1. Introduction to Electromagnetic Induction

What Is Electromagnetic Induction?

Electromagnetic induction is the production of an electromotive force (EMF), or induced voltage, when the magnetic environment of a conductor changes.

A simple example is a bar magnet moving toward or away from a coil of wire.

When the magnet is moving relative to the coil, the magnetic field passing through the coil changes.

This changing magnetic field produces an induced EMF.

If the coil forms part of a complete electrical circuit, the induced EMF can drive an induced current through the circuit.

The central idea is:

changing magnetic field through a conductor → induced EMF

EMF Does Not Mean a Mechanical Force

The term electromotive force, usually written as EMF, can be misleading.

An EMF is not a pushing force measured in newtons.

It is a voltage-like quantity measured in:

volts (V)

An induced EMF provides energy per unit charge that can cause charges to move through a circuit.

So in introductory work, it is often helpful to think of:

induced EMF ≈ induced voltage

A Classic Induction Experiment

Imagine a coil of wire connected to a sensitive voltmeter or galvanometer.

A bar magnet is moved toward the coil.

As the magnet approaches:

• the magnetic field through the coil changes
• an EMF is induced
• the meter shows a deflection

If the circuit is closed, a current flows.

If the magnet is pulled away:

• the magnetic field changes in the opposite way
• an EMF is again induced
• the induced current reverses direction

If the magnet is held still relative to the coil:

no changing magnetic field

so:

no induced EMF.

Relative Motion Is What Matters

It is not necessary for the magnet itself to move.

You could keep the magnet stationary and move the coil instead.

What matters is:

relative motion between the conductor and the magnetic field

For example:

Magnet moves, coil stationary

Induction occurs.

Coil moves, magnet stationary

Induction occurs.

Both remain stationary relative to one another

No induction occurs from that static arrangement.

OpenStax notes that the same induction effect occurs whether the magnet moves or the coil moves; it is the relative motion that matters.

Static Magnetic Field vs Changing Magnetic Field

A magnetic field can exist without producing electromagnetic induction.

This is one of the most important distinctions in this topic.

Static magnetic field

A magnetic field that remains constant through the conductor.

Example:

A magnet is placed beside a stationary coil and neither moves.

The coil may be inside a magnetic field, but if the field through the coil is not changing:

no induced EMF is produced.

Changing magnetic field

If the amount or direction of magnetic field passing through the conductor changes:

an EMF can be induced.

This can happen because:

• a magnet moves
• a conductor moves
• the magnetic-field strength changes
• the conductor rotates
• the conductor's orientation changes
• an alternating current produces a changing magnetic field

The Key Rule

A useful rule is:

A magnetic field alone is not enough.

There must be a:

change in magnetic flux through the conductor

for electromagnetic induction to occur.

At an introductory level, we can think of magnetic flux as:

how much magnetic field passes through a loop or coil.

Magnetic Flux

Magnetic flux is related to:

• magnetic-field strength
• area of the coil
• angle of the coil relative to the magnetic field

The symbol for magnetic flux is:

Φ

For a uniform magnetic field:

Φ = BA cos θ

where:

• Φ = magnetic flux
• B = magnetic-field strength
• A = area
• θ = angle between the magnetic field and the perpendicular to the surface

You do not need this equation to understand the basic concept, but it explains why induction can occur even if neither the magnet nor coil moves closer or farther apart.

For example, simply rotating a coil in a magnetic field changes the angle and therefore changes the magnetic flux.

Faraday's Law

Michael Faraday discovered that induced EMF depends on how quickly magnetic flux changes.

In simplified form:

induced EMF ∝ rate of change of magnetic flux

For a coil of N turns:

ε = −N ΔΦ/Δt

where:

• ε = induced EMF
• N = number of turns
• ΔΦ = change in magnetic flux
• Δt = time taken for the change

The negative sign is associated with Lenz's law, which describes the direction of the induced EMF.

For this introductory topic, the main idea is:

faster change in magnetic flux → larger induced EMF.

Moving a Magnet Faster

Suppose you push a magnet slowly into a coil.

The magnetic field through the coil changes slowly.

The induced EMF is relatively small.

Now push the same magnet quickly into the same coil.

The magnetic field changes more rapidly.

Therefore:

larger induced EMF

The same principle applies when pulling the magnet out.

OpenStax describes that faster relative motion produces a greater induced EMF.

Moving Magnet: Three Situations

Consider these three cases.

Magnet moving into coil

Magnetic flux changes.

EMF induced

Magnet held still inside coil

Magnetic flux remains constant.

No EMF induced

Magnet moving out of coil

Magnetic flux changes again.

EMF induced in the opposite direction

The direction of the induced current changes when the direction of motion changes.

Changing the Magnet's Pole

Suppose the north pole of a magnet is pushed toward a coil.

The induced current flows in one direction.

Now repeat the experiment using the south pole.

The induced current reverses direction.

Why?

Because the direction of the magnetic-field change through the coil has reversed.

This observation helped scientists establish that both the size and direction of the magnetic-field change matter.

How Can a Conductor Moving Through a Magnetic Field Produce Voltage?

Electromagnetic induction can also occur when a straight conductor moves through a magnetic field.

Imagine a metal rod moving across magnetic field lines.

Charges inside the conductor experience magnetic effects.

Positive and negative charges become separated.

This produces a potential difference across the ends of the conductor.

That potential difference is an:

induced EMF

So another useful description is:

when a conductor cuts across magnetic field lines, an EMF can be induced.

Moving Conductor Example

Suppose a straight wire moves through a uniform magnetic field.

If the wire cuts across the magnetic field:

charges in the wire experience magnetic forces

which can lead to:

charge separation

and therefore:

induced voltage

If the conductor is connected into a complete circuit, that voltage can drive a current.

This idea forms the basis of an electrical generator.

Factors That Affect the Induced EMF

The induced EMF can generally be increased by:

• moving the magnet faster
• moving the conductor faster
• using a stronger magnetic field
• increasing the number of turns in the coil
• increasing the area through which the magnetic field changes
• changing the magnetic flux more rapidly

The unifying idea is:

increase the rate of change of magnetic flux

Example: Slow vs Fast Magnet Motion

Experiment A:

A magnet moves into a coil in:

2.0 s

Experiment B:

The same magnet moves through the same distance in:

0.5 s

The change in magnetic flux is similar, but Experiment B produces the change in a shorter time.

Therefore:

Experiment B produces a larger induced EMF.

More Coil Turns

Imagine two coils.

Coil A:

50 turns

Coil B:

500 turns

The same changing magnetic flux passes through each turn.

Because the effects from all the turns add together, Coil B can produce a larger induced EMF.

This is reflected in Faraday's law:

ε = −N ΔΦ/Δt

A larger N can produce a larger EMF.

Stronger Magnetic Field

Suppose the same coil is used with two magnets.

Magnet A produces a relatively weak field.

Magnet B produces a much stronger field.

If both magnets move through the coil at the same speed, the stronger magnet generally creates a larger change in magnetic flux.

Therefore:

larger induced EMF

Static Does Not Mean "No Magnetism"

A common misconception is:

"If no EMF is induced, there must be no magnetic field."

Incorrect.

A coil can sit inside a strong magnetic field and still have:

zero induced EMF

if the magnetic flux through it remains constant.

Electromagnetic induction depends on:

change

not merely the existence of a magnetic field.

Example: Magnet Sitting Inside a Coil

Imagine a bar magnet placed halfway inside a coil.

The magnetic field through the coil may be strong.

But if the magnet stays completely still:

magnetic flux is constant

Therefore:

no induced EMF

Now pull the magnet out.

The flux changes.

Therefore:

an EMF is induced.

This difference between field strength and change in field is fundamental.

How Can the Magnetic Field Change Without Moving a Magnet?

A magnet does not always have to physically move.

A changing current can produce a changing magnetic field.

For example, an alternating current in one coil creates a continuously changing magnetic field around that coil.

If another coil is nearby, that changing field can induce an EMF in the second coil.

This is the operating principle of a:

transformer

Transformers

A transformer contains two coils:

• primary coil
• secondary coil

usually wound around a magnetic core.

Alternating current in the primary coil produces a changing magnetic field in the core.

That changing magnetic flux passes through the secondary coil.

An EMF is therefore induced in the secondary coil.

Transformers can increase or decrease AC voltage depending on the number of turns in the two coils.

Why Direct Current Does Not Continuously Work in a Transformer

If a steady direct current flows through the primary coil:

the magnetic field becomes steady after switching.

A steady field means:

no continuous change in magnetic flux

Therefore:

no continuous induced EMF in the secondary coil

There may be a brief induced effect when the current is switched on or off because the magnetic field is changing during those moments.

This is why ordinary transformers depend on:

alternating current

rather than steady DC.

Electromagnetic Induction in Generators

One of the most important applications of electromagnetic induction is the:

electrical generator

A generator converts:

mechanical energy → electrical energy

It usually does this by creating relative motion between:

• a conductor or coil
• a magnetic field

When a coil rotates in a magnetic field, the magnetic flux through the coil changes continuously.

This induces an EMF.

That is how mechanical motion can be converted into electrical output.

Electromagnetic induction is the basic principle behind electric generators.

How a Generator Works

A simple generator can contain:

• a rotating coil
• magnetic poles
• an axle
• electrical contacts

As the coil rotates:

the angle between the coil and magnetic field changes.

Therefore:

magnetic flux changes

so:

an EMF is induced

Continued rotation produces a continuously changing induced voltage.

Where Does the Energy Come From?

Induction does not create electrical energy from nothing.

In a generator, someone or something must provide mechanical energy.

Examples include:

• falling water
• steam turbines
• wind
• a hand crank
• an engine

The generator converts this mechanical energy into electrical energy.

This is an important conservation-of-energy idea.

Power Stations

Most large-scale electrical generation ultimately relies on electromagnetic induction.

A power station may use:

• steam
• water
• wind
• gas turbines
• nuclear-generated steam

to rotate a turbine.

The turbine turns a generator.

The generator uses electromagnetic induction to produce electrical energy.

So despite having very different energy sources, many power stations share the same final step:

rotate generator → change magnetic flux → induce EMF

Hydroelectric Power

In a hydroelectric station:

1. water falls or flows
2. the water turns a turbine
3. the turbine rotates a generator
4. magnetic flux through coils changes
5. an EMF is induced
6. electrical energy is produced

Electromagnetic induction provides the connection between:

mechanical rotation

and:

electricity generation

Wind Turbines

Wind turbines use the same fundamental principle.

Wind provides kinetic energy.

The blades rotate.

That rotation drives a generator.

Inside the generator:

relative motion between magnets and conductors produces changing magnetic flux

which induces an EMF.

So a wind turbine is not "making electricity directly from wind."

Instead:

wind → mechanical rotation → electromagnetic induction → electrical energy

Bicycle Dynamo

A bicycle dynamo is a small generator.

As the bicycle wheel turns, the dynamo rotates a magnet or coil.

The resulting change in magnetic flux induces an EMF.

This can power:

bicycle lights

It is a simple everyday example of electromagnetic induction.

Induction Cooktops

Electromagnetic induction is also used for heating.

An induction cooktop contains a coil carrying alternating current.

This creates a rapidly changing magnetic field.

The changing magnetic field induces electrical currents in suitable metal cookware.

These currents produce heating because of the electrical resistance of the cookware.

So:

changing current

→ changing magnetic field

→ induced currents

→ heating

Wireless Charging

Many wireless chargers also use electromagnetic induction.

A coil in the charging pad carries alternating current.

This creates a changing magnetic field.

A second coil inside the device experiences the changing magnetic flux.

An EMF is induced in the device's coil.

That electrical energy can then be used to charge the battery.

The device does not need direct metal-to-metal electrical contact between charger and phone.

Electric Guitar Pickups

Electromagnetic induction is also used in many electric guitars.

The guitar strings affect the magnetic field near a pickup coil.

When a magnetized string vibrates, the magnetic flux through the pickup coil changes.

This induces a small voltage.

The changing electrical signal corresponds to the vibration of the string and can be amplified into sound.

OpenStax includes guitar pickup coils among devices using electromagnetic induction.

Dynamic Microphones

Some microphones use electromagnetic induction.

A sound wave vibrates a diaphragm.

The diaphragm moves a coil relative to a magnetic field.

This motion changes magnetic flux through the coil.

A small EMF is induced.

The changing voltage becomes an electrical representation of the sound wave.

Why Electromagnetic Induction Is So Important

Modern electrical technology depends heavily on induction.

It allows us to:

• generate electricity
• change AC voltage
• transfer electrical energy without direct contact
• convert sound into electrical signals
• detect metal
• produce heating
• operate sensors
• convert mechanical motion into electrical signals

Without electromagnetic induction, modern power distribution would look completely different.

Induction and the Electrical Grid

Generators at power stations produce electrical energy.

Transformers then change voltage levels.

High voltages are useful for long-distance transmission because they help reduce energy losses in power lines.

Other transformers later reduce the voltage for homes, businesses, and electronic devices.

Both generators and transformers depend fundamentally on:

changing magnetic flux

This makes electromagnetic induction one of the central ideas behind the modern electrical grid.

Static vs Changing Field: Comparison

The crucial column is:

Magnetic flux changing?

Predicting Whether Induction Occurs

Consider each situation.

Situation 1

A magnet rests motionless next to a coil.

Prediction:

No induced EMF

Reason:

The magnetic field through the coil is not changing.

Situation 2

The magnet is pushed into the coil.

Prediction:

EMF induced

Reason:

The magnetic flux through the coil changes.

Situation 3

The coil is pulled away from a stationary magnet.

Prediction:

EMF induced

Reason:

Relative motion changes the magnetic flux.

Situation 4

A coil rotates continuously between magnetic poles.

Prediction:

Continuously changing induced EMF

Reason:

The angle between the coil and field changes continuously.

Situation 5

A nearby electromagnet is supplied with alternating current.

Prediction:

EMF can be induced in the coil

Reason:

The electromagnet produces a changing magnetic field.

Predicting the Size of the Induced EMF

Suppose the same magnet and coil are used.

Slow movement

Small rate of change of magnetic flux.

Smaller EMF

Fast movement

Large rate of change of magnetic flux.

Larger EMF

This gives another useful relationship:

faster flux change → larger induced EMF

Example Calculation Idea

Suppose the magnetic flux through a single loop changes from:

0.020 Wb

to:

0.005 Wb

in:

0.50 s

Change in flux magnitude:

ΔΦ = 0.015 Wb

Approximate induced EMF magnitude:

ε = ΔΦ/Δt

ε = 0.015 / 0.50

ε = 0.030 V

For a coil of 100 identical turns:

ε = NΔΦ/Δt

ε = 100(0.015/0.50)

ε = 3.0 V

This demonstrates why coils with many turns can generate larger voltages.

What Is an Induced Current?

An induced EMF can exist even if the circuit is open.

If the circuit is closed:

the induced EMF can cause charges to flow.

This flow of charge is called:

induced current

So it is useful to distinguish:

EMF = induced voltage

from:

current = movement of charge caused when a complete conducting path exists

Example: Open vs Closed Circuit

Suppose a magnet moves through a coil.

Coil connected to voltmeter only

An induced voltage can be detected.

Coil connected as part of a closed circuit

The induced EMF can drive a current.

Therefore, induction first describes the creation of:

EMF

A current requires:

a complete circuit

Direction of Induced Current

The direction of the induced current depends on how the magnetic flux is changing.

For example:

magnet moving into coil → current in one direction

magnet moving out → current in opposite direction

This is explained in more detail by:

Lenz's law

which states that the induced effect acts in a direction that opposes the change that produced it.

A Preview of Lenz's Law

Suppose a north pole approaches a coil.

The increasing magnetic flux induces a current.

The magnetic field created by that current acts to oppose the increase in flux.

If the magnet is pulled away:

the induced current reverses so that its magnetic effect opposes the decrease in flux.

This does not mean the system somehow "knows" what is happening.

It follows from electromagnetic laws and conservation of energy.

Why Must the Induced Effect Oppose the Change?

Imagine instead that the induced current strengthened the original change.

A magnet moving toward a coil would cause a current that pulled it in even faster.

The increasing motion would create more current, causing even more motion.

Energy could appear without an external energy source.

That would violate conservation of energy.

Lenz's law prevents this.

Common Misconception: A Magnet Automatically Produces Electricity

Incorrect.

A stationary magnet beside a stationary wire does not automatically create a continuous induced voltage.

What is required is:

changing magnetic flux

A magnet is useful because moving it can create that change.

Common Misconception: The Magnet Must Move

Incorrect.

The conductor can move instead.

Or neither has to translate through space if the magnetic field itself changes.

For example:

• rotating coil
• alternating electromagnet
• changing current in a nearby coil

can all produce induction.

Common Misconception: A Stronger Field Always Means Induction

A strong but perfectly constant magnetic field may produce:

no induced EMF

A weaker magnetic field that changes rapidly can produce an induced EMF.

Therefore, the key idea is not simply:

how strong is the field?

but:

how quickly is the magnetic flux changing?

Common Misconception: EMF and Current Are the Same

They are related but different.

EMF is measured in volts.

Current is measured in amperes.

An induced EMF can exist without a current if the circuit is incomplete.

Common Misconception: Generators Create Energy

Generators do not create energy.

They convert:

mechanical energy

into:

electrical energy

Energy must be supplied to turn the generator.

Common Misconception: A Transformer Creates Extra Energy

A transformer may increase voltage, but this does not mean it creates energy.

In an ideal transformer, increasing voltage is accompanied by a decrease in current.

Real transformers also lose some energy, usually as heat.

The role of a transformer is:

energy transfer and voltage transformation

through electromagnetic induction.

A Useful Cause-and-Effect Chain

For induction problems, use:

something changes the magnetic environment

↓

magnetic flux through conductor changes

↓

EMF is induced

↓

current flows if circuit is complete

This chain is extremely useful.

Technology Example Summary

Electromagnetic induction links magnetism and electricity in an extraordinarily useful way.

Real-World Connection: From Motion to Electricity

Consider a hand-cranked generator.

Your muscles provide chemical energy.

That becomes:

mechanical energy

as you turn the handle.

The spinning generator changes magnetic flux.

Electromagnetic induction produces an EMF.

Electrical energy can then power a lamp.

Energy pathway:

chemical → mechanical → electrical → light + thermal

Induction is the key step between mechanical and electrical energy.

Real-World Connection: Charging Without Wires

In inductive charging:

one coil creates a changing magnetic field.

A second coil experiences the changing field.

This induces a voltage.

Because the coils do not need direct conductive contact, energy can cross a small gap.

Applications include:

• phones
• electric toothbrushes
• some medical devices
• some electric vehicle charging systems

Did You Know?

Michael Faraday demonstrated electromagnetic induction in 1831.

His work established a direct connection between changing magnetic fields and electrical effects.

This discovery became one of the foundations of modern electrical technology.

Generators and transformers now allow electrical energy to be produced and distributed on enormous scales.

A Simple Classroom Demonstration

A basic induction demonstration requires:

• coil of insulated wire
• bar magnet
• sensitive galvanometer or voltmeter

Try:

1. Hold the magnet still near the coil.
2. Move the north pole rapidly into the coil.
3. Hold it still.
4. Pull it rapidly out.
5. Repeat slowly.
6. Reverse the magnet.

Expected observations:

• stationary magnet → no meter deflection
• moving magnet → meter deflection
• faster movement → larger deflection
• reversing movement → opposite deflection
• reversing pole → opposite deflection

These observations provide direct evidence for electromagnetic induction.

Interpreting Experimental Results

Suppose a student records:

We can conclude:

• induction requires changing magnetic flux
• faster change produces larger EMF
• reversing the change reverses the EMF direction

This is a strong experimental summary of the topic.

Predicting Changes

Move magnet faster

Induced EMF increases

Use stronger magnet

Induced EMF usually increases

Add more turns to coil

Induced EMF increases

Stop the magnet

Induced EMF falls to zero

Reverse magnet direction

Induced EMF reverses direction

Rotate coil faster

Induced EMF generally increases

Static and Changing Fields: Final Comparison

A static magnetic field can exert magnetic forces.

However:

static magnetic field through a stationary coil → no continuous induced EMF

A changing magnetic field can produce:

induced EMF

This is the defining idea of electromagnetic induction.

Key Terms

Electromagnetic induction – Production of an EMF due to a changing magnetic flux through a conductor.

EMF – Electromotive force; a voltage associated with energy supplied per unit charge.

Induced EMF – Voltage produced by electromagnetic induction.

Induced current – Current caused by an induced EMF in a complete circuit.

Magnetic field – Region in which magnetic effects can act.

Static magnetic field – Magnetic field that remains constant with time in the relevant situation.

Changing magnetic field – Magnetic field whose magnitude or direction changes.

Magnetic flux – Measure of magnetic field passing through a surface.

Coil – Wire wound into loops, often used to increase induction effects.

Generator – Device that converts mechanical energy into electrical energy using electromagnetic induction.

Transformer – Device that transfers electrical energy between coils using changing magnetic flux.

Faraday's law – Relationship between induced EMF and the rate of change of magnetic flux.

Lenz's law – Rule stating that the induced effect opposes the change in magnetic flux that caused it.

Key Takeaways

• Electromagnetic induction is the production of an EMF by changing magnetic flux.
• A static magnetic field alone does not necessarily produce an induced EMF.
• Moving a magnet relative to a coil can produce an EMF.
• Moving the coil relative to the magnet produces the same basic effect.
• Relative motion is what matters.
• A conductor moving through a magnetic field can also develop an induced voltage.
• A changing current can produce a changing magnetic field and induce an EMF in another coil.
• Faster changes in magnetic flux generally produce larger induced EMFs.
• Stronger fields and more coil turns can increase induced EMF.
• Reversing the magnetic-field change reverses the induced EMF direction.
• An induced current flows only if there is a complete conducting circuit.
• Generators use electromagnetic induction to convert mechanical energy into electrical energy.
• Transformers use changing magnetic fields to transfer electrical energy between coils.
• Wireless chargers, microphones, guitar pickups, dynamos, and induction cookers also use electromagnetic induction.
• Modern electricity generation and distribution depend heavily on this principle.
• The essential relationship is:

changing magnetic flux → induced EMF → induced current if the circuit is complete.

2. Magnetic Flux

What Is Magnetic Flux?

Magnetic flux is a measure of the amount of magnetic field passing through a surface.

The symbol for magnetic flux is:

Φ (Greek letter phi)

A useful way to picture magnetic flux is to imagine magnetic field lines passing through a loop of wire.

More field passing through the surface means:

greater magnetic flux

Less field passing through the surface means:

smaller magnetic flux

Magnetic flux depends on three main factors:

• magnetic field strength
• area of the surface
• orientation of the surface relative to the magnetic field

For a uniform magnetic field through a flat surface:

Φ = BA cos θ

where θ is measured between the magnetic field and the normal to the surface.

---

What Does Magnetic Flux Represent?

Magnetic flux helps us describe how strongly a magnetic field passes through a particular area.

Imagine holding a hoop in a magnetic field.

If the hoop faces directly into the field, many field lines pass through it.

If you turn the hoop sideways, fewer field lines pass through it.

Turn it completely edge-on and no field lines pass through the surface enclosed by the hoop.

This gives us an important idea:

magnetic flux is about the field passing through a surface, not merely the magnetic field existing around it.

---

Magnetic Field Lines and Flux

Magnetic fields are often represented using magnetic field lines.

The arrows show the direction of the magnetic field.

The density of the lines helps us visualize field strength:

closer field lines → stronger magnetic field

more widely spaced field lines → weaker magnetic field

 

 

 

4

Field lines are a visual model; they are not physical strings in space. Their spacing is used to represent the strength of the magnetic field.

---

The Magnetic Flux Equation

For a uniform magnetic field:

Φ = BA cos θ

where:

Φ = magnetic flux, measured in webers (Wb)

B = magnetic flux density, measured in teslas (T)

A = area, measured in square metres (m²)

θ = angle between the magnetic field and the normal to the surface

This equation tells us exactly how field strength, area and orientation combine to determine magnetic flux.

---

The Unit of Magnetic Flux

The SI unit of magnetic flux is the:

weber (Wb)

Since:

Φ = BA

we can also write:

1 Wb = 1 T·m²

So if a magnetic field of 1 T passes perpendicularly through an area of 1 m²:

Φ = 1 Wb

---

What Is the Normal?

The angle in the flux equation is one of the most common sources of mistakes.

The angle θ is normally measured between:

the magnetic field B

and:

an imaginary line perpendicular to the surface

This perpendicular line is called the:

normal

Notice that θ is not normally measured between the magnetic field and the surface itself.

This distinction matters because the equation uses:

cos θ

---

Maximum Magnetic Flux

Magnetic flux is maximum when the magnetic field passes straight through the surface.

In this orientation:

θ = 0°

Therefore:

cos 0° = 1

So:

Φ = BA

The surface itself is perpendicular to the magnetic field.

Its normal is parallel to the field.

---

Zero Magnetic Flux

Now rotate the loop until its surface is parallel to the magnetic field.

The field lines travel along the plane of the loop rather than through it.

In this orientation:

θ = 90°

Since:

cos 90° = 0

then:

Φ = 0

The magnetic field can still be present and strong.

But none of it passes through the surface in the flux sense.

---

Orientation and Flux

As the loop rotates between these two positions, the magnetic flux changes continuously.

Angle between B and normal	cos θ	Magnetic Flux
0°	1.00	Maximum
30°	0.866	Large
45°	0.707	Moderate-large
60°	0.500	Half maximum
90°	0	Zero

This relationship is extremely important in electrical generators.

As a generator coil rotates:

θ changes

therefore:

Φ changes

and that changing flux produces an induced EMF.

---

Effect of Magnetic Field Strength

Suppose area and orientation remain constant.

From:

Φ = BA cos θ

we can see that:

Φ ∝ B

Therefore:

stronger magnetic field → greater magnetic flux

 

If B doubles while everything else remains unchanged:

Φ doubles

If B triples:

Φ triples

This is a direct proportional relationship.

---

Example: Increasing Field Strength

A loop has:

A = 0.20 m²

and is perpendicular to the magnetic field.

Initially:

B = 0.40 T

Since θ = 0°:

Φ = BA

Φ = (0.40)(0.20)

Φ = 0.080 Wb

Now increase the field to:

B = 0.80 T

Then:

Φ = (0.80)(0.20)

Φ = 0.160 Wb

Doubling B doubled the magnetic flux.

---

Effect of Area

Magnetic flux also depends on the area exposed to the magnetic field.

From:

Φ = BA cos θ

if B and θ remain constant:

Φ ∝ A

Therefore:

larger area → greater magnetic flux

Imagine placing a small hoop and a large hoop in the same uniform magnetic field.

 

 

 

5

The larger loop encloses a larger surface, so more of the magnetic field passes through it.

---

Example: Increasing Area

A loop is perpendicular to a:

0.50 T

magnetic field.

Loop A has:

A = 0.10 m²

Therefore:

Φ = (0.50)(0.10)

Φ = 0.050 Wb

Loop B has:

A = 0.30 m²

Therefore:

Φ = (0.50)(0.30)

Φ = 0.150 Wb

Tripling the area tripled the magnetic flux.

---

Effect of Orientation

Orientation is slightly more complicated because it involves:

cos θ

Suppose B and A remain constant.

When:

θ = 0°

flux is maximum.

When:

θ = 60°

cos 60° = 0.5

so:

Φ = 0.5BA

The flux is half its maximum value.

When:

θ = 90°

flux becomes zero.

This changing orientation is one of the easiest ways to produce continuously changing magnetic flux.

---

The Three Ways to Change Magnetic Flux

From:

Φ = BA cos θ

we can immediately identify three ways to change magnetic flux.

Change B

Change the magnetic-field strength.

Change A

Change the area through which the field passes.

Change θ

Change the orientation of the surface relative to the field.

These are the three fundamental ways of changing flux.

---

Worked Example 1: Perpendicular Field

A loop has an area of:

0.040 m²

It is placed perpendicular to a uniform magnetic field of:

0.60 T

Calculate the magnetic flux.

Because the surface is perpendicular to the field:

θ = 0°

Use:

Φ = BA cos θ

Φ = (0.60)(0.040)(cos 0°)

Φ = 0.024 Wb

Answer:

Φ = 0.024 Wb

---

Worked Example 2: Field at an Angle

A loop has:

A = 0.20 m²

and is in a magnetic field:

B = 0.50 T

The magnetic field makes an angle of:

60°

with the normal.

Calculate the magnetic flux.

Use:

Φ = BA cos θ

Φ = (0.50)(0.20)(cos 60°)

Since:

cos 60° = 0.5

then:

Φ = (0.50)(0.20)(0.5)

Φ = 0.050 Wb

---

Worked Example 3: Zero Flux

A loop with area:

0.40 m²

is positioned so its surface is parallel to a:

2.0 T

magnetic field.

The normal is therefore at:

90°

to the field.

Use:

Φ = BA cos θ

Φ = (2.0)(0.40)(cos 90°)

Φ = 0 Wb

Notice:

B is not zero.

The magnetic field is actually quite strong.

The flux is zero because of the loop's orientation.

---

Worked Example 4: Finding Magnetic Field Strength

A loop has:

Φ = 0.060 Wb

and:

A = 0.20 m²

The field passes perpendicularly through the loop.

Since θ = 0°:

Φ = BA

Rearrange:

B = Φ/A

Substitute:

B = 0.060/0.20

B = 0.30 T

---

Worked Example 5: Finding Area

A magnetic flux of:

0.12 Wb

passes perpendicularly through a surface in a:

0.80 T

field.

Find the area.

Use:

Φ = BA

Rearrange:

A = Φ/B

Substitute:

A = 0.12/0.80

A = 0.15 m²

---

Be Careful with Area Units

Area must be measured in:

m²

Suppose a coil has area:

50 cm²

You cannot substitute 50 directly into the flux equation.

Because:

1 cm = 0.01 m

then:

1 cm² = 0.0001 m²

Therefore:

50 cm² = 50 × 10⁻⁴ m²

= 0.0050 m²

This conversion is a frequent source of mistakes.

---

Worked Example 6: Area Conversion

A loop has an area of:

200 cm²

and sits perpendicular to a:

0.30 T

field.

Convert:

200 cm² = 200 × 10⁻⁴ m²

= 0.020 m²

Now:

Φ = BA

Φ = (0.30)(0.020)

Φ = 0.0060 Wb

---

Interpreting Magnetic Flux Using Field Diagrams

Field diagrams provide a visual way to estimate magnetic flux.

Imagine three identical loops.

Loop A

Many field lines pass directly through it.

Large flux

Loop B

The loop is tilted.

Fewer lines effectively pass through its surface.

Smaller flux

Loop C

The loop is edge-on to the field.

No lines pass through its surface.

Zero flux

This visual interpretation matches:

Φ = BA cos θ

---

Field Lines Are a Model

It is tempting to think magnetic flux literally counts individual magnetic field lines.

It does not.

Magnetic field lines are a drawing convention used to represent the field.

Magnetic flux is a measurable physical quantity.

However, the field-line picture is extremely useful:

more field lines drawn through an area → larger represented flux

provided the diagrams use the same field-line scale.

---

Stronger Field in a Diagram

Compare two equal loops.

If one region has field lines that are more closely packed, it represents a stronger magnetic field.

Because:

Φ = BA cos θ

a stronger B gives a larger flux when A and θ are unchanged.

 

 

 

4

This allows us to compare flux qualitatively even without numerical values.

---

Magnetic Flux and Electromagnetic Induction

Magnetic flux becomes especially important because of Faraday's law of electromagnetic induction.

A changing magnetic flux through a conducting loop can produce an:

induced EMF

The key relationship is:

change in magnetic flux → induced EMF

A constant flux does not produce a continuous induced EMF.

A changing flux does.

---

Faraday's Law

For a single loop, the magnitude of the average induced EMF can be related to:

EMF = |ΔΦ/Δt|

For a coil containing N turns:

EMF = N|ΔΦ/Δt|

A more complete form is:

ε = −N ΔΦ/Δt

The negative sign represents Lenz's law and tells us about the direction of the induced EMF.

For now, the important magnitude relationship is:

larger rate of change of flux → larger induced EMF

---

Flux Linkage

If a coil contains many turns, the same magnetic flux may pass through each turn.

We therefore define:

magnetic flux linkage = NΦ

where:

N = number of turns

Φ = magnetic flux through one turn

 

A coil with more turns can therefore produce a larger induced EMF for the same rate of flux change.

---

How Moving a Magnet Changes Flux

Imagine a magnet approaching a coil.

 

 

 

6

As the magnet approaches:

magnetic field through coil increases

therefore:

magnetic flux increases

therefore:

an EMF is induced

When the magnet stops:

flux stops changing

therefore:

induced EMF falls to zero

When the magnet moves away:

flux decreases

therefore:

an EMF is induced in the opposite sense.

---

Static Flux vs Changing Flux

This distinction is essential.

Suppose a loop sits in a constant magnetic field.

It may have:

Φ = 0.50 Wb

That is a substantial magnetic flux.

But if it remains:

0.50 Wb

then:

ΔΦ = 0

Therefore:

no continuous induced EMF

Induction depends on:

change in flux

not simply:

having flux

---

Example: Changing Field Strength

A stationary coil is placed around an electromagnet.

Initially:

B = 0.20 T

Later:

B = 0.80 T

Even though neither the coil nor electromagnet moves, the magnetic flux changes because:

B changes

Therefore an EMF can be induced.

This is one reason alternating currents are so useful in electromagnetic devices.

---

Example: Changing Area

Suppose a conducting loop sits in a constant magnetic field.

If the loop is stretched so that its area increases:

A increases

therefore:

Φ increases

An EMF can therefore be induced even though:

• B stays constant
• orientation stays constant

Changing area alone can change flux.

---

Example: Changing Orientation

Suppose:

B = constant

and:

A = constant

but the loop rotates.

Then:

θ changes

so:

cos θ changes

therefore:

Φ changes

This produces an induced EMF.

 

This is the central principle behind an AC generator.

---

Flux in an Electrical Generator

In a simple generator, a coil rotates between magnetic poles.

 

 

 

6

As the coil rotates:

θ continuously changes

therefore:

Φ continuously changes

therefore:

an EMF is continuously induced

This is how mechanical rotation can be converted into electrical energy.

---

Flux During One Rotation

Imagine a coil rotating through 360°.

At:

0° → maximum positive flux

At:

90° → zero flux

At:

180° → maximum flux in the opposite direction

At:

270° → zero flux

At:

360° → back to maximum positive flux

So the magnetic flux varies continuously as the coil rotates.

 

 

 

4

This repeating change in flux is what allows a generator to produce alternating EMF.

---

Flux and EMF Are Not the Same Thing

This is an important distinction.

Magnetic flux tells us how much magnetic field passes through a surface.

Induced EMF depends on how quickly that flux changes.

So:

large flux does not automatically mean large EMF

For example, a stationary loop could have maximum flux but:

EMF = 0

because the flux is not changing.

---

Maximum Flux Can Mean Zero EMF at That Instant

This can initially seem surprising.

Suppose a rotating coil reaches an orientation where:

Φ is maximum

At that instant, the flux is momentarily changing at its slowest rate.

So the induced EMF can be:

zero at the instant of maximum flux

When the flux passes through zero, it can be changing most rapidly.

Then the induced EMF can be:

maximum

This becomes important when studying AC generators in more detail.

---

Worked Example 7: Change in Flux

A loop initially has:

Φ₁ = 0.080 Wb

Later:

Φ₂ = 0.020 Wb

Calculate the change in flux.

ΔΦ = Φ₂ − Φ₁

ΔΦ = 0.020 − 0.080

ΔΦ = −0.060 Wb

The negative sign indicates that the flux decreased.

The magnitude of the change is:

0.060 Wb

---

Worked Example 8: Induced EMF

The magnetic flux through a single loop changes from:

0.080 Wb

to:

0.020 Wb

in:

0.30 s

Magnitude of flux change:

|ΔΦ| = 0.060 Wb

Use:

|ε| = |ΔΦ|/Δt

|ε| = 0.060/0.30

|ε| = 0.20 V

The average induced EMF has magnitude:

0.20 V

---

Worked Example 9: Coil with Many Turns

A 200-turn coil experiences a flux change per turn of:

0.015 Wb

in:

0.50 s

Use:

|ε| = N|ΔΦ|/Δt

Substitute:

|ε| = 200(0.015)/0.50

|ε| = 6.0 V

This shows why practical generators and transformers often use coils containing many turns.

---

Comparing Two Flux Situations

Consider identical loops.

Loop A

B = 0.50 T
A = 0.20 m²
θ = 0°

Φ = 0.50 × 0.20 × 1

Φ = 0.10 Wb

Loop B

B = 0.50 T
A = 0.20 m²
θ = 60°

Φ = 0.50 × 0.20 × 0.5

Φ = 0.050 Wb

Loop B has half the magnetic flux because of its orientation.

---

Predicting Flux Without Calculating

Sometimes you only need to predict whether flux increases, decreases or stays the same.

Increase B

Flux increases.

Decrease B

Flux decreases.

Increase A

Flux increases.

Decrease A

Flux decreases.

Rotate from θ = 0° toward 90°

Flux decreases.

Rotate from θ = 90° toward 0°

Flux increases.

This follows directly from:

Φ = BA cos θ

---

What If Several Factors Change?

Suppose:

B doubles

but:

A halves

and θ stays the same.

Original:

Φ = BA cos θ

New:

Φ = (2B)(A/2)cos θ

The 2 and 1/2 cancel.

Therefore:

flux stays the same

Even though both B and A changed.

This is why it is important to consider all parts of the equation.

---

Challenge Example

A loop has:

A = 0.080 m²

and sits in a:

0.75 T

magnetic field.

The angle between B and the normal is:

40°

Calculate the magnetic flux.

Use:

Φ = BA cos θ

Φ = (0.75)(0.080)(cos 40°)

Φ ≈ (0.060)(0.766)

Φ ≈ 0.046 Wb

---

Challenge Example: Finding the Angle

A loop has:

B = 0.50 T

A = 0.20 m²

and magnetic flux:

Φ = 0.050 Wb

Use:

Φ = BA cos θ

Substitute:

0.050 = (0.50)(0.20)cos θ

0.050 = 0.10 cos θ

Therefore:

cos θ = 0.50

So:

θ = 60°

Remember:

This is the angle between the magnetic field and the normal to the surface.

---

Interpreting Field Diagrams

When looking at a field diagram, ask three questions.

1. How strong is the field?

Look at field-line density.

2. How large is the area?

A larger enclosed area can produce greater flux.

3. How is the surface oriented?

Look at how directly the field passes through the surface.

 

 

 

6

These correspond directly to:

B, A and θ

in:

Φ = BA cos θ

---

A Useful Visual Analogy: Rain Through a Window

Imagine rain falling straight downward.

A horizontal opening catches a large amount of rain.

Tilt the opening and less rain passes through it.

Turn it vertically and almost no rain passes through the opening.

Magnetic flux behaves similarly.

The "rain" represents:

magnetic field

The opening represents:

area

The tilt represents:

orientation

The amount passing through represents:

flux

The analogy is not perfect, but it helps visualize why the cosine term appears.

---

Magnetic Flux Is Not Magnetic Field Strength

Another important distinction:

B = magnetic field strength / magnetic flux density

Φ = magnetic flux

They are related but not the same.

B describes the magnetic field at a region.

Φ describes how that field passes through a particular surface.

A small area in a strong field may have the same flux as a larger area in a weaker field.

---

Example

Situation A:

B = 2 T

A = 0.10 m²

Flux:

Φ = 0.20 Wb

Situation B:

B = 0.5 T

A = 0.40 m²

Flux:

Φ = 0.20 Wb

The magnetic fields are different, but the magnetic fluxes are equal.

---

Common Misconception: Flux Is the Number of Field Lines

Field lines are only a visual representation.

Magnetic flux is not literally obtained by counting drawn lines.

The actual quantity is determined from:

Φ = BA cos θ

for a uniform field and flat surface.

---

Common Misconception: Maximum Flux Occurs at 90°

This depends on which angle you are measuring.

In:

Φ = BA cos θ

θ is measured between:

B and the normal

Therefore:

θ = 0° → maximum flux

θ = 90° → zero flux

Always identify the normal before using the equation.

---

Common Misconception: Strong Magnetic Field Means Large EMF

Not necessarily.

A strong but constant magnetic field can produce:

zero induced EMF

if the flux is not changing.

EMF depends on:

rate of change of magnetic flux

not simply magnetic-field strength.

---

Common Misconception: Moving a Magnet Always Produces an EMF

Motion matters only if it changes the magnetic flux through the circuit.

The deeper principle is not simply:

movement → EMF

It is:

movement → changing flux → EMF

This distinction becomes important in more complicated induction situations.

---

Common Misconception: Flux Must Change Because the Coil Moves

Not necessarily.

Imagine moving a coil sideways through a perfectly uniform magnetic field while:

• B stays constant
• A stays constant
• θ stays constant
• the entire loop remains inside the same uniform field

Then:

Φ = BA cos θ

remains unchanged.

Therefore that motion alone does not necessarily produce an induced EMF around the loop.

It is changing flux, rather than motion by itself, that matters.

---

Real-World Connection: Electrical Generators

Generators deliberately change magnetic flux.

Mechanical energy rotates coils or magnets.

This changes:

θ

and therefore:

Φ

The changing flux produces an induced EMF.

 

This principle is used in generators driven by:

• wind turbines
• hydroelectric turbines
• steam turbines
• gas turbines
• hand cranks

---

Real-World Connection: Transformers

Transformers change magnetic flux in a different way.

The coils do not need to rotate.

Instead, alternating current in the primary coil creates a:

changing magnetic field

This changes B.

Therefore the magnetic flux through the secondary coil changes.

The changing flux induces an EMF in the secondary coil.

So generators and transformers use the same fundamental principle but change flux in different ways.

---

Real-World Connection: Wireless Charging

Wireless charging also depends on changing magnetic flux.

Alternating current in the charging pad creates a changing magnetic field.

That changing field passes through a receiving coil inside the device.

Therefore:

Φ changes

which produces:

induced EMF

which allows electrical energy to be transferred to the device.

---

Did You Know?

The weber is named after German physicist Wilhelm Eduard Weber.

One weber is equivalent to:

1 T·m²

and also:

1 V·s

The second relationship connects magnetic flux directly to electromagnetic induction.

A change of magnetic flux of 1 Wb occurring uniformly in 1 second through a single loop corresponds to an average induced EMF magnitude of:

1 V

---

A Strong Magnetic Flux Explanation

Suppose the question asks:

Explain why rotating a coil in a magnetic field can produce an induced EMF.

A weak answer:

"The coil moves through the magnetic field."

A stronger answer:

"Rotating the coil changes the angle between the magnetic field and the normal to the coil. Since Φ = BA cos θ, the magnetic flux through the coil changes. According to Faraday's law, a changing magnetic flux produces an induced EMF."

This gives the complete chain:

rotation

→ θ changes

→ magnetic flux changes

→ EMF induced

---

A Strong Field-Diagram Explanation

Suppose two identical loops are shown in the same field.

Loop A is face-on to the field.

Loop B is edge-on.

A strong explanation is:

"Loop A has greater magnetic flux because the magnetic field passes perpendicularly through its surface. Its normal is parallel to the field, so θ = 0° and Φ = BA. Loop B has zero flux because its surface is parallel to the field, making θ = 90° and Φ = 0."

This combines:

diagram interpretation + geometry + equation

---

Key Terms

Magnetic flux (Φ) – Measure of magnetic field passing through a surface.

Magnetic flux density (B) – Measure of magnetic-field strength, measured in teslas.

Weber (Wb) – SI unit of magnetic flux.

Tesla (T) – SI unit of magnetic flux density.

Area (A) – Surface area through which magnetic flux is measured.

Normal – Imaginary line perpendicular to a surface.

Orientation – Direction of a surface relative to the magnetic field.

Magnetic field line – Visual representation of magnetic-field direction and relative strength.

Flux linkage – Product of magnetic flux and number of turns: NΦ.

Induced EMF – Voltage produced by changing magnetic flux.

Faraday's law – Relationship between induced EMF and the rate of change of magnetic flux linkage.

---

Key Takeaways

• Magnetic flux describes how much magnetic field passes through a surface.
• Magnetic flux is represented by Φ.
• The SI unit of magnetic flux is the weber (Wb).
• For a uniform field and flat surface:

Φ = BA cos θ

• B is magnetic flux density in teslas.
• A must be measured in square metres.
• θ is the angle between the magnetic field and the normal to the surface.
• Stronger magnetic fields produce greater flux when other factors remain constant.
• Larger areas produce greater flux when other factors remain constant.
• Orientation affects flux through the cosine relationship.
• Flux is maximum when θ = 0°.
• Flux is zero when θ = 90°.
• Magnetic field lines help us visualize magnetic flux but are not physical objects.
• A strong magnetic field does not automatically mean a large induced EMF.
• A constant magnetic flux produces no continuous induced EMF.
• Changing B, A or θ can change magnetic flux.
• Changing magnetic flux can produce an induced EMF.
• Faster flux change produces a larger induced EMF.
• Multiple turns increase flux linkage and can increase induced EMF.
• Generators change flux by rotating coils or magnets.
• Transformers change flux using changing magnetic fields.
• The central relationship is:

B, A or θ changes → magnetic flux changes → EMF is induced.

 

 

 

3. Faraday’s and Lenz’s Laws

From Magnetic Flux to Induced EMF

In the previous topic, we saw that magnetic flux describes how much magnetic field passes through a surface.

For a uniform magnetic field:

Φ = BA cos θ

Electromagnetic induction occurs when that magnetic flux changes.

Faraday’s Law tells us how the size of the induced EMF depends on the rate at which magnetic flux changes. Lenz’s Law tells us the direction of the induced EMF and current.

The basic chain is:

magnetic flux changes

→ EMF is induced

→ current flows if the circuit is complete

Faraday's Law

Faraday's Law states:

The induced EMF is proportional to the rate of change of magnetic flux linkage.

For a coil:

ε = −N(ΔΦ/Δt)

where:

• ε = induced EMF, measured in volts (V)
• N = number of turns in the coil
• ΔΦ = change in magnetic flux, measured in webers (Wb)
• Δt = time over which the change occurs, measured in seconds (s)

The negative sign is connected to Lenz's Law.

What Is Magnetic Flux Linkage?

For one loop:

magnetic flux = Φ

For a coil containing N turns:

magnetic flux linkage = NΦ

If the same flux passes through every turn, increasing the number of turns increases the total flux linkage.

Therefore:

more turns → larger induced EMF

for the same rate of flux change.

This is one reason practical generators and transformers use coils containing many turns.

The Meaning of Rate of Change

Faraday's Law does not simply depend on how much the magnetic flux changes.

It depends on:

how quickly the change occurs

Compare two situations.

Situation A

Flux changes by:

0.20 Wb

in:

4.0 s

Rate of change:

0.20 / 4.0 = 0.050 Wb/s

Situation B

The same flux change occurs in:

0.50 s

Rate of change:

0.20 / 0.50 = 0.40 Wb/s

Situation B produces the larger induced EMF because the change happens much more quickly.

Faster Flux Change Means Greater EMF

Faraday's Law can be written in terms of magnitude as:

|ε| = N|ΔΦ|/Δt

Therefore:

larger ΔΦ → larger EMF

and:

smaller Δt → larger EMF

provided the other quantities remain unchanged.

OpenStax describes induced EMF as directly proportional to the change in flux and inversely proportional to the time over which that change occurs.

Moving a Magnet Faster

Consider a bar magnet moving toward a coil.

Moving it slowly produces a gradual change in magnetic flux.

Moving it quickly produces a much faster change.

Therefore:

faster magnet → faster flux change → larger induced EMF

Experiments with a magnet and galvanometer show greater meter deflection when the magnet is moved more rapidly.

What Happens If the Magnet Stops?

Suppose the magnet is placed inside the coil and then held completely still.

The coil may still have a large magnetic flux.

However:

flux is no longer changing

so:

ΔΦ/Δt = 0

Therefore:

ε = 0

This is a key idea:

large magnetic flux does not automatically mean large EMF.

It is the rate of change of flux that matters.

Three Ways to Change Magnetic Flux

Since:

Φ = BA cos θ

magnetic flux can change if we change:

Magnetic field strength B

For example, move a magnet closer or change the current in an electromagnet.

Area A

Change the area enclosed by a conducting loop.

Orientation θ

Rotate a coil relative to the magnetic field.

Any of these can produce an induced EMF if they cause flux to change with time.

Faraday's Classic Magnet-and-Coil Experiment

A coil is connected to a galvanometer.

A bar magnet is moved toward the coil.

The changing magnetic flux induces an EMF, causing current to flow.

If the magnet stops:

current falls to zero

If the magnet is pulled away:

the current reverses direction

If the magnet is moved faster:

the induced EMF becomes larger

Faraday's experiments established that it is the change in magnetic flux, rather than simply the presence of a magnetic field, that produces induction.

Worked Example 1: Single Loop

The magnetic flux through a single loop changes from:

0.080 Wb

to:

0.020 Wb

in:

0.30 s

First calculate the change in flux:

ΔΦ = 0.020 − 0.080

ΔΦ = −0.060 Wb

For the magnitude:

|ΔΦ| = 0.060 Wb

Use Faraday's Law:

|ε| = N|ΔΦ|/Δt

For one loop:

N = 1

Therefore:

|ε| = (1)(0.060)/0.30

|ε| = 0.20 V

The induced EMF has a magnitude of:

0.20 V

Worked Example 2: Coil with Many Turns

A coil has:

N = 250 turns

The flux through each turn changes by:

0.012 Wb

in:

0.40 s

Use:

|ε| = N|ΔΦ|/Δt

Substitute:

|ε| = 250(0.012)/0.40

|ε| = 7.5 V

Answer:

7.5 V

Notice how the large number of turns increases the induced EMF.

Worked Example 3: Faster Change

A 100-turn coil experiences a flux change of:

0.020 Wb

Case A: change occurs in 2.0 s

|ε| = 100(0.020)/2.0

|ε| = 1.0 V

Case B: change occurs in 0.50 s

|ε| = 100(0.020)/0.50

|ε| = 4.0 V

The same flux change produced four times the EMF because it happened four times faster.

Lenz's Law

Faraday's Law tells us the size of the induced EMF.

Lenz's Law helps us determine its direction.

Lenz's Law states:

The induced current flows in a direction that creates a magnetic field opposing the change in magnetic flux that caused it.

This does not mean that the induced field always opposes the original magnetic field.

It opposes the change in magnetic flux.

Opposing the Change

This wording is extremely important.

Consider an external magnetic field through a loop.

External field increasing

The induced magnetic field points in the opposite direction.

It tries to oppose the increase.

External field decreasing

The induced magnetic field points in the same direction as the original field.

It tries to oppose the decrease.

So Lenz's Law acts as though the circuit tries to resist the change in flux, not simply oppose every magnetic field.

Why Does the Induced Current Oppose the Change?

Lenz's Law is closely connected to the:

conservation of energy

Suppose a magnet approaching a coil produced an induced magnetic field that helped pull the magnet in faster.

Then:

• motion would create current
• current would pull the magnet faster
• faster motion would create even more current
• energy would continually increase without an external energy source

That would violate conservation of energy.

Instead, the induced magnetic field resists the change that produced it. Mechanical work must be done to keep moving the magnet, and that energy can be transferred into electrical and thermal energy.

Magnet Approaching a Coil

Suppose the north pole of a magnet moves toward a coil.

The magnetic flux through the coil increases.

The coil responds by creating a magnetic field that opposes this increase.

Therefore the near face of the coil becomes:

a north pole

The approaching north pole is therefore:

repelled

This opposition does not prevent the magnet from moving.

You can still push it toward the coil, but you must perform work against the magnetic resistance.

Magnet Moving Away

Now suppose the north pole moves away from the coil.

The magnetic flux through the coil decreases.

The induced current creates a magnetic field that tries to maintain the original flux.

The near face of the coil therefore becomes:

a south pole

The receding north pole is:

attracted

The coil is again opposing the change — this time it opposes the decrease in flux.

Lenz's Law and the Minus Sign

Faraday's Law is:

ε = −N ΔΦ/Δt

The minus sign does not mean the EMF is simply "negative."

It represents the directional relationship described by Lenz's Law:

the induced effect opposes the change in magnetic flux.

When only the magnitude is required, we often use:

|ε| = N|ΔΦ|/Δt

and determine the direction separately using Lenz's Law.

Determining Induced Current Direction

A reliable method uses two ideas:

Lenz's Law

and:

the right-hand grip rule

The steps are:

1. Determine the direction of the external magnetic field through the loop.
2. Decide whether the magnetic flux is increasing or decreasing.
3. Determine which direction the induced magnetic field must point to oppose that change.
4. Use the right-hand grip rule to determine the current direction.

This is essentially the problem-solving method recommended by OpenStax.

Right-Hand Grip Rule for a Current Loop

Curl the fingers of your right hand in the direction of conventional current around the loop.

Your thumb points in the direction of the magnetic field through the centre of the loop.

Another useful shortcut when looking directly at one face of a coil:

anticlockwise current → that face behaves as a north pole

clockwise current → that face behaves as a south pole

Example 1: Increasing Field Into the Page

Suppose the external magnetic field points:

into the page

and its strength is increasing.

Lenz's Law says the induced field must oppose the increase.

Therefore the induced field points:

out of the page

Use the right-hand rule.

To create a field out of the page, the current must flow:

anticlockwise

So:

increasing field into page → anticlockwise induced current

Example 2: Decreasing Field Into the Page

Now the field still points into the page, but its strength is decreasing.

The loop tries to prevent the decrease.

Therefore the induced field must also point:

into the page

Using the right-hand rule:

clockwise current

creates a field into the page.

So:

decreasing field into page → clockwise induced current

Example 3: Increasing Field Out of the Page

External flux:

out of page and increasing

The induced field must point:

into the page

Therefore:

clockwise induced current

Example 4: Decreasing Field Out of the Page

External flux:

out of page and decreasing

The induced field must try to maintain it:

out of the page

Therefore:

anticlockwise induced current

Direction Summary

This table is a useful checking tool, but understanding the reasoning is much more valuable than memorising it.

Approaching North Pole: Direction Method

Consider a north pole approaching a loop.

Step 1:

The magnetic field from the north pole points away from the magnet and through the loop.

Step 2:

The magnet is approaching, so that flux is:

increasing

Step 3:

The induced field must oppose that increase.

Step 4:

The near face of the coil must become:

north

Step 5:

Viewed from the magnet side, a north face requires:

anticlockwise current

Approaching South Pole

Now reverse the magnet so the south pole approaches.

The direction of the external field through the coil reverses.

The induced field must again oppose the increase in flux.

Therefore the near side of the coil becomes:

south

Viewed from the magnet side, this requires:

clockwise current

So reversing the magnet reverses the induced current.

Pulling the Magnet Away Reverses the Current

If the direction of motion reverses:

increasing flux becomes decreasing flux

or vice versa.

Lenz's Law then requires the induced magnetic field to reverse.

Therefore:

the induced current reverses direction

This matches what is observed in magnet-and-galvanometer experiments.

Magnetic Flux and EMF Graphs

Faraday's Law says induced EMF depends on the slope of a magnetic-flux-versus-time graph.

If flux changes rapidly:

steep flux graph → large EMF

If flux is constant:

horizontal flux graph → zero EMF

If the direction of the flux change reverses:

the sign of the EMF reverses

Flux and EMF in a Rotating Generator

In a simple AC generator, a coil rotates in a magnetic field.

The magnetic flux varies approximately as:

Φ = BA cos θ

As the coil rotates:

θ changes continuously

so flux changes continuously.

Faraday's Law therefore produces an alternating EMF.

The induced EMF is greatest when the magnetic flux is changing most rapidly, not when the flux itself is greatest.

Maximum Flux Does Not Mean Maximum EMF

This is an important point.

When magnetic flux reaches its maximum value, its graph is momentarily flat.

That means:

rate of change of flux = 0

Therefore:

induced EMF = 0

When the flux passes through zero, it can be changing most rapidly.

At that moment:

induced EMF is maximum

Worked Example 4: Finding the Time Interval

A 50-turn coil experiences a flux change of:

0.030 Wb

An average EMF of:

3.0 V

is induced.

Find the time interval.

Use:

|ε| = N|ΔΦ|/Δt

Rearrange:

Δt = N|ΔΦ|/|ε|

Substitute:

Δt = 50(0.030)/3.0

Δt = 0.50 s

Worked Example 5: Finding Flux Change

A 200-turn coil produces an average EMF of:

12 V

for:

0.25 s

Find the change in flux through each turn.

Use:

|ε| = N|ΔΦ|/Δt

Rearrange:

|ΔΦ| = |ε|Δt/N

Substitute:

|ΔΦ| = (12)(0.25)/200

|ΔΦ| = 0.015 Wb

Worked Example 6: Changing Magnetic Field

A 100-turn coil has an area of:

0.020 m²

The magnetic field passes perpendicular to the coil and increases from:

0.10 T

to:

0.40 T

in:

0.50 s

Since the field is perpendicular to the surface:

Φ = BA

Change in magnetic field:

ΔB = 0.40 − 0.10

ΔB = 0.30 T

Change in flux through one turn:

ΔΦ = AΔB

ΔΦ = (0.020)(0.30)

ΔΦ = 0.0060 Wb

Now use Faraday's Law:

|ε| = 100(0.0060)/0.50

|ε| = 1.2 V

Worked Example 7: Rotating a Coil

A 40-turn coil has an area of:

0.050 m²

in a magnetic field of:

0.20 T

Initially, the normal to the coil is parallel to the field.

The coil rotates until its normal is perpendicular to the field.

Initial flux:

Φ₁ = BA cos 0°

Φ₁ = (0.20)(0.050)

Φ₁ = 0.010 Wb

Final flux:

Φ₂ = BA cos 90°

Φ₂ = 0

Therefore:

|ΔΦ| = 0.010 Wb

If the rotation occurs in:

0.20 s

then:

|ε| = 40(0.010)/0.20

|ε| = 2.0 V

Why Number of Turns Matters

Suppose two coils experience exactly the same magnetic flux change.

Coil A:

20 turns

Coil B:

200 turns

The rate of change per turn is the same.

But Coil B has ten times as many turns.

Therefore:

Coil B produces ten times the EMF

according to:

ε = −NΔΦ/Δt

Faraday's Law in Transformers

A transformer has:

• a primary coil
• an iron core
• a secondary coil

Alternating current in the primary produces a changing magnetic flux in the core.

That changing flux passes through the secondary coil.

Faraday's Law then produces an induced EMF in the secondary.

This is one of the most important practical applications of Faraday's Law.

Faraday's Law in Generators

A generator changes magnetic flux by rotating a coil or magnet.

Mechanical energy causes the rotation.

The rotation changes magnetic flux.

Faraday's Law produces an EMF.

So the energy pathway is approximately:

mechanical energy

→ changing magnetic flux

→ electrical energy

Faraday's Law in Wind Turbines

Wind turns the blades of a turbine.

The blades drive a shaft.

The shaft turns part of a generator.

Inside the generator, magnets and conducting coils move relative to one another.

This changes magnetic flux and induces an EMF.

Faraday's Law is therefore directly involved in converting wind's kinetic energy into electrical energy.

Lenz's Law in Generators

Lenz's Law explains why turning a generator requires effort.

As the generator produces current, the induced current creates its own magnetic field.

That field opposes the change that caused it.

Therefore the generator resists being turned.

The greater the electrical power being produced, the more mechanical input is generally required.

This is another consequence of conservation of energy.

A Useful Four-Step Lenz's Law Method

For almost any direction question, use:

Step 1: Find the external field direction

Is it into the page, out of the page, left, right, etc.?

Step 2: Decide whether the flux is increasing or decreasing

Ask what is changing.

Step 3: Find the required induced magnetic field

The induced field must oppose the change.

Step 4: Use the right-hand rule

Convert the required induced magnetic field into a current direction.

This approach is much more reliable than trying to memorize individual cases.

Example Direction Problem

A magnetic field points out of the page through a circular loop.

The field strength is increasing.

Step 1

External field:

out of page

Step 2

Flux:

increasing

Step 3

The induced field must oppose the increase:

into the page

Step 4

Right-hand rule:

clockwise current

Answer:

The induced current flows clockwise.

Another Direction Problem

A magnetic field points out of the page and is becoming weaker.

Step 1

External field:

out of page

Step 2

Flux:

decreasing

Step 3

The induced field tries to maintain the flux:

out of page

Step 4

Right-hand rule:

anticlockwise current

Answer:

The induced current flows anticlockwise.

What If the Circuit Is Open?

A changing magnetic flux can still induce an:

EMF

even if the circuit is open.

But without a complete conducting path:

no continuous current flows

This is why it is more accurate to say Faraday's Law describes an induced EMF, rather than automatically describing an induced current.

Common Misconception: More Flux Means More EMF

Not necessarily.

A coil may have a very large but constant magnetic flux.

Then:

ΔΦ = 0

so:

ε = 0

It is the change in flux per unit time, not simply the flux itself, that determines induced EMF.

Common Misconception: Lenz's Law Opposes the Magnetic Field

Not always.

Lenz's Law opposes:

the change in flux

If an external magnetic field is decreasing, the induced field can point in the same direction as the external field in an attempt to maintain the flux.

Common Misconception: The Induced Current Stops the Change

Lenz's Law does not mean the change cannot happen.

You can still:

• push the magnet into the coil
• pull it out
• rotate the generator
• change the magnetic field

The induced effect simply acts in a direction that resists the change.

External work can overcome that resistance.

Common Misconception: Faster Magnet Means More Magnetic Flux

Moving the magnet faster does not necessarily change the total amount of flux change.

It changes how quickly that flux change occurs.

For example:

Same:

ΔΦ

but smaller:

Δt

means larger:

ΔΦ/Δt

and therefore larger:

EMF

That distinction is central to Faraday's Law.

Common Misconception: The Minus Sign Is Used in Every Numerical Calculation

If the question asks only for the magnitude of induced EMF, use:

|ε| = N|ΔΦ|/Δt

Then report the positive magnitude.

Use Lenz's Law to determine direction separately.

The negative sign in the full equation represents direction, not a negative amount of voltage in an absolute sense.

Faraday vs Lenz

Faraday:

|ε| = N|ΔΦ|/Δt

Lenz:

induced effect opposes the change in magnetic flux

Together they give both:

magnitude + direction

A Complete Induction Explanation

Suppose a magnet moves rapidly toward a coil.

A strong explanation would be:

"As the magnet approaches, the magnetic flux through the coil increases. According to Faraday's Law, this changing flux induces an EMF, and because the flux is changing rapidly, the induced EMF is relatively large. According to Lenz's Law, the induced current produces a magnetic field that opposes the increase in flux, so the near side of the coil forms the same magnetic pole as the approaching pole and repels it."

This combines:

• magnetic flux
• rate of change
• Faraday's Law
• Lenz's Law
• induced magnetic field
• current direction
• energy conservation

Did You Know?

Michael Faraday's experiments in 1831 showed that changing magnetic fields could produce electrical effects. Heinrich Lenz later clearly formulated the rule governing the direction of the induced current. Their work became fundamental to generators, transformers, electric power systems, microphones, induction cookers, and many other technologies.

Key Terms

Faraday's Law – The induced EMF is proportional to the rate of change of magnetic flux linkage.

Lenz's Law – The induced current produces a magnetic field that opposes the change in magnetic flux that caused it.

Magnetic flux (Φ) – Measure of magnetic field passing through a surface.

Flux linkage (NΦ) – Magnetic flux multiplied by the number of turns in a coil.

Induced EMF – Voltage produced by changing magnetic flux.

Induced current – Current produced when an induced EMF acts in a complete circuit.

Rate of change – How quickly a quantity changes with time.

Right-hand grip rule – Method for relating current direction in a loop to the direction of its magnetic field.

Conservation of energy – Principle that energy cannot be created or destroyed, only transferred or transformed.

Key Takeaways

• Faraday's Law states that induced EMF depends on the rate of change of magnetic flux linkage.
• For a coil:

ε = −NΔΦ/Δt

• More coil turns generally produce a larger induced EMF.
• A larger flux change produces a larger EMF if the time remains constant.
• A faster flux change produces a larger EMF.
• Constant magnetic flux produces no continuous induced EMF.
• Magnetic flux can change because B, A, or θ changes.
• Lenz's Law determines the direction of the induced effect.
• The induced magnetic field opposes the change in magnetic flux, not necessarily the original magnetic field.
• Increasing flux produces an induced field that acts against the increase.
• Decreasing flux produces an induced field that acts against the decrease.
• Use the right-hand rule to convert induced field direction into induced current direction.
• The minus sign in Faraday's Law represents Lenz's Law.
• Lenz's Law is consistent with conservation of energy.
• Faster magnet motion generally produces a greater induced EMF because the flux changes more rapidly.
• When a magnet reverses direction, the induced current reverses.
• When the magnet stops relative to the coil, the induced EMF falls to zero.
• Generators and transformers are major applications of Faraday's Law.
• A reliable direction strategy is:

external field → increasing or decreasing flux → required induced field → right-hand rule → induced current direction.

4. Factors Affecting Induced EMF

5. Practical Demonstrations of Induction