PETROVIETNAM UNIVERSITY
FACULTY OF FUNDAMENTAL SCIENCES
Fundamental Physics 2
Chapter 2
Pham Hong Quang
E-mail: quangph@pvu.edu.vn
Vungtau 2012
CHAPTER 2
Capacitance , Current and
Resistance, Direct Current
Circuits
Pham Hong Quang
2
PetroVietnam University
2
CHAPTER 2
2.1 Definition of Capacitance
2.2 Combinations of Capacitors
2.3 Energy Stored in a Charged Capacitor
2.4 Electric Current
2.5 The Resistance
2.6 Ohm’s law
2.7 Electric Power in Electric Circuits
2.8 Direct Current
2.9 Resistors in Series and Parallel
2.10 Kirchhoff ’s Rules
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Faculty of Fundamental Sciences
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2.1 Definition of Capacitance
Definition of Capacitor
A capacitor consists of 2
conductors of any shape placed
near one another without
touching. It is common; to fill
up the region between these 2
conductors with an insulating
material called a dielectric. We
charge these plates with
opposing charges to set up an
electric field.
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2.1 Definition of Capacitance
Electric Potential for Conducting Sheets
 E  dA 
EA 
qenc
o
Q
o
Q
A
  , EA 
A
o

E
o
Pham Hong Quang
V    E dr

)dr
a 
o
a 
V (b)  V (a )   ( )dr
b 
o

V (b)  V (a )  (a  b), a  b  d
o

Qd
V  d  Ed 
o
o A
b
V (b)  V (a )    (
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2.1 Definition of Capacitance
V  Ed ,
V E , if d  constant
E Q Therefore
Q V
C  contant of proportion ality
C  Capacitanc e
Q  CV
Q
C
V
The unit for capacitance is the FARAD, F.
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2.1 Definition of Capacitance
The Capacitance of a parallel plate capacitor
(1) Calculate q:
(2) Calculate V
(3) Calculate C:
q  0 EA
C 
V
Ed
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2.1 Definition of Capacitance
Dielectric
Remember, the dielectric is an insulating material placed
between the conductors to help store the charge.
A
C  k o
d
k  Dielectric
dielectric constant k, which is
the ratio of the field magnitude
E0 without the dielectric to the
field magnitude E inside the
dielectric:
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2.2 Combinations of Capacitors
Let’s say you decide that 1
capacitor will not be
enough to build what you
need to build.You may
need to use more than 1.
There are 2 basic ways to
assemble them together
Series – One after another
Parallel – between a set of
junctions and parallel to
each other.
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2.2 Combinations of Capacitors
Parallel Combination
The total charge on capacitors
connected in parallel is the sum of the
charges on the individual capacitors
for the equivalent capacitor
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2.2 Combinations of Capacitors
If we extend this treatment to three or more
capacitors connected in parallel, we find the
equivalent capacitance to be
Thus, the equivalent capacitance of a parallel
combination of capacitors is the algebraic sum
of the individual capacitances and is greater
than any of the individual capacitances.
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2.2 Combinations of Capacitors
Series Combination
•The charges on capacitors connected in series are the same.
•The total potential difference across any number of
capacitors connected
in series is the sum of the potential differences across the
individual capacitors.
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2.2 Combinations of Capacitors
When this analysis is applied to three or more capacitors
connected in series, the relationship for the equivalent
capacitance is
the inverse of the equivalent capacitance is the algebraic
sum of the inverses of the individual capacitances and the
equivalent capacitance of a series
combination is always less than any individual capacitance in
the combination.
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2.3 Energy Stored in a Charged Capacitor
The potential energy of a charged capacitor may be viewed
as being stored in the electric field between its plates.
Suppose that, at a given instant, a
charge q′ has been transferred from
one plate of a capacitor to the other.
The potential difference V′ between the
plates at that instant will be q′/C. If an
extra increment of charge dq′ is then
transferred, the increment of work
required will be,
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2.3 Energy Stored in a Charged Capacitor
The work required to bring the total capacitor
charge up to a final value q is
This work is stored as potential energy U in the
capacitor, so that
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2.3 Energy Stored in a Charged Capacitor
Energy Density
The potential energy per unit volume
between parallel-plate capacitor is
V/d equals the electric field magnitude E
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2.4 Electric current
Electric current I is the rate of the flow of
charge Q through a cross-section A in a
unit of time t.
The SI unit for current is a coulomb per
second (C/s), called as an ampere (A)
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Faculty of Fundamental Sciences
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2.4 Electric current
Direction of current
A current arrow is drawn in
the direction in which
positive charge carriers
would move, even if the
actual charge carriers are
negative and move in the
opposite direction.
The direction of conventional
current is always from a point
of higher potential toward a
point of lower potential—
that is, from the positive
toward the negative terminal.
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2.4 Electric current
Current Density
J
• Current density is to study the flow of charge through a
cross section of the conductor at a particular point
• It is a vector which has the same direction as the velocity
of the moving charges if they are positive and the opposite
direction if they are negative.
• The magnitude of J is equal to the current per unit area
through that area element.
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2.4 Electric current
Drift Speed
When a conductor does not have a current through it, its
conduction electrons move randomly, with no net motion in any
direction. When the conductor does have a current through it,
these electrons actually still move randomly, but now they tend to
drift with a drift speed vd in the direction opposite that of the
applied electric field that causes the current
Here the product ne, whose SI unit is the coulomb per
cubic meter (C/m3), is the carrier charge density
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Faculty of Fundamental Sciences
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2.5 The Resistance
The resistance (R) is defined as the ratio of
the voltage V applied across a piece of
material to the current I through the
material: R=V/i
SI Unit of Resistance:
volt/ampere (V/A)=ohm(Ω)
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Faculty of Fundamental Sciences
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2.5 the Resistance
The Resistivity
Resistivity of a material is:
The unit of ρ is ohm-meter (Ωm):
The conductivity σ of a material is
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2.5 the Resistance
Calculating Resistance from
Resistivity
•Resistance is a property of an object. It may vary
depending on the geometry of the material.
•Resistivity is a property of a material.
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2.6 Ohm’s law
Ohm’s law states that the current I through a given
conductor is directly proportional to the potential
difference V between its end points.
Ohm ' s law : I  V
Ohm’s law allows us to define resistance R and
to write the following forms of the law:
V
I ;
R
Pham Hong Quang
V  IR;
Faculty of Fundamental Sciences
V
R
I
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2.7 Electric Power in Electric Circuits
The amount of charge dq that moves
from terminals a to b in time interval dt
is equal to i dt.
Its electric potential energy decreases in
magnitude by the amount
•The decrease in electric potential energy from a to b is
accompanied by a transfer of energy to some other form. The
power P VIt
associated with that transfer is the rate of transfer
d U/dt,
P which is
t
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Faculty of Fundamental Sciences
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2.7 Electric Power in Electric Circuits
The transfer of electric potential
energy to thermal energy
The rate of electrical energy dissipation due to a resistance is
Caution:
• P=iV applies to electrical energy transfers of all kinds;
•P=i2R and P=V2/R apply only to the transfer of electric potential
energy to thermal energy in a device with resistance.
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Faculty of Fundamental Sciences
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2.8 Direct Current
When the current in a circuit has a constant
direction, the current is called direct current
Most of the circuits analyzed will be assumed to
be in steady state, with constant magnitude and
direction
Because the potential difference between the
terminals of a battery is constant, the battery
produces direct current
The battery is known as a source of emf
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Faculty of Fundamental Sciences
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2.8 Direct Current
Electromotive Force
The electromotive force (emf), e, of a battery is the
maximum possible voltage that the battery can
provide between its terminals
The battery will normally be the source of energy
in the circuit
The positive terminal of the battery is at a higher
potential than the negative terminal
We consider the wires to have no resistance
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Faculty of Fundamental Sciences
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2.8 Direct Current
Internal Battery Resistance
If the internal resistance is zero, the
terminal voltage equals the emf
In a real battery, there is internal
resistance, r
The terminal voltage, DV = e – Ir
Use the active figure to vary
the emf and resistances and see
the effect on the graph
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Faculty of Fundamental Sciences
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2.8 Direct Current
The emf is equivalent to the open-circuit
voltage
This is the terminal voltage when no
current is in the circuit
This is the voltage labeled on the
battery
The actual potential difference between
the terminals of the battery depends on
the current in the circuit
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Faculty of Fundamental Sciences
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2.8 Direct Current
Load Resistance
The terminal voltage also equals the voltage across the
external resistance
This external resistor is called the load resistance
In the previous circuit, the load resistance is just
the external resistor
In general, the load resistance could be any
electrical device
These resistances represent loads on the
battery since it supplies the energy to operate
the device containing the resistance
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Faculty of Fundamental Sciences
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2.8 Direct Current
Power
The total power output of the battery is
  I V  I
This power is delivered to the external
resistor (I 2 R) and to the internal resistor
(I2 r)
 I R I r
2
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2
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2.9 Resistors in Series and Parallel
Resistors in Series
When two or more resistors are connected end-to-end,
they are said to be in series
For a series combination of resistors, the currents are the
same in all the resistors because the amount of charge that
passes through one resistor must also pass through the
other resistors in the same time interval
The potential difference will divide among the resistors
such that the sum of the potential differences across the
resistors is equal to the total potential difference across the
combination
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2.9 Resistors in Series and Parallel
•The equivalent resistance has the same
effect on the circuit as the original
combination of resistors
•
Req = R1 + R2 + R3 + …
•The equivalent resistance of a series
combination of resistors is the algebraic
sum of the individual resistances and is
always greater than any individual
resistance
•If one device in the series circuit creates
an open circuit, all devices are inoperative
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Faculty of Fundamental Sciences
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2.9 Resistors in Series and Parallel
Resistors in Parallel
•The potential difference across each resistor is the
same because each is connected directly across the
battery terminals
•A junction is a point where the current can split
•The current, I, that enters a point must be equal to the
total current leaving that point
I = I 1 + I 2
The currents are generally not the same
Consequence of Conservation of Charge
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Faculty of Fundamental Sciences
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2.9 Resistors in Series and Parallel
Equivalent Resistance
1
1
1
1




Req R1 R2 R3
The inverse of the equivalent
resistance of two or more resistors
connected in parallel is the algebraic
sum of the inverses of the individual
resistance
The equivalent is always less than
the smallest resistor in the group
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Faculty of Fundamental Sciences
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2.9 Resistors in Series and Parallel
•In parallel, each device operates independently of the
others so that if one is switched off, the others remain on
•In parallel, all of the devices operate on the same voltage
•The current takes all the paths
The lower resistance will have higher currents
Even very high resistances will have some currents
•Household circuits are wired so that electrical devices are
connected in parallel
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Faculty of Fundamental Sciences
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2.10 Kirchhoff’s Rules
There are ways in which resistors can be
connected so that the circuits formed cannot
be reduced to a single equivalent resistor
Two rules, called Kirchhoff’s rules, can be
used instead
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2.10 Kirchhoff’s Rules
Kirchhoff’s Junction Rule
•The sum of the currents at any
junction must equal zero
Currents directed into the junction
are entered into the -equation as +I
and those leaving as -I
A statement of Conservation of
Charge
•Mathematically,
 I 0
junction
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Faculty of Fundamental Sciences
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2.10 Kirchhoff’s Rules
Kirchhoff’s Loop Rule
•Loop Rule
The sum of the potential differences
across all elements around any closed
circuit loop must be zero
•Mathematically,
 V  0
closed
loop
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Faculty of Fundamental Sciences
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2.10 Kirchhoff’s Rules
More about the Loop Rule
Traveling around the loop
from a to b
In (a), the resistor is
traversed in the direction of
the current, the potential
across the resistor is – IR
In (b), the resistor is
traversed in the direction
opposite of the current, the
potential across the resistor
is is + IR
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Faculty of Fundamental Sciences
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2.10 Kirchhoff’s Rules
In (c), the source of emf
is traversed in the
direction of the emf
(from – to +), and the
change in the electric
potential is +ε
In (d), the source of emf
is traversed in the
direction opposite of the
emf (from + to -), and
the change in the electric
potential is -ε
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Faculty of Fundamental Sciences
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2.10 Kirchhoff’s Rules
Problem-Solving Strategy –
Kirchhoff’s Rules
Conceptualize
Study the circuit diagram and identify all the elements
Identify the polarity of the battery
Imagine the directions of the currents in each battery
Categorize
Determine if the circuit can be reduced by combining
series and parallel resistors
If so, proceed with those techniques
If not, apply Kirchhoff’s Rules
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2.10 Kirchhoff’s Rules
Analyze
Assign labels and symbols to all known and
unknown quantities
Assign directions to the currents
The direction is arbitrary, but you must
adhere to the assigned directions when
applying Kirchhoff’s rules
Apply the junction rule to any junction in the
circuit that provides new relationships among
the various currents
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Faculty of Fundamental Sciences
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2.10 Kirchhoff’s Rules
Analyze, cont
Apply the loop rule to as many loops as are needed to solve
for the unknowns
To apply the loop rule, you must choose a direction in
which to travel around the loop
You must also correctly identify the potential difference as
you cross various elements
Solve the equations simultaneously for the unknown quantities
Finalize
Check your numerical answers for consistency
If any current value is negative, it means you guessed the
direction of that current incorrectly
The magnitude will still be correct
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Faculty of Fundamental Sciences
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Thank you!
Pham Hong Quang
46
PetroVietnam
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