Resistance – Learning Outcomes Define resistance and give its unit.

Solve problems about resistance.

State Ohm’s Law.

HL: Derive the formulas for resistors in series and parallel.

Solve problems about resistors in series and parallel.

Give the factors that affect the resistance of a

conductor.

Use an ohmmeter.

Solve problems about resistivity.

Resistance – Learning Outcomes Discuss light-dependent resistors (LDRs) and thermistors.

Demonstrate LDRs and thermistors.

HL: Describe wheatstone bridges.

HL: Solve problems about wheatstone bridges.

HL: Discuss uses of a wheatstone bridges.

HL: Use a metre bridge.

Resistance The resistance of a conductor is the ratio of the voltage

across it to the current flowing through it.

Formula: 𝑅 =𝑉

𝐼

Resistance is a scalar quantity measured in ohms (Ω).

A conductor has a resistance of 1 ohm if the current

through it is 1 ampere when the voltage across it is 1 volt.

Resistance is measured using an ohmmeter or

multimeter set to measure resistance – alternatively

calculate it by measuring current and voltage, then

using the formula.

Resistance

Resistance e.g. Find the resistance of a conductor if it carries a

current of 4 A when the voltage across it is 20 V.

e.g. What potential difference will produce a current of

5 A in a 12 Ω resistor?

e.g. At a certain temperature, the current through a

conductor is 3 A when the voltage across it is 24 V. Find

the resistance of the conductor.

When the temperature of the conductor is raised, the

same voltage causes a current of 2 A to flow through it.

Find the increase in its resistance.

Ohm’s LawOhm’s Law states the current flowing through a

conductor is proportional to the voltage across it at

constant temperature.

Formula: 𝑉 ∝ 𝐼

Constant temperature is required since resistance varies

with temperature – more on this later.

Some conductors will also vary their resistance with

voltage.

For conductors which obey Ohm’s Law, the constant of

proportionality is resistance:

Formula: 𝑉 = 𝐼𝑅 (the same formula we covered earlier)

Types of Resistor

Resistors in Series For two or more resistors in series, their total resistance is

the sum of their resistances.

Formula: RTotal = 𝑅1 + 𝑅2 + 𝑅3 +⋯

Resistors in Series To prove: RT = 𝑅1 + 𝑅2 + 𝑅3

Let 𝑉1, 𝑉2, 𝑉3 be the voltages across each resistor.

Let 𝐼 be the current through each resistor.

By formula, 𝑉1 = 𝐼𝑅1, 𝑉2 = 𝐼𝑅2, 𝑉3 = 𝐼𝑅3

But VT = 𝑉1 + 𝑉2 + 𝑉3

⇒ 𝑉𝑇 = 𝐼𝑅1 + 𝐼𝑅2 + 𝐼𝑅3

⇒ 𝐼𝑅𝑇 = 𝐼𝑅1 + 𝐼𝑅2 + 𝐼𝑅3

⇒ 𝐼𝑅𝑇 = 𝐼(𝑅1 + 𝑅2 + 𝑅3)

⇒ 𝑅𝑇 = 𝑅1 + 𝑅2 + 𝑅3

Resistors in Series e.g. Calculate the total resistance of the following

resistors:

Resistors in Parallel For two or more resistors in parallel, their total resistance

is given by:

Formula: 1

𝑅𝑇=

1

𝑅1+

1

𝑅2+

1

𝑅3+⋯

Resistors in Parallel

To prove: 1

𝑅𝑇=

1

𝑅1+

1

𝑅2+

1

𝑅3

Let V be the voltage across each resistor.

Let 𝐼1, 𝐼2, 𝐼3 be the current through each resistor.

By formula, 𝐼1 =𝑉

𝑅1, 𝐼2 =

𝑉

𝑅2, 𝐼3 =

𝑉

𝑅3

But V

RT= 𝐼 = 𝐼1 + 𝐼2 + 𝐼3

So 𝑉

𝑅𝑇=

𝑉

𝑅1+

𝑉

𝑅2+

𝑉

𝑅3

So 1

𝑅𝑇=

1

𝑅1+

1

𝑅2+

1

𝑅3

Resistors in Parallel e.g. Calculate the total resistance of the following

resistors:

Resistance in Circuits e.g. What is the total resistance in this circuit? What is the

potential difference across the 9Ω resistor?

Resistance in Circuits e.g. What is the total resistance of this circuit? What is

the current flowing through the 3Ω resistor?

Resistance in Circuits e.g. If the bulb has resistance 4Ω, what is the total

resistance of this circuit? What is the current flowing

through the bulb?

Resistance in Circuits e.g. What is the total resistance of the following resistors?

Factors Affecting Resistance – Temperature

We already know that the resistance of a conductor

depends on temperature.

Increased temperature has two effects:

Heat releases extra electrons from the atoms, decreasing

resistance.

Heat causes atoms to vibrate more, increasing resistance.

For metallic conductors, very few electrons are released,

so resistance increases with increasing temperature.

For insulators and semiconductors, lots of electrons are

released, so resistance decreases with increasing

temperature.

Factors Affecting Resistance The resistance of a conductor also depends on:

Length, 𝑙,

Cross-sectional area, 𝐴,

Resistivity of the material, 𝜌.

Factors Affecting Resistance - LengthConsider a cuboid resistor:

What is the effect on resistance if a second identical

resistor is added in series?

It doubles ⇒ 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∝ 𝑙𝑒𝑛𝑔𝑡ℎ

Factors – Cross-Sectional AreaWhat if the second resistor is instead added in parallel?

Using 1

𝑅𝑇=

1

𝑅1+

1

𝑅2,

⇒1

𝑅𝑇=

2

𝑅1

⇒ 𝑅𝑇 =1

2𝑅1

The same is true if the resistors

are in contact.

⇒ 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∝1

𝐴𝑟𝑒𝑎

Factors - Resistivity Different materials come with a natural level of

resistance – we normalise this using resistivity.

The resistivity of a material is the resistance of a 1𝑚 ×1𝑚 × 1𝑚 cube of the material.

The relationship is: r𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∝ 𝑟𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑖𝑡𝑦

Factors Affecting Resistance Putting each of these factors into a single formula, we

get:

Formula: 𝑅 =𝜌𝑙

𝐴

We also get a definition for resistivity from this:

Formula: 𝜌 =𝑅𝐴

𝑙

Factors Affecting Resistance e.g. A uniform wire of length 2 m has a resistance of 12Ω.

Find the resistance of a piece of identical wire of length

14 m.

e.g. What length of copper wire of cross-sectional area

2 mm2 is needed to make a resistor of resistance 10 Ω?

Resistivity of copper = 1.7 × 10−8 Ω𝑚.

e.g. A coil of copper wire 20 m long has uniform

composition and uniform cross-sectional area. The

diameter of the wire is 0.055 mm. Calculate the

resistance of the coil if 𝜌𝑐𝑜𝑝𝑝𝑒𝑟 = 1.7 × 10−8 Ω𝑚.

Light-Dependent Resistor A light-dependent resistor (LDR) is

a semiconductor that decreases

its resistance when light shines on it.

Light hitting the resistor releases

electrons from the molecules,

allowing them to conduct electricity.

Thermistor A thermistor is a semiconductor

designed to decrease its resistance

as its temperature increases.

The heat energy frees electrons

from the material, allowing them

to be used for conduction.

Wheatstone Bridge In a wheatstone bridge, four

resistors are arranged around a

galvanometer such that no current

flows through the galvanometer.

This happens when the ratio of the

resistors is given by:

𝑅1

𝑅2=

𝑅3

𝑅4

Wheatstone Bridge – Uses Usually, one resistor is variable and

set to monitor something, while

the galvanometer is replaced by

some circuit.

e.g. A thermistor can be used to

monitor room / oven temperature

and the galvanometer can be

replaced by a heater.

When the thermistor unbalances

the circuit due to falling

temperature, a current flows and

activates the heater.

Also used in fail-safe devices.

Wheatstone Bridge e.g. If the bridge pictured is balanced, what is the value

of R?

Metre Bridge A metre bridge replaces one

side of a wheatstone bridge

with a resistive wire.

The galvanometer

connection can be made

anywhere on the wire and a

balance point will exist where:

𝑅1

𝑅2=

𝑙1

𝑙2

Usually one resistor is unknown

and this formula can be used

to find it.