Download Current and Resistance

Current and Resistance
FCI
21-10-2013
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Define the current.
Understand the microscopic description of
current.
Discuss the rat at which the power transfer
to a device in an electric current.
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2-1 Electric current

2-2 Resistance and Ohm’s Law

2-3 Current density, conductivity and
resistivity

2-4 Electrical Energy and Power
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Whenever electric charges of like signs
move, 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
I 


Q
t
The SI unit of current is
Ampere (A) 1 A = 1 C/s
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∆Q is the amount of charge that
passes through this area in a
time interval ∆ t,
the average current Iav is equal to
the charge that passes through A
per unit time

We define the instantaneous current I as
the differential limit of average current:
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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. For example, the mobile charge carriers in
a metal are electrons.
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Charged particles
move through a
conductor of crosssectional area A
n is the number of
charge carriers per
unit volume
n A Δx is the total
number of charge
carriers
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The total charge is the number of carriers times the
charge per carrier, q
◦ ΔQ = (n A Δx) q

The drift speed, vd, is the speed at which the
carriers move
◦ vd = Δx/ Δt

Rewritten: ΔQ = (n A vd Δt) q

Finally, current, I = ΔQ/Δt = nqvdA

OR
the average current in the
conductor
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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
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
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

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Consider a conductor of cross-sectional area
A carrying a current I. The current density J in
the conductor is defined as the current per unit
area. Because the current I = nqvdA,
the current density is:
the current density is proportional to the electric field:
Where σ the constant of
proportionality & is called the
conductivity of the conductor.
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If the field is assumed to be uniform,
the potential difference is related to
the field through the relationship
express the magnitude of the current
density in the wire as
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,

Where ,J = I/A, we can write the potential
difference as
The quantity R = ℓ/σA is called the resistance of
the conductor. We can define the resistance as
the ratio of the potential difference across a
conductor to the current in the conductor:
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
In a conductor, the voltage applied across the
ends of the conductor is proportional to the
current through the conductor

The constant of proportionality is the
resistance of the conductor
V
R
I
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Units of resistance are ohms (Ω)
◦ 1Ω=1V/A

Resistance in a circuit arises due to collisions
between the electrons carrying the current with
the fixed atoms inside the conductor
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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
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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
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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
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
The resistance of an ohmic conductor is
proportional to its length, L, and inversely
proportional to its cross-sectional area, A
L
A
◦ ρ is the constant of proportionality and is called
the resistivity of the material
R  
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
For most metals, resistivity increases with
increasing temperature
◦ With a higher temperature, the metal’s
constituent atoms vibrate with increasing
amplitude
◦ The electrons find it more difficult to pass
through the atoms
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
For most metals, resistivity increases
approximately linearly with temperature
over a limited temperature range
  o [1  (T  To )]
◦ ρ is the resistivity at some temperature T
◦ ρo is the resistivity at some reference temperature
To
 To is usually taken to be 20° C
  is the temperature coefficient of resistivity
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
Since the resistance of a conductor with
uniform cross sectional area is proportional
to the resistivity, you can find the effect of
temperature on resistance
R  Ro [1 (T  To )]
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(T)  o 1    T  To  
R

A
R(T)  R o 1    T  To  
1 

= temperature coefficient of resistivity
o T

A class of materials
and compounds
whose resistances
fall to virtually zero
below a certain
temperature, TC
◦ TC is called the
critical
temperature

The graph is the
same as a normal
metal above TC,
but suddenly drops
to zero at TC
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In a circuit, as a charge moves through the
battery, the electrical potential energy of
the system is increased by ΔQΔV
As the charge moves through a resistor, it
loses this potential energy during collisions
with atoms in the resistor
◦ The temperature of the resistor will
increase
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
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Consider the circuit
shown
Imagine a quantity of
positive charge, Q,
moving around the
circuit from point A
back to point A
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Point A is the reference point
◦ It is grounded and its potential
is taken to be zero

As the charge moves
through the battery from A
to B, the potential energy of
the system increases by QV
◦ The chemical energy of the
battery decreases by the same
amount
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As the charge moves
through the resistor, from
C to D, it loses energy in
collisions with the atoms
of the resistor
The energy is transferred
to internal energy
When the charge returns to A, the net result is that
some chemical energy of the battery has been
delivered to the resistor and caused its temperature
to rise
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The rate at which the energy is lost is the
power
From Ohm’s Law, alternate forms of power
are
Q

t
V  I V
V
 I R 
R
2
2
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The SI unit of power is Watt (W)
◦ I must be in Amperes, R in ohms and V
in Volts
The unit of energy used by electric
companies is the kilowatt-hour
◦ This is defined in terms of the unit of
power and the amount of time it is
supplied
◦ 1 kWh = 3.60 x 106 J
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The same potential difference
is applied to the two
lightbulbs shown in Figure
.Which one of the following
statements is true?
 (a) The 30-W bulb carries
the greater current and has
the higher resistance.
 (b) The 30-W bulb carries
the greater current, but the
60-W bulb has the higher
resistance.
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(c) The 30-W bulb
has
the
higher
resistance, but the
60-W bulb carries the
greater current.
(d) The 60-W bulb
carries the greater
current and has the
higher resistance.
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
(c). Because the potential difference ∆V is the
same across the two bulbs and because the
power delivered to a conductor is P= I V, the
60-W bulb, with its higher power rating, must
carry the greater current.

The 30-W bulb has the higher resistance
because it draws less current at the same
potential difference.
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1.
The electric current I in a conductor is defined as
where dQ is the charge that passes through a cross
section of the conductor in a time interval dt. The SI unit
of current is the ampere (A), where 1 A = 1 C/s.
The average current in a conductor is related to the motion
of the charge carriers through the relationship
where n is the density of charge carriers, q is the charge
on each carrier, vd is the drift speed, and A is the crosssectional area of the conductor.
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2. The current density in an ohmic conductor is
proportional to the electric field according to the
expression
The proportionality constant σ is called the conductivity of the
material of which the conductor is made. The inverse of & is
known as resistivity ρ (that is, ρ = 1/ σ).
The last equation is known as Ohm’s law, and a material is said
to obey this law if the ratio of its current density J to its applied
electric field E is a constant that is independent of
the applied field.
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3. The resistance R of a conductor is defined as
where ∆V is the potential difference across it, and I is the
current it carries. The SI unit of resistance is volts per
ampere, which is defined to be 1 ohm; that is, 1Ω = 1 V/A. If
the resistance is independent of the applied potential
difference, the conductor obeys Ohm’s law.
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4. If a potential difference ∆V is maintained across a circuit
element, the power, or rate at which energy is supplied
to the element, is
Because the potential difference across a
resistor is given by ∆V = IR, we can express
the power delivered to a resistor in the form
The energy delivered to a resistor by electrical transmission appears in
the form of internal energy in the resistor.
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1. The charge carriers in metals are
A. electrons.
B. positrons.
C. protons.
D. a mix of protons and electrons.
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2. A battery is connected to a resistor.
Increasing the resistance of the resistor
will
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A. increase the current in the circuit.
B. decrease the current in the circuit.
C. not affect the current in the circuit.
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3. A battery is connected to a resistor. As
charge flows, the chemical energy of the
battery is dissipated as
 A. current.
 B. voltage.
 C. charge.
 D. thermal energy
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