PHY 111
Class #14 , 2/12/16
Spring 2016
1. Drift velocity of current carriers
Last time we ended with a calculation of current carriers (charged particles) flowing through a conducting wire:
Current density:
Current:
J = n q vd
I =JA
where n is the number of charge carriers per unit volume, q is the charge on each, and vd is the average drift velocity
of the particles carrying the current. These generalize in a straightforward manner:
If there are N charge species carrying current:
Current is the flux of current density:
J = ∑Ni=1 ni qi vdi
I = ∫ J·ⅆ A
S
Here, the flux integral is over a surface S relevant to the particular problem (usually not a closed surface). Examples
of current carrier species are:
electrons
(in metallic conductors)
electrons and “holes” (in semiconductors)
electrons and ions
(in plasmas and ionic solutions)
2. Resistivity and Resistance
If there is a uniform electric field in the wire, then the force on a charge carrier q would be F = q E and the charge
would accelerate uniformly. In our derivation last time I said the charges carrying the current had a constant drift
velocity. How is this possible? The answer is via collisions. Free charges carrying current are subject to collisions,
which slow them down and even reverse their direction temporarily. ICBS: if you average over All carriers and
collisions, the net drift is constant and given by
vd = ⟨v⟩charges & collisions ≈
qEτ
m
where τ is the mean time between collisions. Using the above equation for current density
2
J = n q vd =  n qm τ  E
The constant of proportionality is called the electrical conductivity σ. In general, conductors in which current carriers
have a well defined average drift velocity satisfy the vector version of this equation:
J=σE=
1
ρ
E
This is known as Ohm’s Law for conducting materials. The new constant ρ is called the resistivity of the material.
Both conductivity and resistivity have been measured for many materials and can be looked up in a handbook or
online. They are properties of the material the conductor is made of, and don’t depend on the specific device (e.g. σ
and ρ do not depend on the length of a particular wire, or its cross-sectional area). If you’re interested in how good of
PHY a
111
Spring 2016or a resistor of current a specific device is, we can derive that as follows:
R. Martin
conductor
I = current = flux of J = J ·A = σ E A
J=σE=
1
ρ
E
2
This is known as Ohm’s Law for conducting materials. The new constant ρ is called the resistivity of the material.
Both conductivity and resistivity have been measured for many materials and can be looked up in a handbook or
online. They are properties of the material the conductor is made of, and don’t depend on the specific device (e.g. σ
and ρ do not depend on the length of a particular wire, or its cross-sectional area). If you’re interested in how good of
a conductor or a resistor of current a specific device is, we can derive that as follows:
I = current = flux of J = J ·A = σ E A
If the device obeys the parallel plate equation, E = V /∕L where V is the voltage across the device and L is its length, so
I=
σV A
L
=  σLA  V =  ρAL  V
or
V =I R
The constant of proportionality is called the resistance R of the device (which is made of a material of resistivity ρ,
length L and cross-sectional area A). The inverse of resistance is the conductance: conductance = 1/∕R = σLA . Mostly
the resistance is used in circuit analysis, but conductance is just as meaningful physically. The above equation is also
called Ohm’s Law, since it really tells us the same physics as the current density-electric field relation.
3. Power dissipated in a resistor
Ohm’s Law tells us that the voltage changes across a resistor by the amount V = I R. For a positive test charge q
flowing with the current, the voltage decreases and therefor the potential energy of the charge decreases:
Δ U = U f -− Ui = q Δ V = q(-−I R)
and the instantaneous power “lost” is P = ⅆ U /∕ⅆ t = -−(ⅆ q/∕ⅆ t) I R = -−I 2 R. Thus, in absolute value
P = I2 R
is the power dissipated or gained through a resistor. It is dissipated (converted to heat) when the test charge passes
through R in the direction of the current and is gained if we force it through in the direction opposite to the current (i.e.
we must do work on it). Engineers usually talk about “I 2 R” losses in a resistor, effectively using the above equation
to label the power lost.
PHY 111 Spring 2016
R. Martin