Vhigh
Vlow
q
E
vd
conducting material
q
battery
A battery is connected to a conducting material to produce a potential
difference ∆V across the material. The battery “pumps” the positive
charges q from the low potential to the high potential side of the
conductor. The charges inside the material are moving from high to low
potential with an average drift speed vd.
J
silver
copper
gold
q
slope = σ
vd
J
E
iron
E
For many conductors it is found experimentally that the current density
J is directly proportional to the electric field E. The proportionality
constant is referred to as the conductivity “σ” of the material and its
value depends on the conductor’s material.
r
r
J = σE
Ohm’s Law
1
Ohm’s Law Applied to a Cylindrical Conductor
E
A
J
vd
q
Vhigh
Vlow
L
The conducting cylinder has a length L and cross sectional area A.
The conductivity is assumed to be uniform and the electric field E
is related to the potential difference by:
∆V = Vlow − Vhigh = − ∫ Eds = − EL
If we now let ∆V be the magnitude of the potential difference across the
cylinder we have:
E=
∆V
L
Ohm’s Law Applied to a Cylindrical Conductor
E
A
J
q
Vhigh
vd
Vlow
L
The current density is related to the current and cross sectional area by:
J =
I
A
Ohm’s Law applied to the cylinder gives:
J = σE ⇒
I
∆V
=σ
A
L
 L 
⇒ ∆V = I  
 σA 
2
Ohm’s Law Applied to a Cylindrical Conductor
E
A
J
vd
q
Vhigh
Vlow
L
 L 
∆V = I  
 σA 
The electrical resistance, R, of the cylindrical conductor is:
R=
L ρL
=
σA A
ρ is the reciprocal of σ and is called the resistivity.
∆V = IR
Alternate form of Ohm’s Law
Summary
E
A
q
Vhigh
vd
Vlow
L
The field E is in the direction from high to low potential.
The charge moves from high to low potential with a constant drift speed.
The resistance of the conductor is:
The potential difference, current
and resistance are related by
Ohm’s Law:
R=
L ρL
=
σA A
∆V = IR
3