Everglades High SchoolCambridge Physics – AS Level

Kirchhoff’s Laws

(Chapter 10)

Alberto Dominguez

Learning outcomes

• Recall and apply Kirchhoff’s Laws

• Use Kirchhoff’s Laws to derive the formulae for the combined

resistance of two or more resistors in series and in parallel

• Recognize that ammeters are connected in series within a circuit

and therefore should have low resistance

• Recognize that voltmeters are connected in parallel across a

component, or components, and therefore should have high

resistance

Coursebook, p. 143

CHAPTER OUTLINE

Learning outcome

• Recall and apply Kirchhoff’s Laws

• Use Kirchhoff’s Laws to derive the formulae for the combined

resistance of two or more resistors in series and in parallel

• Recognize that ammeters are connected in series within a circuit

and therefore should have low resistance

• Recognize that voltmeters are connected in parallel across a

component, or components, and therefore should have high

resistance

Kirchhoff’s First Law

• Also known as Kirchhoff’s Junction Rule

• The sum of the currents entering any point (junction) in a circuit is equal to the sum of the currents leaving the same point (junction)

• Follows from conservation of charge

• Written as ∑ Iin = ∑ Iout or ∑ I = 0

Coursebook, p. 144

Kirchhoff’s Second Law

• Also known as Kirchhoff’s Loop Rule

• The sum of the emf’s around any closed loop in a circuit is equal to the sum of the p.d.’s around the same loop

• Follows from conservation of energy

• Written as ∑ 𝜀 = ∑ p.d. or ∑ V = 0

Coursebook, p. 145

Learning outcome

• Recall and apply Kirchhoff’s Laws

• Use Kirchhoff’s Laws to derive the formulae for the combined

resistance of two or more resistors in series and in parallel

• Recognize that ammeters are connected in series within a circuit

and therefore should have low resistance

• Recognize that voltmeters are connected in parallel across a

component, or components, and therefore should have high

resistance

Resistors in Series

• The combined resistance in a

series circuit is the sum of the

circuit’s individual resistances

• Req = R1 + R2 + R3 + … + Rn

Coursebook, p. 148

• emfbattery = pdresistor 1 + pdresistor 2

• V = V1 + V2

• V = I1R1 + I2R2 = I(R1 + R2)

• V = IReq

• Therefore Req = R1 + R2

• More generally Req = R1 + R2 + R3 + … + Rn

• The combined resistance of series resistors is

always greater than any individual resistance

Coursebook, p. 148

Resistors in Series

Resistors in Parallel

• The sum of currents in parallel

resistors equals the total current

• The combined resistance in a

parallel circuit can be calculated

using a reciprocal relationship

•1

𝑅𝑒𝑞=

1

𝑅1+

1

𝑅2+

1

𝑅3+⋯+

1

𝑅𝑛

Coursebook, p. 149

• I = I1 + I2

•𝑉

𝑅𝑒𝑞=

𝑉1

𝑅1+

𝑉2

𝑅2

• V = V1 = V2

• Therefore 1

𝑅𝑒𝑞=

1

𝑅1+

1

𝑅2

• More generally 1

𝑅𝑒𝑞=

1

𝑅1+

1

𝑅2+

1

𝑅3+⋯+

1

𝑅𝑛

• The combined resistance of parallel resistors is

always less than the smallest individual resistance

Resistors in Parallel

Coursebook, p. 149

Series v Parallel – Summary

Property Series Circuit Parallel Circuit

Combined Resistance R = R1 + R2 + R3 +… 1/R = 1/R1 + 1/R2 + 1/R3 +…

Larger than largest Ri Smaller than smallest Ri

Voltage Vtot = Sum of all Vi V is the same across all Ri

Current Same across all Ri Itot = Sum of all Ii

Learning outcome

• Recall and apply Kirchhoff’s Laws

• Use Kirchhoff’s Laws to derive the formulae for the combined

resistance of two or more resistors in series and in parallel

• Recognize that ammeters are connected in series within a

circuit and therefore should have low resistance

• Recognize that voltmeters are connected in parallel across a

component, or components, and therefore should have high

resistance

Ammeters

• Ammeters measure the current in a circuit

• Therefore, they need to be connected in series

• To minimize the impact on the circuit, they should have as low

resistance as possible

• The ideal resistance is zero; digital ammeters have very low

resistances

Coursebook, p. 151

Voltmeters

• Voltmeters measure the p.d. between two points in a circuit

• Therefore, they need to be connected in parallel between the two

points

• To minimize the impact on the circuit, they should have as high

resistance as possible

• The ideal resistance is infinite; voltmeters have resistances of 1

MΩ or more

Coursebook, p. 151

Additional Resources

www.flashscience.com/electricity/kirchhoff.htm

Kirchhoff’s laws presented in a slightly different way to further your

understanding

http://labs.physics.dur.ac.uk/skills/skills/kirchhoff.php

Short notes on Kirchhoff’s laws, with a link to animated examples

of both laws in circuits

www.youtube.com/watch?v=Z2QDXjG2ynU

Walkthrough solution to a circuit problem using Kirchhoff’s laws

Teacher Book, Recommended Resources, Chapter 10

QUESTIONS

Learning outcome

• Recall and apply Kirchhoff’s Laws

• Use Kirchhoff’s Laws to derive the formulae for the combined

resistance of two or more resistors in series and in parallel

• Recognize that ammeters are connected in series within a circuit

and therefore should have low resistance

• Recognize that voltmeters are connected in parallel across a

component, or components, and therefore should have high

resistance

Question 1

Deduce the value of the current I

4.5 ACoursebook, p. 145

Question 2

Calculate the current in the wire X and state the

direction of this current.

1.5 A, towards P

Coursebook, p. 145

Question 3

Is Kirchhoff’s first law satisfied?

Yes

Coursebook, p. 145

Question 4

Deduce the value and direction of the current I

2.0 A, towards P

Coursebook, p. 145

Question 5

Deduce the p.d. across the resistor in the circuit

and hence find the value of R.

Potential Difference = 8.0 V

R = 80 Ω

Coursebook, p. 146

Question 6

(a) Choose the loop containing

the 5 V cell at the top, the 10 Ω

resistor with current I, and the

central 5 V cell, as the only

current involved is I.

(b) 1.0 A

Coursebook, p. 147

You can use Kirchhoff’s second

law to find the current I.

Choosing the best loop can

simplify the problem.

a) Which loop in the circuit

should you choose?

b) Calculate the current I.

Question 7

18 Ω

Coursebook, p. 147

Use Kirchhoff’s second

law to deduce the

resistance R.

Question 8

Explain why two 6 V batteries connected in series can

give an e.m.f. of 12 V or 0 V, but connected in parallel

they can give an e.m.f. of 6 V.

In series, the 1 C charge passes through both batteries

and gains or loses 6 J in each. If the batteries are

connected so that both of them move the charge in the

same direction, the total emf = 6 V+ 6 V = 12V. If the

batteries are connected back to front, the charge gains

energy in one cell but loses it in the other, so total emf

= 0 V.

In parallel, half the charge flows through one battery and

half through the other, so the total energy gained is 6J,

meaning the total emf = 6V. Coursebook, p. 148

Question 9

Apply Kirchhoff’s

laws to the circuit to

determine the

current that will be

shown by ammeters

A1, A2 and A3.

I1 = 0.50 A

I2 = 0.25 A

I3 = 0.25 ACoursebook, p. 148

Question 10

Calculate the combined resistance of two 5 Ω resistors

and a 10 Ω resistor connected in series.

20 Ω

Coursebook, p. 149

Question 11

The cell provides an

emf of 2.0 V. The

p.d. across one

lamp is 1.2 V.

Determine the p.d.

across the other

lamp.

0.8 V

Coursebook, p. 149

Question 12

You have five 1.5 V cells. How would you connect all five of

them in series to give an emf of

a) 7.5 V

All five in series and pointing the same way

b) 1.5 V

In series, with two facing in the opposite direction

c) 4.5 V

In series, with one facing in the opposite direction

Alternate answer: two in parallel to give emf of 1.5 V,

connected in series to two more in parallel, then connected

in series to the single remaining cell

Follow-up question: Is there any way to get 3.0 V or 6.0 V?

Not by connecting them only in series Coursebook, p. 149

Question 13

Calculate the combined resistance of four 10 Ω resistors

connected in parallel.

2.5 Ω

Coursebook, p. 150

Question 14

Calculate the combined resistances of the following

combinations.

a) 100 Ω and 200 Ω in series

300 Ω

b) 100 Ω and 200 Ω in parallel

67 Ω

c) 100 Ω and 200 Ω in series and this in parallel with 200 Ω

120 Ω

Coursebook, p. 150

Question 15

Calculate the current drawn from a 12 V battery of

negligible internal resistance connected to the following.

a) 500 Ω resistor

0.024 A

b) 500 Ω and 1000 Ω resistors in series

0.008 A

c) 500 Ω and 1000 Ω resistors in parallel

0.036 A

Coursebook, p. 150

Question 16

You are given one 200 Ω resistors and two 100 Ω resistors.

What total resistances can you obtain by connecting some

or all of these resistors in various combinations.

Total resistances possible are 40 Ω, 50 Ω, 67 Ω, 75 Ω, 100

Ω, 167 Ω, 200 Ω, 250 Ω, 300 Ω, and 400 Ω.

Coursebook, p. 150

Question 17

Three resistors of resistances 20 Ω, 30 Ω and 60 Ω are

connected together in parallel. Select which of the

following gives their combined resistance.

(a) 110 Ω (b) 50 Ω (c) 20 Ω (d) 10 Ω

Answer: (d)

No calculation required (answer must be less than 20 Ω)

Coursebook, p. 150

Questions 18 & 19

18) The battery of emf 10 V

has negligible internal

resistance. Calculate the

current in the 20 Ω resistor.

0.50 A

19) Determine the current

drawn from the battery.

0.95 A

Coursebook, p. 151

Question 20

What resistor must be connected in parallel with a 20 Ω

so that their combined resistance is 10 Ω?

20 Ω

Coursebook, p. 151

Question 21

You are supplied with a number of 100 Ω resistors.

Describe how you could combine the minimum number of

these to make a 250 Ω resistor.

Two combined in parallel (yielding a combined resistance

of 50 Ω), connected in series with a further two.

Coursebook, p. 151

Question 22

Determine the

current at each point

A-E.

Point A = 6.0 A

Point B = 6.0 A

Point C = 1.0 A

Point D = 5.0 A

Point E = 6.0 A

Coursebook, p. 151

Question 23

(a) A 10 V power supply of negligible internal resistance is

connected to a 100 Ω resistor. Calculate the current in

the resistor.

0.10 A

(b) An ammeter is now connected in the circuit to

measure the current. The resistance of the ammeter is 5

Ω. Calculate the ammeter reading.

0.095 A

Coursebook, p. 152