + + + ⋯+
= ( − )
( − ) : average free time between collisions.
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Mean Free Time
− ≡
: Mean free time, mean time between collisions, or mean scattering time
• τ is directly related to the microscopic processes that cause the scattering of
the electrons in the metal — lattice vibrations, crystal imperfections, and
impurities, to name a few.
• 1∕τ represents the mean frequency of collisions or scattering events. During a
small time interval δt, the probability of scattering will be δt∕τ.
Drift Mobility
=
: drift mobility
• represents the ease of electron conduction under an electric
field.
• If the electron is not highly scattered, then the mean free time
between collisions will be long, τ will be large, and will also
be large; the electrons will therefore be highly mobile and be
able to “respond” to the field.
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Ohm’s Law and Conductivity
=
• Using the expression for drift velocity vdx : =
• Ohm’s Law: =
• = Conductivity
• A large does not necessarily imply high conductivity,
because σ also depends on the concentration of conduction
electrons n.
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