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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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

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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

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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

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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

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Direction Summary

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

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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

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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.

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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.

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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

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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.

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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

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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

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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

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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

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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

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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

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Faraday vs Lenz

Faraday:

|ε| = N|ΔΦ|/Δt

Lenz:

induced effect opposes the change in magnetic flux

Together they give both:

magnitude + direction

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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

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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.

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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.

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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.