fundamental physics 2 chapter 2

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


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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Electric Potential for Conducting Sheets

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The unit for capacitance is the FARAD, F.

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The Capacitance of a parallel plate capacitor

(1) Calculate q:

(2) Calculate V

(3) Calculate C:

0EAqCV Ed

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DielectricRemember, the dielectric is an insulating material placed between the conductors to help store the charge.

Dielectric

kdAkC 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


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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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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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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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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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


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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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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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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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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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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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


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


The ResistivityResistivity 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 VI V IR RR I

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2.7 Electric Power in Electric Circuits


VItPt

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


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


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


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


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


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


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


PowerThe 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


Resistors in Series

When two or more resistors are connected end-to-end, they are said to be in seriesFor 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 intervalThe 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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•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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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 sameConsequence of Conservation of Charge

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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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•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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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 resistorTwo rules, called Kirchhoff’s rules, can be used instead

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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

junctionI

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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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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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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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Problem-Solving Strategy – Kirchhoff’s Rules

ConceptualizeStudy the circuit diagram and identify all the elementsIdentify the polarity of the batteryImagine 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 techniquesIf not, apply Kirchhoff’s Rules

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AnalyzeAssign 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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Analyze, contApply 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!

fundamental physics 2 chapter 2

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