### Chapter 17

```Current
and
Resistance
The Starting Point: Elements,
Atoms and Charge
Electrons and protons
their mass, a quantity
called charge
Charge (unlike mass) can
be either positive
(protons) or negative
(electrons)
Like charges repel, unlike charges attract
Free Electrons
An electron that is not bound to any particular
atom

Ions
 External force can cause an electron to
leave its orbit -atom is referred to as a
positive ion
 External force can cause an atom to
gain an electron -atom is referred to as
a negative ion
Electric Current


Whenever electric charges, an electric
current is said to exist
The current is the rate at which the
charge flows through this surface


Look at the charges flowing perpendicularly
to a surface of area A
Q
I 
t
The SI unit of current is Ampere (A)

1 A = 1 C/s
Electrical Current

Electron Flow Versus Conventional
Current
Insert Figure 1.10
Electric Current, cont

The direction of the current is the
direction positive charge would flow

This is known as conventional current
direction


In a common conductor, such as copper, the
current is due to the motion of the negatively
charged electrons
It is common to refer to a moving
charge as a mobile charge carrier

A charge carrier can be positive or negative
Current and Drift Speed


If the conductor is isolated, the
electrons undergo random motion
When an electric field is set up in
the conductor, it creates an electric
force on the electrons and hence a
current
Charge Carrier Motion in a
Conductor

The zig-zag black
line represents the
motion of charge
carrier in a
conductor



The net drift speed is
small
The sharp changes in
direction are due to
collisions
The net motion of
electrons is opposite
the direction of the
electric field
Current and Drift Speed



Charged particles
move through a
conductor of crosssectional area A
n is the number of
charge carriers per
unit volume
nAΔx is the total
number of charge
carriers
Current and Drift Speed

The total charge is the number of
carriers times the charge per carrier, q


The drift speed, vd, is the speed at
which the carriers move



ΔQ = (n A Δx) q
vd = Δx/ Δt
Rewritten: ΔQ = (n A vd Δt) q
Finally, current, I = ΔQ/Δt = nqvdA
Quiz 1







Consider positive and negative charges moving
horizontally through the four regions in the following
Figure. Rank the magnitudes of the currents in these four
regions from lowest to highest.
(a) Id , Ia , Ic , Ib
(b) Ia , Ic , Ib , Id
(c) Ic , Ia , Id , Ib
(d) Id , Ib , Ic , Ia
(e) Ia , Ib , Ic , Id
(f) none of these
(d). Negative charges moving in one
direction are equivalent to positive
charges moving in the opposite direction.
Thus, Ia, Ib, Ic and Id
are equivalent to the movement of 5,
3, 4, and 2 charges respectively,
giving Id < Ib < Ic< Ia
Example
A copper wire of cross-sectional area 3.00x10-6
m2 carries a current of 10 A. Assuming that
each copper atom contributes one free electron
to the metal, find the drift speed of the electron
in this wire. The density of copper is 8.95 g/cm3.
I
10.0C / s
vd 

nqA 8.48 1022 electrons m3 1.6 1019 C 3.00 106 m2

 2.46 106 m / s



Electrons in a Circuit



The drift speed is much smaller than
the average speed between collisions
When a circuit is completed, the electric
field travels with a speed close to the
speed of light
Although the drift speed is on the order
of 10-4 m/s the effect of the electric
field is felt on the order of 108 m/s
Quiz 2





Suppose a current-carrying wire has a crosssectional area that gradually becomes smaller
along the wire, so that the wire has the shape
of a very long cone. How does the drift speed
vary along the wire?
(a) It slows down as the cross section becomes
smaller.
(b) It speeds up as the cross section becomes
smaller.
(c) It doesn’t change.
the current is the same in all parts
of the wire. Thus, the drift velocity,
given by vd = I/nqA, is inversely
proportional to the cross-sectional
area
Meters in a Circuit – Ammeter

An ammeter is used to measure current

In line with the bulb, all the charge passing
through the bulb also must pass through
the meter
Meters in a Circuit – Voltmeter

A voltmeter is used to measure voltage
(potential difference)

Connects to the two ends of the bulb
Voltage


The electrical
potential
difference is what
causes current to
flow.
The basic unit of
voltage is the
volts . ~
Current flow-Amperes (Amps)


Ampere: unit of
measurement for
electrical current.
One volt across a
resistance of one
ohm will produce
a flow of one amp.
~
Exercise

The capacity of a car battery is usually
specified in ampere-hours. A battery rated
at, say, 100 A-h should be able to supply
100 A for 1 h, 50 A for 2 h, 25 A for 4 h, 1
A for 100 h, or any other combination
yielding a product of 100 A-h.


a. How many coulombs of charge should we be
able to draw from a fully charged 100 A-h
battery?
a require?
Assumptions:
Battery fully charged.
C
s 


a) 100 A 1hr  100  (1hr )  3600
  360000C
s
hr 


b) charge on electron: -1.602 ´ 10-19 C
no. of electrons =
360 10
22
 224.7 10
19
1.602 10
3
Ohm’s Law

German Physicist – George Simon Ohm




Found that current is inversely proportional
to resistance for a given voltage
Known as Ohm’s law
The Relationship Between Current and
Voltage
The Relationship Between Current and
Resistance
Basic Circuit Calculations

Using Ohm’s Law to Calculate
Current I  V
R
where
R = the circuit resistance
V = the applied voltage
Basic Circuit Calculations

Using Ohm’s Law to Calculate
Voltage
V  IR
where
I = the circuit current
R = the circuit resistance
Basic Circuit Calculations

Using Ohm’s Law to Calculate
V
Resistance R 
I
where
V = the circuit voltage
I = the circuit current
Resistance Element
Ohm’s Law


Experiments show that for many
materials, including most metals, the
resistance remains constant over a wide
range of applied voltages or currents
This statement has become known as
Ohm’s Law


ΔV = I R
Ohm’s Law is an empirical relationship
that is valid only for certain materials

Materials that obey Ohm’s Law are said to
be ohmic
Ohm’s Law
Conductivity and resistivity
A
I V
L
V I
L
V  IR
R
A

A
I
J 
 E
A

1

is current density
is resistivity
Example
A nichrome wire of radius 0.321 mm has resistivity of 1.5 x 10-6
Ωm. If a potential difference of 10.0 V is maintained across a 1
m length of the nichrome wire, what is the current?
Cross section:


2
A   r   0.32110 m  3.24 107 m2
2
Resistance/unit length:
3
R  1.5 10 m

 

4.6
m
7 2
l A 3.24 10 m
6
V 10.0V
I

 2.2 A
R
4.6
Ohm’s Law, cont




An ohmic device
The resistance is
constant over a wide
range of voltages
The relationship
between current and
voltage is linear
The slope is related
to the resistance
Ohm’s Law, final



Non-ohmic materials
are those whose
resistance changes
with voltage or
current
The current-voltage
relationship is
nonlinear
A diode is a common
example of a nonohmic device
Ohmic devices vs. non-Ohmic
Quiz 4

In Figure, does
the resistance of
the diode (a)
increase or (b)
decrease as the
positive voltage
ΔV increases?

(b). The slope of the line tangent to the
curve at a point is the reciprocal of the
resistance at that point. Note that as
increases, the slope (and hence )
increases. Thus, the resistance decreases.
Temperature Dependence of
Resistance
Define temp coefficient of resistivity
1 d

 dT
If  is small and constant
  o 1  T  To 
Units of  is (oC)-1
Example
Platinum Resistance Thermometer
A resistance thermometer, which measures temperature
by measuring the change in the resistance of a conductor,
is made of platinum and has a resistance of 50 Ω at 20oC.
When the device is immersed in a vessel containing
melting indium, its resistance increases to 76.8 Ω. Find
the melting point of Indium.
α=3.92x10-3(oC)-1
R  Ro
o
T  To 
 137 C
 Ro
o
o
Since the To= 20 c then T  157 C
Microscopic view of Ohm’s
Law
m
 2
ne 
Free electron Theory
Resistivity is a
constant
Why do old light bulbs give less
light than when new?

The filament of a light bulb, made of tungsten,
is kept at high temperature when the light
bulb is on.
It tends to evaporate, i.e. to become thinner, thus
decreasing in radius, and cross sectional area.
Its resistance increases with time.
The current going though the filament then
decreases with time – and so does its luminosity.


Tungsten atoms evaporate off the filament
and end up on the inner surface of the bulb.
Over time, the glass becomes less transparent and
therefore less luminous.

Low temperature behavior of
resistance
Semiconductor
Superconducti
vity
Exercise 1

If a current of 80.0 mA exists in a
metal wire, how many electrons
flow past a given cross section of
the wire in 10.0 min? Sketch the
direction of the current and the
direction of the electrons’ motion.
Exercise 2

If 3.25 × 10−3 kg of gold is
deposited on the negative
electrode of an electrolytic cell in a
period of 2.78 h, what is the
current in the cell during that
period? Assume that the gold ions
carry one elementary unit of
positive charge.
Exercise 3

An aluminum wire with a crosssectional area of 4.0 × 10−6 m2
carries a current of 5.0 A. Find the
drift speed of the electrons in the
wire. The density of aluminum is
2.7 g/cm3. (Assume that one
electron is supplied by each atom.)
Exercise 4

If the current carried by a
conductor is doubled, what
happens to (a) the charge carrier
density? (b) the electron drift
velocity?
```