Current and Resistance
October 9, 2006
Notes
New topic today – Current and Resistance
 Quiz on Friday
 Friday 7:30 review sessiomn.
 There is a new WebAssign.

 Is
it AM or PM … only the shadow knows for
sure.
 I know too.
New Topic:
Current and Resistance
Physical Resistors
More …
What Happens?
“+”
REMEMBER, THE ELECTRONS
“+”
ARE ACTUALLY MOVING THE
OTHER WAY!
-
“+”
“+”
DEFINITION

Current is the motion of POSITIVE CHARGE
through a circuit. Physically, it is electrons
that move but …
Conducting material
DQ,Dt
Conducting material
DQ,Dt
CURRENT
DQ
i
Dt
or
dq
i
dt
UNITS

A current of one coulomb per second is
defined as ONE AMPERE.
Fraggen …..
A small sphere that carries a charge q is
whirled in a circle at the end of an
insulating string. The angular frequency
of rotation is ω. What average current
does this rotating charge represent?
ANOTHER DEFINITION
current I
J

area
A
Figure P27.8 represents a section of a circular conductor of
non-uniform diameter carrying a current of 5.00 A. The radius
of cross section A1 is 0.400 cm. (a) What is the magnitude of
the current density across A1? (b) If the current density across
A2 is one-fourth the value across A1, what is the radius of the
conductor at A2?
Ohm





A particular object will
resist the flow of current.
It is found that for any
conducting object, the
current is proportional to
the applied voltage.
STATEMENT: DV=IR
R is called the resistance of
the object.
An object that allows a
current flow of one ampere
when one volt is applied to
it has a resistance of one
OHM.
Graph
A DIODE
Resistance Varies with Applied Voltage
(actually with current)
Let’s look at the atomic level ..




Conduction is via electrons.
They are weak and small and don’t exercise
much.
Positive charge is big and strong and doesn’t
intimidate easily.
It’s an ugly situation … something like ……
+
-
Consider a metal conductor

So far, we have said that a metal is an equipotential
because no charges were moving.



Hence, no electric field in the metal
You can move a charge freely in the metal BECAUSE there
is no electric field.
NOW we have a current.



This can only happen if we allow an electric field to push
the charges.
Thus, the metal is NO LONGER A TRUE
EQUIPOTENTIAL.
But almost …. as we shall see in the next chapter.
Vb  Va
E
l
The Current


Electrons are going the opposite way from
the current. (WHY?)
They probably follow a path like …
Average “drift”
speed - vd
IN
OUT
Notation





vd average drift velocity of the electron
n number of electrons (mobile) per unit
volume.
Dt interval of time
Dx average distance the electron moves in
time Dt.
Q total amount of CHARGE that goes
through a surface of the conductor in time Dt.
The Diagram
DQ  (nAvd Dt )e
DQ
I avg 
 nAvd e
Dt
I avg
J
 nevd
A
J  nev d
Often a Vector
We return to the diagram …..





Consider an electron.
Assume that whenever it
“bumps” into something it loses
its momentum and comes to
rest.
It’s velocity therefore starts at
zero, the electric field
accelerates it until it has
another debilitating collision
with something else.
During the time it accelerates,
its velocity increases linearly .
The average distance that the
electron travels between
collisions is called the “mean
free path”.
Starting when the electron is
at rest:
We showed two slides ago::
v  v0  at  at
F eE
a 
m m
eE
v  vd 

m
Let n= number of charge carriers
per unit volume (mobile electrons)
eE
J  nqvd  nevd  ne 
m
or
ne 2 E
J
  E
m
so
ne 2

m
1


Finally
  vd
Reference
The average drift velocity of an
electron is about 10-4 m/s
Ponder
How can a current go through a
resistor and generate heat
(Power) without decreasing the
current itself?
Loses Energy
Gets it back
Exit
Conductivity
In metals, the bigger the electric field at a
point, the bigger the current density.
J  E
 is the conductivity of the material.
=(1/) is the resistivity of the material
Going to the usual limit …
dI
J
dA
and
I   JdA
Example
A cylindrical conductor of radius R has
a current density given by
(a) J0 (constant)
(b) gr
Find the total current in each case.
  0 1   (T  T0 )
Range of  and 
Ye old RESISTANCE
DV  El
1 DV
1 DV I
J  E 
E

 El
 l
A
l
DV  I
A
l
R
A
DV  V  IR
REMEMBER
R
L
A
DV  IR
A closed circuit
Power
In time Dt, a charge DQ is pushed through
the resistor by the battery. The amount of work
done by the battery is :
DW  VDQ
Power :
DW
DQ
V
 VI
Dt
Dt
Power  P  IV  I IR   I 2 R
2
E
P  I 2 R  IV 
R
Ponder
How can a current go through a
resistor and generate heat
(Power) without decreasing the
current itself?
Loses Energy
Gets it back
Exit
A Circuit
E
I
(R  r)
E=emf