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Simple complicated

J. Manuel Physics 72 2nd Sem AY13-14

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Problem:

How do we analyze such kind of

complicated circuit?


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Direct-Current DC direction of the current does not change with time

Alternating-Current AC current oscillates back and forth

Direct-Current Circuits


26 Direct-Current Circuits

4 J. Manuel Physics 72 2nd Sem AY13-14

Resistors in Series and Parallel

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Objectives 1. Given a network of resistors connected in series and/or

parallel, evaluate the equivalent resistance, current and

voltage

2. Evaluate the voltage drop and current passing thru each

circuit element



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Resistor Resistance R (Ω)



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Four ways to connect 3 resistors

with resistances R1, R2, R3



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8 Equivalent resistance



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Resistors in Series

Current I must be the same in all resistors.

Their potential differences add.

Recall: Equivalent resistance of 3 resistors



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Resistors in Series

Equivalent resistance of n resistors

In general, for n number of resistors,



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Resistors in Parallel

Potential difference is the same for all resistors.

Their currents add.

Recall:

Equivalent resistance of 3 resistors



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Resistors in Parallel

Equivalent resistance of n resistors

In general, for n number of resistors,



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Example: Compute the equivalent resistance of the network

shown, and find the current in each resistor. The

battery has negligible internal resistance.



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Seatwork: In the circuit shown, each resistor represents a light bulb.

Let R1= R2 = R3 = R4 = 1 Ω and ε = 9.00 V.

a) (0.2 pt) Find the equivalent resistance of the network.

b) (0.2 x 4 pt) Find the current (I1, I2, I3, I4) in each bulb.



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Homework (0.5+0.5 pt): Consider the circuit shown. The current through the

6.00-Ω resistor is 4.00 A, in the direction shown. What

are the currents through the 25.0-Ω and 20.0-Ω

resistors?



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Objective 1. Given a circuit diagram, calculate the current through and

voltage across a circuit element using Kirchhoff’s loop and

junction rules



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Many practical resistor networks cannot be reduced to simple

series-parallel combinations.

Kirchhoff’s Rules



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Examples:

A junction in a circuit is a point where three or more conductors meet.

• (node or branch point)

A loop is any conducting path. J. Manuel Physics 72 2nd Sem AY13-14


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1. Kirchhoff’s Junction Rule The algebraic sum of the currents into any junction is zero.

The junction rule is based on conservation of electric charge.



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Example: Consider the circuit fragment shown in the figure. What

is the current IX flowing out of node B?



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Example: Using the circuit diagram below use junction rule to

express the relationship of the assumed current

direction.

I1

I2

I3



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2. Kirchhoff’s Loop Rule The algebraic sum of the potential differences in any loop,

(including those associated with emfs and those of resistive elements),

must equal to zero.

The loop rule is a statement that the electrostatic force is

conservative.



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2. Kirchhoff’s Loop Rule

Sign Conventions:



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Example:

I. -I1 R1 - I2 R2 - V1 = 0

II. I1 R1 + I3 R3 + V2 = 0

III. -I2 R2 - I4 R4 - V2 = 0

Consider the circuit shown with the

assumed direction of each branch

current shown by the labeled arrows.

Assuming each inner loop is

travelled counterclockwise, which

of the following equations follows

Kirchhoff's voltage law?



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Example: In the circuit shown, find

(a) the current in the 3.00-Ω resistor;

(b) the unknown emfs ε1 and ε2;

(c) the resistance R.

Note that three currents are given.



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Seatwork: What is the internal resistance r in the 12-V battery?



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In the circuit shown, find

(a)the current in resistor R;

(b)the resistance R;

(c)the unknown emf ε.

(d)If the circuit is broken at point x,

what is the current in resistor R?

Homework (0.5 x 4 pts):



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Objective 1. Describe the behavior of current, potential, and charge as a

capacitor is charging or discharging in terms of the initial,

transient, and steady-state conditions



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R-C Circuit A circuit that has a resistor and a capacitor in series

1. Current changes with time

2. Voltage changes with time

3. Power changes with time

Charging/Discharging a Capacitor

heart pacemakers

flashing traffic lights

automobile turn signals

electronic flash units

Applications:



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Caution: Lowercase (i, q, v) means time-varying.

Instantaneous current i = i(t)

Instantaneous charge q = q(t)

Instantaneous voltage v = v(t)



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Charging a Capacitor

1. idealized battery (or power supply)

constant emf ε (zero internal resistance, r = 0)

2. connecting conductors has negligible resistance

Assumptions:

Initially, capacitor C is uncharged. At t = 0, switch is closed.



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Charging a Capacitor At t = 0, switch is closed.

vbc = q/C = 0 (since q = 0)

Current Io through the resistor R

ε = IoR

Io = ε/R

After some time t,

Charge q on the capacitor increases.

Current i decreases.



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Charging a Capacitor After some time t,

Current i decreases.

Charge q on the capacitor increases.

Kirchhoff’s Loop Rule:



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Charging a Capacitor



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Time Constant τ measure of how quickly a capacitor charges

If τ is small, capacitor charges quickly.



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Time Constant



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Time Constant



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Discharging a Capacitor Initially, capacitor is charged (q = Qo).

1. Remove the battery from RC circuit.

2. Connect point a and c to an open switch.

At t = 0, switch is closed.

Kirchhoff’s Loop Rule:



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Discharging a Capacitor After some time t,



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Discharging a Capacitor After some time t,



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A 4.60-μF capacitor that is initially uncharged is connected in series with a

7.50-kΩ resistor and an emf source with ε = 125 V and negligible internal

resistance. Just after the circuit is completed, what are

(a) the voltage drop across the capacitor;

(b) the voltage drop across the resistor;

(c) the charge on the capacitor;

(d) the current through the resistor;

(e) the time constant?

(f) A long time after the circuit is completed (after many time constants)

what are the values of the quantities in parts (a)-(d)?

Example:


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