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Direct Current Circuits:

3-1 EMF

3-2 Resistance in series and parallel .

3-3 Kirchhoff’s Rules 3-4 RC circuit

3-5 Electrical instruments

- Weston bridge

-Potentiometer19-11-2014 2FCI

Objectives:Defined some circuits in which resistors

can be combined using simple rules.

Understand the analysis of more complicated circuits is simplified using two rules known as Kirchhoff’s rules.

Describe electrical meters for measuring current and potential difference

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1- Sources of emfThe source that maintains the current in a

closed circuit is called a source of emfAny devices that increase the potential

energy of charges circulating in circuits are sources of emf

Examples include batteries and generators

SI units are VoltsThe emf is the work done per unit charge

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emf and Internal ResistanceA real battery has

some internal resistance “r”

Therefore, the terminal voltage is not equal to the emf

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More About Internal ResistanceThe schematic shows

the internal resistance, r

The terminal voltage is ΔV = Vb-Va

ΔV = ε – IrFor the entire

circuit,

ε = IR + Ir

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Now imagine moving through the

battery from a to b and measuring

the electric potential at various

locations. As we pass from the

negative terminal to the positive

terminal, the potential increases by

an amount Ɛ. As we move through

the resistance r, the potential

decreases by an amount Ir. Constant

potential from b to c. As we move

through the resistance R, the

potential decreases by an amount IR

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Internal Resistance and emf, contε is equal to the terminal voltage

when the current is zero

Also called the open-circuit voltageR is called the load resistance

The current depends on both the resistance external to the battery and the internal resistance

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Internal Resistance and emf, finalWhen R >> r, r can be ignored

Generally assumed in problemsPower relationship

I = I2 R + I2 rWhen R >> r, most of the power delivered by the battery is transferred to the load resistor

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Quiz 1In order to maximize the percentage of the

power that is delivered from a battery to a device, the internal resistance of the battery should be:

(a) as low as possible (b) as high as possible (c) The percentage does not depend on the internal resistance.

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Example 1:

A battery has an emf of 12.0 V and an internal resistance of 0.05 Ω. Its terminals are connected to a load resistance of 3.00 Ω.

(A) Find the current in the circuit and the terminal voltage of the battery.

(B) Calculate the power delivered to the load resistor, the power delivered to the internal resistance of the battery, and the power delivered by the battery.

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2-Resistors in SeriesWhen two or more resistors are connected

end-to-end, they are said to be in seriesThe current is the same in all resistors

because any charge that flows through one resistor flows through the other

The sum of the potential differences across the resistors is equal to the total potential difference across the combination

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Resistors in Series, contPotentials add

ΔV = IR1 + IR2 = I (R1+R2)

Consequence of Conservation of Energy

The equivalent resistance has the effect on the circuit as the original combination of resistors

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Equivalent Resistance – SeriesReq = 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 of the individual resistors

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Equivalent Resistance – Series: An Example

Four resistors are replaced with their equivalent resistance

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Quiz 2:With the switch in the circuit of Fig. A closed, there is no

current in R2, because the current has an alternate zero-resistance path through the switch. There is current in R1 and this current is measured with the ammeter, at the right side of the circuit. If the switch is opened (Fig. B), there is current in R2 . What happens to the reading on the ammeter when the switch is opened? (a) the reading goes up; (b) the reading goes down; (c) the reading does not change.

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(b). When the switch is opened, resistors

R1 and R2 are in series, so that the

total circuit resistance is larger than

when the switch was closed. As a

result, the current decreases.

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Resistors in ParallelThe potential difference across each resistor

is the same because each is connected directly across the battery terminals

The current, I, that enters a point must be equal

to the total current leaving that point

I = I1 + I2

The currents are generally not the same

Consequence of Conservation of Charge

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Equivalent Resistance – Parallel, Example:

Equivalent resistance replaces the two original resistances

Household circuits are wired so the electrical devices are connected in parallelCircuit breakers may be used in series with other

circuit elements for safety purposes19-11-2014 19FCI

Equivalent Resistance – ParallelEquivalent 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

321eq R

1

R

1

R

1

R

1

The equivalent is always less than the smallest resistor in the group

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Problem-Solving Strategy, 1Combine all resistors in series

They carry the same currentThe potential differences across them

are not the sameThe resistors add directly to give the

equivalent resistance of the series combination:

Req = R1 + R2 + …

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Problem-Solving Strategy, 2

Combine all resistors in parallelThe potential differences across them

are the sameThe currents through them are not

the sameThe equivalent resistance of a

parallel combination is found through reciprocal addition:

321eq R

1

R

1

R

1

R

1

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Problem-Solving Strategy, 3

A complicated circuit consisting of several resistors and batteries can often be reduced to a simple circuit with only one resistorReplace any resistors in series or in parallel

using steps 1 or 2. Sketch the new circuit after these changes

have been madeContinue to replace any series or parallel

combinations Continue until one equivalent resistance is

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Problem-Solving Strategy, 4If the current in or the potential difference

across a resistor in the complicated circuit is

to be identified, start with the final circuit

found in step 3 and gradually work back

through the circuits

Use ΔV = I R and the procedures in

steps 1 and 219-11-2014 24FCI

Equivalent Resistance

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Quiz 3:With the switch in the circuit of Fig. A

open, there is no current in R2. There is current in R1 and this current is measured with the ammeter at the right side of the circuit. If the switch is closed Fig.B, there is current in R2. What happens to the reading on the ammeter when the switch is closed?

(a) the reading goes up; (b) the reading goes down; (c) the reading does not change.19-11-2014 26FCI

(a). When the switch is closed, resistors R1

and R2 are

in parallel, so that the total circuit

resistance is smaller than when the switch

was open. As a result, the current

increases.

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More About the Junction RuleI1 = I2 + I3

From Conservation of Charge

Diagram b shows a mechanical analog

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Setting Up Kirchhoff’s Rules

Assign symbols and directions to the currents in all branches of the circuitIf a direction is chosen incorrectly, the

resulting answer will be negative, but the magnitude will be correct

When applying the loop rule, choose a direction for transversing the loopRecord voltage drops and rises as they

occur

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More About the Loop RuleTraveling around the loop

from a to b

In (a), the resistor is transversed in the direction of the current, the potential across the resistor is –IR

In( b), the resistor is transversed in the direction opposite of the current, the potential across the resistor is +IR

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Loop Rule, finalIn (c), the source of emf is

transversed in the direction of the emf (from – to +), the change in the electric potential is +ε

In (d), the source of emf is transversed in the direction opposite of the emf (from + to -), the change in the electric potential is -ε

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Junction Equations from Kirchhoff’s RulesUse the junction rule as often as needed,

so long as, each time you write an equation, you include in it a current that has not been used in a previous junction rule equationIn general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit

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Loop Equations from Kirchhoff’s RulesThe loop rule can be used as often as needed

so long as a new circuit element (resistor or battery) or a new current appears in each new equation

You need as many independent equations as you have unknowns

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Example: (A) Find the current in the circuit.

starting at a, we see that a b represents a potential differenceof + Ɛ1

b c represents a potential difference of -IR1, c d represents a potential difference of - Ɛ2, andd a represents a potential difference of -IR2

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(B) What power is delivered to each resistor? What power is delivered by the 12-V battery?

The negative sign for I indicates that the direction of the

current is opposite the assumed direction.

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