Lecture 24

• current and current density

• conductivity and resistivity

• Resistance and Ohm’s Law

• continue electron current

• creating current: model of conduction

Creating Current by

• electron current is non-equilibrium motion of charges sustained by internal

• establishing in wire

E

E

Creating Current by • model wire with rings of varying

charge: use (i) points away from positive ring; (ii) charge and (iii) decrease with distance

• net inside wire (from non-uniform charge distribution) pushes electron current

• how electron current turns a corner

E!

E

Model of conduction

• micro/marco link (like relating T, p f gas to energy of molecules): conduction electrons are “free” (average kinetic energy )

• : zero average velocity; straight lines between collisions

• : drift in direction opposite to ; net (small) motion (between collisions) superimposed on (larger) thermal motion

32kT

EE != 0

E = 0

Model of conduction (current )• After collision:

• till next collision (transfers energy to ion; raises temperature of wire); rebounds with new velocity (“reset”)

• repeated speeding up (between collisions) non-zero average velocity for single electron (drift speed, )

• average over all electrons:

ax = Fm = eE

m ; vx = vix + ax!t = vix + eEm !t

vd

vd = vx = vix + eEm !t

! ! !t (some shorter, other larger): vd = e!m E ! Using

! E

Current and Current Density

• connect ideas of electron current to conventional definition (before atoms...) of current: rate of flow of charge in a wire (units 1 ampere, A = 1 C/s: 1 A in lightbulb; mA in computers)

• charge delivered in terms of electrons

• current direction defined to be in which positive charges seem to move (opposite to direction of electrons - charge carriers in metals, makes no difference at macroscopic level): current in a wire from positive to negative terminal of battery

I !!

dQdt , in direction of E

"

Q = I!t; Ne = i!t; Q = eNe

! I = Q!t = eNe

!t = ei

Current and Current Density

• current density (same for all wires for given E; units ):

• conservation of charge ...of electron current ....of conventional current, even at junctions:

I = ei = nevdA I = JA

A/m2

Conductivity and Resistivity• characterize material:

• current caused by E exerting forces on charge carriers,

• conductivity decreases with temperature (more collisions...)

• more practical:

• Units:

• small E (very few surface charges) enough to carry considerable current due to huge n

• superconductivity: loss of resistance at low temperature (carry huge currents without heating, create huge magnetic fields)

J = nevd = ne!

e!Em

"= ne2!

m E

conductivity, ! = ne2!m

! E,n, !

AC/N m2 ! !!1 m!1 for !; ! m for " (! is ohm)

resistivity, ! = 1! = m

ne2"

Example: A 2.0-mm-diameter aluminium wire carries a current of 800 mA.What is the electric field strength inside the wire?E = J

! = I!A = I

!"r2 = 0.80 A(3.5!10!7!!1m!1)"(0.0010 m)2

= 0.00072 N/C