Resistance

What Is Electrical Resistance?

Electrical resistance describes how strongly a material or component opposes the flow of electric current.

When a potential difference is applied across a conductor, an electric field causes charge carriers to move. In a metal, these charge carriers are mainly free electrons.

As the electrons move through the material, they interact with the atoms and ions in the material. These interactions make it more difficult for charge to flow.

Resistance is represented by:

R

and its SI unit is the:

ohm (Ω)

A component with a high resistance allows less current to flow for a given voltage than a component with a low resistance.

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Resistance, Voltage and Current

Resistance is closely related to potential difference (voltage) and current.

For an ohmic conductor:

R = V/I

where:

• R = resistance in ohms (Ω)
• V = potential difference in volts (V)
• I = current in amperes (A)

This can also be rearranged:

V = IR

and:

I = V/R

These relationships form Ohm's Law.

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Understanding R = V/I

Resistance tells us how much voltage is required to produce a particular current.

Suppose:

V = 12 V

I = 3 A

Then:

R = V/I

R = 12/3

R = 4 Ω

The component has a resistance of 4 Ω.

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Comparing Different Resistances

Suppose two resistors are connected separately to the same 12 V supply.

Resistor A:

R = 3 Ω

I = V/R

I = 12/3

I = 4 A

Resistor B:

R = 6 Ω

I = 12/6

I = 2 A

The higher-resistance component allows less current to flow.

For the same voltage:

higher resistance → lower current

lower resistance → higher current

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What Causes Resistance?

In a metal, some electrons are free to move throughout the material.

Without an applied electric field, these electrons move randomly.

When a voltage is applied, the electric field causes the electrons to develop an overall drift through the conductor.

However, the electrons do not travel freely from one end of the wire to the other.

They continually interact with the metal's lattice of positive ions.

These interactions oppose the organized movement of charge and produce electrical resistance.

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Microscopic View of Resistance

A useful microscopic model is:

electric field → electron drift → interactions with lattice → resistance

The electrons gain energy from the electric field between interactions.

When they interact with the lattice, some of this energy is transferred to the material.

This increases the internal energy of the conductor.

As a result, the conductor may become warmer.

This is why resistors and wires can heat up when current flows through them.

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Resistance and Energy Transfer

Electrical resistance does not mean that electrons are simply "used up."

Instead, energy is transferred.

Electrical energy can be converted into thermal energy as charges move through a resistive material.

The power transferred can be calculated using:

P = VI

Using V = IR gives:

P = I²R

Another useful form is:

P = V²/R

These equations help explain electrical heating.

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Resistance in Heating Devices

Resistance is deliberately used in many devices to produce thermal energy.

Examples include:

• electric kettles
• toasters
• electric ovens
• hair dryers
• space heaters
• electric blankets

A heating element is designed to have enough resistance to convert electrical energy into thermal energy effectively.

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Factors Affecting Resistance

The resistance of a wire depends mainly on:

• its length
• its cross-sectional area
• the material
• its temperature

For a uniform conductor:

R = ρL/A

where:

• R = resistance (Ω)
• ρ = resistivity (Ω·m)
• L = length (m)
• A = cross-sectional area (m²)

This equation allows us to separate the effects of the material from the dimensions of the conductor.

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Effect of Length

Resistance increases with the length of a wire.

For the same material and cross-sectional area:

R ∝ L

A longer wire provides a longer path through which electrons must travel.

This means more opportunities for interactions with the lattice.

Therefore:

longer wire → greater resistance

If the length doubles while everything else remains constant:

resistance doubles

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Example: Changing Length

A wire has:

R = 5 Ω

If its length is doubled while its material, temperature and cross-sectional area remain unchanged:

R_new = 10 Ω

If its length is tripled:

R_new = 15 Ω

Resistance is directly proportional to length.

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Effect of Cross-Sectional Area

Resistance decreases as the cross-sectional area increases.

R ∝ 1/A

A thicker wire provides more space for charge to flow.

Therefore:

larger area → lower resistance

smaller area → higher resistance

If the cross-sectional area doubles:

resistance halves

provided all other factors remain constant.

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Why Thick Wires Have Lower Resistance

Imagine charge moving through two wires made of the same material.

A thin wire provides a relatively narrow pathway.

A thick wire provides many more possible pathways for charge carriers.

This reduces the resistance.

This is why electrical cables designed to carry large currents generally use relatively thick conductors.

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Diameter and Cross-Sectional Area

For a circular wire:

A = πr²

where r is the radius.

Because area depends on the square of the radius, changing the diameter can have a large effect on resistance.

Suppose the radius doubles.

Then:

A_new = π(2r)²

A_new = 4πr²

The area becomes four times larger.

Since:

R ∝ 1/A

the resistance becomes:

one-quarter of its original value

This is an important distinction: doubling the radius does not simply halve the resistance.

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Worked Example Using R = ρL/A

A wire has:

ρ = 1.7 × 10⁻⁸ Ω·m

L = 5.0 m

A = 1.0 × 10⁻⁶ m²

Calculate its resistance.

Use:

R = ρL/A

R = (1.7 × 10⁻⁸)(5.0)/(1.0 × 10⁻⁶)

R = 8.5 × 10⁻² Ω

Therefore:

R = 0.085 Ω

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What Is Resistivity?

Resistivity is a property of a material that describes how strongly the material resists electric current.

It is represented by:

ρ

Its SI unit is:

Ω·m

Resistance and resistivity are related, but they are not the same thing.

Resistance depends on both the material and the object's dimensions.

Resistivity is a property of the material itself at a given temperature.

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Resistance vs Resistivity

For example, two copper wires can have different resistances because they have different lengths or thicknesses.

However, at the same temperature, the copper in both wires has approximately the same resistivity.

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Comparing Materials

Different materials have very different resistivities.

Materials with low resistivity are good electrical conductors.

Examples include:

• silver
• copper
• gold
• aluminium

Materials with high resistivity resist charge flow much more strongly.

Examples include:

• rubber
• glass
• plastics
• ceramics

This difference is why copper can be used for the conducting core of a cable while plastic is used around it as insulation.

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Conductors

A conductor is a material through which electric charge can move relatively easily.

Metals are good conductors because they contain many mobile electrons.

Copper is widely used in electrical wiring because it combines:

• low resistivity
• good mechanical properties
• relatively reasonable cost
• good durability

Silver has even lower resistivity but is much more expensive.

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Insulators

An insulator has very few mobile charge carriers under normal conditions.

Examples include:

• rubber
• plastic
• glass
• dry wood
• ceramic

Insulators are important for electrical safety because they prevent unwanted current from flowing.

Electrical cables therefore commonly combine:

conducting metal core + insulating outer layer

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What Is a Resistor?

A resistor is an electrical component designed to provide a particular amount of resistance.

Resistors can be used to:

• limit current
• divide voltage
• protect components
• control circuit behavior
• produce heating
• establish operating conditions in electronic circuits

A conductor is a material, while a resistor is a component designed to have useful electrical resistance.

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Conductors Still Have Resistance

Calling something a conductor does not mean its resistance is zero.

Copper wire has a low resistivity, but a long, thin copper wire can still have significant resistance.

The equation:

R = ρL/A

shows why.

Even if ρ is small:

• increasing L increases R
• decreasing A increases R

Real conductors normally have some resistance.

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Effect of Temperature on Metals

For most metals:

temperature increases → resistance increases

As a metal becomes hotter, its lattice ions vibrate more strongly.

The moving electrons then experience more frequent or stronger interactions with the lattice.

This makes organized electron flow more difficult.

Therefore:

higher temperature → greater resistance

for ordinary metallic conductors.

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Microscopic Explanation of Temperature Effects

At lower temperature, lattice ions vibrate less.

Electrons can move through the material with fewer disruptions.

At higher temperature, lattice vibrations become larger.

The moving electrons are scattered more frequently.

Therefore, resistance increases.

This microscopic model connects:

temperature → lattice vibration → electron scattering → resistance

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Resistance and Temperature Equation

Over a limited temperature range, the resistance of many metals can be approximated by:

R = R₀[1 + α(T − T₀)]

where:

• R = resistance at temperature T
• R₀ = resistance at reference temperature T₀
• α = temperature coefficient of resistance
• T − T₀ = temperature change

This relationship is approximately linear over suitable temperature ranges.

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Temperature Sensors

The relationship between resistance and temperature can be useful.

A resistance temperature detector (RTD) measures temperature by measuring changes in electrical resistance.

Some other sensors, called thermistors, also change resistance strongly with temperature.

These devices are used in:

• thermostats
• digital thermometers
• appliances
• vehicles
• laboratory equipment
• electronic systems

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Ohmic Conductors

An ohmic conductor obeys Ohm's Law when its physical conditions, especially temperature, remain constant.

For an ohmic conductor:

V ∝ I

and:

V/I = constant

Therefore:

R = constant

A fixed resistor may behave approximately this way over an appropriate operating range.

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Non-Ohmic Components

Not every electrical component has constant resistance.

Examples include:

• filament lamps
• thermistors
• light-dependent resistors
• diodes

A filament lamp becomes extremely hot when current flows.

As its temperature rises, its resistance increases.

Therefore, voltage and current are not directly proportional across its entire operating range.

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Resistance in a Filament Lamp

A lamp filament is usually made from a material such as tungsten.

When the lamp is switched on:

• current flows
• electrical energy is transferred
• the filament heats up
• lattice vibrations increase
• resistance increases

The filament becomes hot enough to emit visible light.

This provides a real-world example of the microscopic explanation of resistance.

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Resistance and Electrical Safety

Resistance plays an important role in electrical safety.

Electrical wires carrying large currents can heat up because of:

P = I²R

Even a relatively small resistance can produce significant heating if the current is large.

For example, if:

I = 10 A

R = 0.5 Ω

then:

P = I²R

P = 10² × 0.5

P = 50 W

This energy transfer can heat the wire significantly.

This is one reason correct wire thickness is important.

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Transmission of Electrical Energy

Resistance also matters when electrical energy is transmitted over long distances.

Power lost as thermal energy in transmission wires is approximately:

P_loss = I²R

Reducing the current greatly reduces these losses.

This is one reason electrical power is transmitted at very high voltages: for a given transmitted power, increasing voltage allows the current to be reduced.

Lower current means less energy is wasted heating the transmission wires.

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Superconductors

Some materials behave very differently at extremely low temperatures.

Below a particular critical temperature, certain materials become superconductors.

Their electrical resistance falls to essentially zero.

This allows current to flow with extremely small resistive energy losses.

Superconductors have applications in areas such as:

• powerful electromagnets
• MRI systems
• particle accelerators
• magnetic levitation
• scientific research

Superconductivity cannot be explained simply by the ordinary classical collision model used for normal metallic resistance.

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Comparing a Conductor and a Resistor

Suppose we have a short copper wire and a 100 Ω resistor.

The copper wire is designed mainly to carry current with minimal resistance.

The resistor is designed to provide a controlled resistance.

Both conduct electricity, but they serve different purposes.

The distinction is therefore not:

conductor = no resistance

and:

resistor = resistance

Instead:

conductors are chosen to carry charge efficiently

while:

resistors are designed to provide a specific resistance

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Worked Example: Comparing Wires

Wire A and Wire B are made from the same material and have the same cross-sectional area.

Wire B is three times as long as Wire A.

Since:

R ∝ L

Wire B has:

three times the resistance

of Wire A.

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Worked Example: Changing Area

A wire has resistance:

R = 12 Ω

Its cross-sectional area is doubled while its length, material and temperature remain unchanged.

Since:

R ∝ 1/A

the new resistance is:

R_new = 12/2

R_new = 6 Ω

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Worked Example: Length and Area Change Together

A wire has resistance:

R

A second wire of the same material has:

• twice the length
• twice the cross-sectional area

Using:

R = ρL/A

the length change would multiply resistance by 2.

The area change would divide resistance by 2.

Therefore:

R_new = R

The two effects cancel.

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Common Mistakes

Mistake: Thinking resistance is the same as resistivity.
Resistance depends on the dimensions of an object. Resistivity is primarily a material property at a given temperature.

Mistake: Thinking good conductors have zero resistance.
Ordinary conductors still have resistance.

Mistake: Saying thicker wires have greater resistance.
For the same material and length, thicker wires have lower resistance.

Mistake: Saying longer wires have lower resistance.
Longer wires have greater resistance.

Mistake: Thinking current gets "used up" by resistance.
Charge is conserved. Resistance is associated with opposition to charge flow and energy transfer.

Mistake: Assuming all components have constant resistance.
Some components are non-ohmic, and resistance may change with temperature, light or other conditions.

Mistake: Saying electrons simply crash into stationary atoms.
A more accurate model describes electrons being scattered by the material's lattice, impurities, defects and lattice vibrations.

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Key Terms

Electrical resistance — measure of how strongly a material or component opposes electric current.

Ohm (Ω) — SI unit of electrical resistance.

Ohm's Law — relationship V = IR for an ohmic conductor under constant physical conditions.

Resistivity (ρ) — material property describing opposition to electrical conduction.

Conductor — material containing mobile charge carriers that allow charge to move relatively easily.

Insulator — material in which charge carriers cannot move easily under ordinary conditions.

Resistor — component designed to provide a particular resistance.

Charge carrier — mobile charged particle responsible for electric current.

Drift — overall directed motion of charge carriers caused by an electric field.

Lattice — ordered arrangement of particles in a solid.

Scattering — interaction that changes the motion of charge carriers and contributes to resistance.

Cross-sectional area — area of a slice perpendicular to the length of a conductor.

Ohmic conductor — conductor for which V is proportional to I under constant physical conditions.

Temperature coefficient — measure of how strongly resistance changes with temperature.

Superconductor — material that can exhibit essentially zero electrical resistance below a critical temperature.

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Key Takeaways

• Electrical resistance measures opposition to the flow of electric current.
• Resistance is measured in ohms (Ω).
• For an ohmic conductor, R = V/I.
• Higher resistance produces less current for the same voltage.
• The resistance of a uniform wire can be calculated using R = ρL/A.
• Longer wires have greater resistance.
• Thicker wires have lower resistance.
• Resistivity describes the electrical properties of the material itself.
• Low-resistivity materials such as copper are good conductors.
• Resistance and resistivity are related but are not the same quantity.
• Metals conduct because they contain mobile electrons.
• Electrical resistance can be understood microscopically through the scattering of moving electrons within the material.
• For most metals, increasing temperature increases resistance because lattice vibrations increase.
• Resistive components transfer electrical energy into thermal energy.
• Resistors deliberately provide resistance for controlling current and voltage in circuits.
• Conductors are normally selected to carry current with as little unwanted resistance as practical.
• Ordinary conductors do not have zero resistance.
• Electrical heating can be calculated using P = I²R.
• Resistance is important in circuit design, electrical heating, transmission systems and electrical safety.

Check Your Understanding

1. Define electrical resistance.
2. State the SI unit of resistance.
3. A resistor has 12 V across it and carries 3 A. Calculate its resistance.
4. A 20 Ω resistor is connected across 10 V. Calculate the current.
5. State four factors that can affect the resistance of a wire.
6. Explain why increasing the length of a wire increases its resistance.
7. Explain why increasing the cross-sectional area decreases resistance.
8. A wire's length doubles while everything else remains constant. What happens to its resistance?
9. A wire's cross-sectional area triples. What happens to its resistance?
10. Explain the difference between resistance and resistivity.
11. State the equation relating resistance, resistivity, length and cross-sectional area.
12. Explain why copper is commonly used for electrical wiring.
13. Explain microscopically why metals have electrical resistance.
14. Explain why the resistance of most metals increases when temperature increases.
15. Compare the purpose of a copper connecting wire with the purpose of a resistor.
16. Explain why a filament lamp does not maintain constant resistance while operating.
17. A wire is doubled in both length and cross-sectional area. Predict the change in resistance.
18. Explain why thick cables are used when large electrical currents must be carried.
19. Explain why electrical resistance can cause a wire to become warm.
20. Explain why reducing current reduces energy losses in electrical transmission wires.