fundamental physics 2 chapter 2 petrovietnam university faculty of fundamental sciences vungtau 2012...
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Fundamental Physics 2Chapter 2
PETROVIETNAM UNIVERSITYFACULTY OF FUNDAMENTAL SCIENCES
Vungtau 2012
Pham Hong QuangE-mail: [email protected]
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CHAPTER 2
Pham Hong Quang 2 PetroVietnam University
Capacitance , Current and Resistance, Direct Current
Circuits
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CHAPTER 2
Pham Hong Quang Faculty of Fundamental Sciences
2.1 Definition of Capacitance2.2 Combinations of Capacitors2.3 Energy Stored in a Charged Capacitor2.4 Electric Current2.5 The Resistance2.6 Ohm’s law2.7 Electric Power in Electric Circuits2.8 Direct Current2.9 Resistors in Series and Parallel2.10 Kirchhoff’s Rules
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2.1 Definition of Capacitance
Pham Hong Quang Faculty of Fundamental Sciences
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.
Definition of Capacitor
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2.1 Definition of Capacitance
Pham Hong Quang Faculty of Fundamental Sciences
Electric Potential for Conducting Sheets
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2.1 Definition of Capacitance
Pham Hong Quang Faculty of Fundamental Sciences
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The unit for capacitance is the FARAD, F.
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2.1 Definition of Capacitance
Pham Hong Quang Faculty of Fundamental Sciences
The Capacitance of a parallel plate capacitor
(1) Calculate q:
(2) Calculate V
(3) Calculate C:
0EAqC
V Ed
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2.1 Definition of Capacitance
Pham Hong Quang Faculty of Fundamental Sciences
Dielectric
Remember, the dielectric is an insulating material placed between the conductors to help store the charge.
Dielectric
kd
AkC o
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
Pham Hong Quang Faculty of Fundamental Sciences
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 togetherSeries – One after anotherParallel – between a set of junctions and parallel to each other.
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2.2 Combinations of Capacitors
Pham Hong Quang Faculty of Fundamental Sciences
Parallel CombinationThe 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
Pham Hong Quang Faculty of Fundamental Sciences
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
Pham Hong Quang Faculty of Fundamental Sciences
Series Combination
•The charges on capacitors connected in series are the same.•The total potential difference across any number of capacitors connectedin series is the sum of the potential differences across the individual capacitors.
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2.2 Combinations of Capacitors
Pham Hong Quang Faculty of Fundamental Sciences
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 seriescombination is always less than any individual capacitance in the combination.
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2.3 Energy Stored in a Charged Capacitor
Pham Hong Quang Faculty of Fundamental Sciences
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
Pham Hong Quang Faculty of Fundamental Sciences
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
Pham Hong Quang Faculty of Fundamental Sciences
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
Pham Hong Quang Faculty of Fundamental Sciences
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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2.4 Electric current
Pham Hong Quang Faculty of Fundamental Sciences
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.
Direction of current
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2.4 Electric current
Pham Hong Quang Faculty of Fundamental Sciences
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
Pham Hong Quang Faculty of Fundamental Sciences
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
Drift Speed
Here the product ne, whose SI unit is the coulomb per cubic meter (C/m3), is the carrier charge density
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2.5 The Resistance
Pham Hong Quang Faculty of Fundamental Sciences
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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2.5 the Resistance
Pham Hong Quang Faculty of Fundamental Sciences
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
Pham Hong Quang Faculty of Fundamental Sciences
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
Pham Hong Quang Faculty of Fundamental Sciences
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 V
I V IR RR I
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2.7 Electric Power in Electric Circuits
Pham Hong Quang Faculty of Fundamental Sciences
VItP
t
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 associated with that transfer is the rate of transfer d U/dt, which is
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2.7 Electric Power in Electric Circuits
Pham Hong Quang Faculty of Fundamental Sciences
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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2.8 Direct Current
Pham Hong Quang Faculty of Fundamental Sciences
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 currentThe battery is known as a source of emf
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2.8 Direct Current
Pham Hong Quang Faculty of Fundamental Sciences
Electromotive Force
The electromotive force (emf), e, of a battery is the maximum possible voltage that the battery can provide between its terminalsThe battery will normally be the source of energy in the circuitThe positive terminal of the battery is at a higher potential than the negative terminalWe consider the wires to have no resistance
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2.8 Direct Current
Pham Hong Quang Faculty of Fundamental Sciences
Internal Battery Resistance
If the internal resistance is zero, the terminal voltage equals the emfIn a real battery, there is internal resistance, rThe 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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2.8 Direct Current
Pham Hong Quang Faculty of Fundamental Sciences
The emf is equivalent to the open-circuit voltage
This is the terminal voltage when no current is in the circuitThis 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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2.8 Direct Current
Pham Hong Quang Faculty of Fundamental Sciences
Load Resistance
The terminal voltage also equals the voltage across the external resistance
This external resistor is called the load resistanceIn the previous circuit, the load resistance is just the external resistorIn 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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2.8 Direct Current
Pham Hong Quang Faculty of Fundamental Sciences
Power
The total power output of the battery is
This power is delivered to the external resistor (I 2 R) and to the internal resistor (I2 r)
I V I
2 2I R I r
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2.9 Resistors in Series and Parallel
Pham Hong Quang Faculty of Fundamental Sciences
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
Pham Hong Quang Faculty of Fundamental Sciences
•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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2.9 Resistors in Series and Parallel
Pham Hong Quang Faculty of Fundamental Sciences
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 pointI = I 1 + I 2
The currents are generally not the sameConsequence of Conservation of Charge
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2.9 Resistors in Series and Parallel
Pham Hong Quang Faculty of Fundamental Sciences
Equivalent Resistance
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
1 2 3
1 1 1 1
eqR R R R
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2.9 Resistors in Series and Parallel
Pham Hong Quang Faculty of Fundamental Sciences
•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 pathsThe lower resistance will have higher
currentsEven very high resistances will have some
currents
•Household circuits are wired so that electrical
devices are connected in parallel
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2.10 Kirchhoff’s Rules
Pham Hong Quang Faculty of Fundamental Sciences
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
Pham Hong Quang Faculty of Fundamental Sciences
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, 0
junction
I
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2.10 Kirchhoff’s Rules
Pham Hong Quang Faculty of Fundamental Sciences
Kirchhoff’s Loop Rule
•Loop RuleThe sum of the potential differences across all elements around any closed circuit loop must be zero
•Mathematically, closedloop
0V
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2.10 Kirchhoff’s Rules
Pham Hong Quang Faculty of Fundamental Sciences
Traveling around the loop from a to bIn (a), the resistor is traversed in the direction of the current, the potential across the resistor is – IRIn (b), the resistor is traversed in the direction opposite of the current, the potential across the resistor is is + IR
More about the Loop Rule
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2.10 Kirchhoff’s Rules
Pham Hong Quang Faculty of Fundamental Sciences
In (c), the source of emf is traversed in the direction of the emf (from – to +), and the change in the electric potential is +εIn (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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2.10 Kirchhoff’s Rules
Pham Hong Quang Faculty of Fundamental Sciences
Problem-Solving Strategy – Kirchhoff’s Rules
ConceptualizeStudy the circuit diagram and identify all the
elements
Identify the polarity of the battery
Imagine the directions of the currents in each
battery
CategorizeDetermine 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
Pham Hong Quang Faculty of Fundamental Sciences
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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2.10 Kirchhoff’s Rules
Pham Hong Quang Faculty of Fundamental Sciences
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 loopYou must also correctly identify the potential difference as you cross various elements
Solve the equations simultaneously for the unknown quantities
FinalizeCheck your numerical answers for consistencyIf any current value is negative, it means you guessed the direction of that current incorrectly
The magnitude will still be correct
46Pham Hong Quang 46 PetroVietnam University
Thank you!