Current, Resistance, and
Electromotive Force
Physics 231
Lecture 5-1
Fall 2008
Current
Current is the motion of any charge, positive or
negative, from one point to another
Current is defined to be the amount of charge that
passes a given point in a given amount of time
dQ
I=
dt
Current has units of
Physics 231
1Coulomb
Ampere =
1 sec
Lecture 5-2
Fall 2008
Drift Velocity
Assume that an external electric field E has been
established within a conductor
Then any free charged particle in the
conductor will experience a force given by
r
r
F = qE
The charged particle will experience frequent
collisions, into random directions, with the particles
compromising the bulk of the material
There will however be a net overall motion
Physics 231
Lecture 5-3
Fall 2008
Drift Velocity
There is net displacement given by vd∆t where
vd is known as the drift velocity
Physics 231
Lecture 5-4
Fall 2008
Drift Velocity
Consider a conducting wire of
cross sectional area A having n
free charge-carrying particles
per unit volume with each
particle having a charge q with
particle moving at vd
The total charge moving past a given point is then given by
dQ = n q vd A dt
the current is then given by
dQ
I=
= n q vd A
dt
Physics 231
Lecture 5-5
Fall 2008
Current Density
dQ
I=
= n q vd A
This equation
dt
is still arbitrary because of the area still being
in the equation
We define the current density J to be
I
J = = n q vd
A
Physics 231
Lecture 5-6
Fall 2008
Current Density
Current density can also be defined to be a vector
r
r
J = n q vd
Note that this vector definition gives the same
direction for the current density whether we are
using the positive or negative charges as the
current carrier
Physics 231
Lecture 5-7
Fall 2008
Resistivity
The current density in a wire is not only dependent upon the
external electric field that is imposed but
It is also dependent upon the material that is being used
Ohm found that J is proportional to E and in an idealized
situation it is directly proportional to E
The resistivity is this proportionality constant and is given by
E
ρ=
J
The greater the resistivity for a given electric field,
the smaller the current density
Physics 231
Lecture 5-8
Fall 2008
Resistivity
The inverse of resistivity is defined to be the conductivity
The resistivity of a material is temperature dependent with
the resistivity increasing as the temperature increases
This is due to the increased vibrational motion of the
atoms the make up the lattice further inhibiting the
motion of the charge carriers
The relationship between the resistivity and temperature is
given approximately by
ρ (T ) = ρ 0 [1 + α (T − T0 )]
Physics 231
Lecture 5-9
Fall 2008
Resistivity
Let us take a length of conductor
having a certain resistivity
r
r
We have that E = ρ J
But E and the length of the wire, L, are related to
potential difference across the wire by V = E L
I
We also have that J =
A
Putting this all together, we then have
V ρI
=
L
A
Physics 231
or
V=
Lecture 5-10
ρL
A
I
Fall 2008
Resistance
We take the last equation
V=
ρL
and rewrite it as V = I R
with
R=
ρL
A
A
I
being the resistance
The resistance is proportional to the length of the material
and inversely proportional to cross sectional area
V = I R is often referred to as Ohm’s Law
The unit for R is the ohm or Volt / Ampere
Physics 231
Lecture 5-11
Fall 2008
Example
Two cylindrical resistors, R1 and R2, are made
of identical material. R2 has twice the length of
R1 but half the radius of R1. These resistors are
then connected to a battery V as shown:
I1
I2
V
What is the relation between I1, the current flowing in R1 , and I2 , the
current flowing in R2?
(b) I1 = I2
(a) I1 < I2
(c) I1 > I2
The resistivity of both resistors is the same (r). Therefore the resistances
are related as:
R2 = ρ
L2
2 L1
L
= ρ
= 8 ρ 1 = 8 R1
A2
( A1 / 4 )
A1
The resistors have the same voltage across them; therefore
I2
Physics 231
V
V
1
=
=
= I1
R2
8 R1
8
Lecture 5-12
Fall 2008
Resistance
Because the resistivity is temperature dependent,
so is the resistance
R(T ) = R0 [1 + α (T − T0 )]
This relationship really only holds if the the length and
the cross sectional area of the material being used does
not appreciably change with temperature
Physics 231
Lecture 5-13
Fall 2008
Electromotive Force
A steady current will exist in a conductor only if it is part
of a complete circuit
For an isolated conductor that has an external field
impressed on it
Physics 231
Lecture 5-14
Fall 2008
Electromotive Force
To maintain a steady current in an external circuit we
require the use of a source that supplies electrical energy
Whereas in the external circuit the current flows
from higher potential to lower potential, in this
source the current must flow from lower potential to
higher potential, even though the electrostatic force
within the source is in fact trying to do the opposite
In order to do this we must have an electromotive
force, emf, within such a source
The unit for emf is also Volt
Physics 231
Lecture 5-15
Fall 2008
Electromotive Force
Ideally, such a source would have a constant potential
difference, ε, between its terminals regardless of
current
Real sources of emf have an internal resistance which has
to be taken into account
The potential difference across the terminals of the source
is then given by
Vab = ε − I rinternal
Physics 231
Lecture 5-16
Fall 2008
Energy
As a charge “moves” through a circuit, work is done
that is equal to
qVab
This work does not result in an increase in the kinetic
energy of the charge, because of the collisions that
occur
Instead, this energy is transferred to the circuit or
circuit element within the complete circuit
Physics 231
Lecture 5-17
Fall 2008
Power
We usually are not interested in the amount of work done
but in the rate at which work is done
This given by
P = Vab I
If we have a pure resistance, we also have from before that
Vab = I R
giving us the additional relations
2
V
P = I 2 R = ab
R
Physics 231
Lecture 5-18
Fall 2008