17.5 series and parallel resistors ;Kirchhoff’s rules

Statement of Kirchhoff’s Rules• Junction Rule ( I = 0)– The sum of the currents entering any point must

equal the sum of the currents leaving that junction• A statement of Conservation of Charge

• I1 = I2 + I3

outin II

– You must go around the loop in one direction– The sum of the measured will equal zero

Loop Rule ( V = 0)The sum of the potential changes around any

closed circuit loop must be zero

(a) The voltage across a battery is taken to be positive (a voltage rise) if traversed from – to + and negative if traversed in the opposite direction.

(b) The voltage across a resistor is taken to be negative (a drop) if the loop is traversed in in the direction of the assigned current and positive if traversed in the opposite direction

Vba = - IR

Vba = IR

ExampleCalculate the current I flowing into the node

outin II

(3+ I ) A = 2 AI = 2 -3 = -1 A

The current flowing into the node is – 1 A which is the same as +1 A flowing out of the node

ExampleCalculate the current I defined in the diagram

I +2 A = - 4 AI = (- 4 – 2 ) A = - 6 AI is in the opposite directionI + I = 6 AI = ( 6 – 6 ) A = A

Kirchhoff’s rules :The sum of the potential changes around any closed circuit must be zero

There are “two” ways to connect circuit elements.

1) Series combination: R3R2R1

+ -

ε

I

III

V1 V3V2

( a )

Rs

+ -

V

I

ε ( b )

) 12 . 17 (

0

0

321

321

321

321

RRRR

RRRI

RIRIRIVVV

RIRI

s

ss

Apply the Loop Rule

Figure 17- 10 (a) three resistors in series ( b) the equivalent resistance Rs leads to the same current I,

The current is the same in resistors because any charge that flows through one resistor flows through the other but the potential differences across them are not the same

2) Parallel combination

(a )

( b )

13 17. 1111

111

111

OR

leaving that equalmust A point entering

321

321

321321

3

3

2

2

1

1

321

RRRR

RRRR

RRRRRRI

RI

RI

RI

RI

IIII

currentthe

p

p

p

I

Figure 17 .11 ( a ) three resistors in parallel . ( b ) the equivalent single resistance Rp produces the same current I

V

Rp

+ -

I

IA B

Iε

R3

R2

R1

+ -

I

I3

I1

I2

A B

Iε

Example 17.10(a ) find the equivalent resistance of the resistors in figure 17.10 a ( b ) the current I in each resistor

(a )

(b )

( c )

Solved in the text book

T.Norah Ali Almoneef 8

V20 10

I 30

From the circuit with source voltage V and Total current I, which resistor will have the greatest voltage across it?The resistor with the largest resistance (30 )

Which resistor has the greatest current flow through it?Same for all because series circuit

If we re-ordered the resistors, what if any of this would change?Nothing would change

Conceptoal question

A) find the current in the circuit shown in the figure .B ) find the potential difference across each circuit element

In the figure, we had a 3kΩ, 10 kΩ, and 5 kΩ resistor in series,

AI 5.0

VV

VV

VV

5.2

5

5.1

3

2

1

Example

18kRs

ExampleFrom the figure find ( a ) I ( total current ) , Rp ( total resistance ) ( b ) I 1 , I 2 , I 3

AI 23 AI 32

AI 61

AI 11

64.1 R p

Four resistors are connected as shown in figure. Find the equivalent resistance between points a and c.

A. 4 R.

B. 3 R.

C. 2.5 R.

D. 0.4 R.

E. Cannot determine from information given

.

Example

V2010

I 30

From the circuit with source voltage V and Total current I, which resistor will have the greatest voltage across it?The resistor with the largest resistance (30 )

Which resistor has the greatest current flow through it?Same for all because series circuit

If we re-ordered the resistors, what if any of this would change?Nothing would change

Conceptual questions

I

Total resistance would increaseTotal current would decreaseVoltage across each resistor would decrease (All voltage drops must still sum to total in series circuit; Kirchhoff’s law of voltages)Current through each resistor would be lower (total current decreased, but same through each one)

• If we added a resistor in series with these, what would happen to the total resistance, total current, voltage across each resistor, and current through each resistor?

from the circuit with source voltage V and Total current I, which resistor will have the greatest voltage across it?All the same in parallel branches Which resistor has the greatest current flow through it?The “path of least resistance” (10) What else can you say about the current through each branch?They will sum to the total I (currents sum in parallel circuits; Kirchhoff’s law of current)

Conceptual questions

• If we added a resistor in parallel with these, what would happen to the total resistance, total current, voltage across each resistor, and current through each resistor?Total resistance would decreaseTotal current would increaseVoltage across each resistor would still be VCurrent through each resistor would be higher and would sum to new total

17.12 Kirchhoff’s rules in complex circuits Kirchhoff’s rules permit us to analyze any dc circuit .including circuits too complex Using the two rules (1) the sum of all the potential drops around any closed path in a circuit is equal to zero. (2) The current entering any point = The current leaving.

Example 17.15

Find the current in the circuit shown in the figure

Solved in the text book

A) 2 A

B) 3 A

C) 5 A

D) 6 A

E) 10 A

5 A

8 A

2 A

P

What is the current in branch P?

The current entering the junction in red is 8 A, so the current leaving must also be 8 A. One exiting branch has 2 A, so the other branch (at P) must have 6 A.

5 A

8 A

2 A

P

Junction

6 A

SS

Conceptual questions

2 V

2

6 V

4 V

3 1

1

I1 I3

I2

Which of the equations is valid for the circuit below?

A) 2 – I1 – 2I2 = 0

B) 2 – 2I1 – 2I2 – 4I3 = 0

C) 2 – I1 – 4 – 2I2 = 0

D) I3 – 4 – 2I2 + 6 = 0

E) 2 – I1 – 3I3 – 6 = 0

Conceptual questions

+

+

+

9 V

9 V

9 V

ΔVab if one battery is reversed?

a

b

ΔVab= 27V

Δvab= 9v

quiz

Calculate ΔVab

AI 625.0

quiz

Calculate the current in the circuit.

Find the current I, r and ε.

I = 3 Ar =2 =-5 V

quiz

Calculate the currents I1, I2, and I3 in the three branches of the circuit in the figure.

Quiz

I1 = - 0.87 A. I2 = 2.6 A. I3 = 1.7 A.

summary

Kirchhoff’s Rules

outin II1-

2- Loop Rule

Series combination: 321 RRRRs

Parallel combination 1111

321

RRRRp

0i

iV

Home work 45,46,71