Download Current and Resistance

PHY 2049
Chapter 26
Current and Resistance
Chapter 26
Current and Resistance
In this chapter we will introduce the following new concepts:
-Electric current ( symbol i )
-Electric current density vector (symbol J )
-Drift speed (symbol vd )
-Resistance (symbol R ) and resistivity (symbol ρ ) of a conductor
-Ohmic and non-Ohmic conductors
We will also cover the following topics:
-Ohm’s law
-Power in electric circuits
(26 - 1)
Physical Resistors
What Happens?
“+”
REMEMBER, THE ELECTRONS
“+”
ARE ACTUALLY MOVING THE
“+”
OTHER WAY!
“+”
What’s Moving?
What is making the charged
move??
Battery
KEEP IN MIND


A wire is a conductor
We will assume that the conductor is essentially an
equi-potential


It really isn’t.
Electrons are moving in a conductor if a current is
flowing.




This means that there must be an electric field in the
conductor.
This implies a difference in potential since E=DV/d
We assume that the difference in potential is small and that
it can often be neglected.
In this chapter, we will consider this difference and what
causes it.
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
(26 - 3)
conductor
v
+q
dq
i
dt
i
conductor
v
-q
Current direction :
An electric current is represented by an arrow which has
i
the same direction as the charge velocity. The sense of the
current arrow is defined as follows:
1. If the current is due to the motion of positive charges
the current arrow is parallel to the charge velocity v
2. If the current is due to the motion of negative charges
the current arrow is antiparallel to the charge velocity v
UNITS

A current of one coulomb per second is
defined as ONE AMPERE.
ANOTHER DEFINITION
current I
J

area
A
Drift speed
When a current flows through a conductor
the electric field causes the charges to move
with a constant drift speed vd . This drift speed
is superimposed on the random motion of the
charges.
J  nvd e
J  nevd
Consider the conductor of cross sectional area A shown in the figure. We assume
that the current in the conductor consists of positive charges. The total charge
q within a length L is given by: q   nAL  e. This charge moves through area A
in a time t 
L
q nALe
. The current i  
 nAvd e
vd
t L / vd
The current density J 
i nAvd e

 nvd e
A
A
In vector form: J  nevd
(26 - 5)
i
-
+
V
R
R
V
i
Resistance
If we apply a voltage V across a conductor (see figure)
a current i will flow through the conductor.
V
We define the conductor resistance as the ratio R 
i
V
SI Unit for R :
 the Ohm (symbol )
A
A conductor across which we apply a voltage V = 1 Volt
and results in a current i = 1 Ampere is defined as
having resistance of 1 
Q : Why not use the symbol "O" instead of " "
A : Suppose we had a 1000  resistor.
We would then write: 1000 O which can easily
be mistaken read as 10000 .
A conductor whose function is to provide a
specified resistance is known as a "resistor"
The symbol is given to the left.
(26 - 6)
Ohm’s Law
DV  IR
Graph
DV  IR
Vb  Va
E
l
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.
i
Resistivity
E
Unlike the electrostatic case, the electric field in the
-
+
V
J E
E  J
conductor of the figure is not zero. We define as
E
resistivity  of the conductor the ratio  
J
In vector form: E   J
SI unit for ρ :
E
V/m V
 m  m
2
A/m
A
The conductivity  is defined as:  
1

Using  the previous equation takes the form: J   E
R
L
A
Consider the conductor shown in the figure above. The electric field inside the
V
i
conductor E  . The current density J 
We substitute E and J into
L
A
E
V /L V A
A
L
equation   and get:  

R R
J
i/ A
i L
L
A
(26 -7)
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
  0 1   (T  T0 )
Range of  and 
REMEMBER
R
L
A
DV  IR
Temperature
Effect
D
DT
  0 (1  DT )
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
Pi R
2
V
V2
P
R
If the device connected to the battery is a resistor R then the energy transfered by the
battery is converted as heat that appears on R. If we combine the equation P  iV
V
, we get the following two equivalent expressions for
R
the rate at which heat is dissipated on R.
with Ohm's law: i 
Pi R
2
and
V2
P
R
(26 - 13)