2. Induction in Loops

Induction in Loops

Electromagnetic induction occurs when a changing magnetic environment produces an electromotive force (EMF) in a conductor.

For a conducting loop, the key idea is magnetic flux. If the magnetic flux through the loop changes, an EMF is induced. If the loop forms a complete conducting circuit, the induced EMF causes an induced current.

Magnetic flux can change because:

• the magnetic field strength changes
• the loop moves into or out of a magnetic field
• the loop rotates
• the area of the loop changes
• the angle between the loop and the magnetic field changes

This principle is the basis of generators, transformers, induction cooktops, wireless charging systems, and many other electrical technologies.

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Magnetic Flux

Magnetic flux describes how much magnetic field passes through a surface.

For a flat loop in a uniform magnetic field:

Φ = BA cos θ

where:

• Φ = magnetic flux in webers (Wb)
• B = magnetic field strength in teslas (T)
• A = area of the loop in square metres (m²)
• θ = angle between the magnetic field and a line perpendicular to the loop

That last point is important.

The angle is measured from the normal to the surface, not from the surface itself.

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The Normal to a Loop

The normal is an imaginary line drawn perpendicular to the plane of the loop.

If the magnetic field points in the same direction as the normal:

θ = 0°

Therefore:

cos 0° = 1

and:

Φ = BA

The magnetic flux is maximum.

If the magnetic field is parallel to the plane of the loop:

θ = 90°

so:

cos 90° = 0

and:

Φ = 0

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Maximum Magnetic Flux

Maximum magnetic flux occurs when the magnetic field is:

perpendicular to the plane of the loop

This means the field is parallel to the loop's normal.

Then:

θ = 0°

and:

Φ_max = BA

It is easy to confuse this with maximum induced EMF.

Maximum flux and maximum induced EMF do not necessarily occur at the same orientation.

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Zero Magnetic Flux

Magnetic flux is zero when the field lies parallel to the plane of the loop.

Then:

θ = 90°

and:

Φ = 0

Again, zero flux does not necessarily mean zero induced EMF.

If the loop is rotating rapidly through this position, the flux may be changing very quickly even though its instantaneous value is zero.

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Faraday's Law

Faraday's law states that an EMF is induced whenever the magnetic flux through a circuit changes.

For a single loop, the magnitude can be written:

|ε| = |ΔΦ/Δt|

For a coil containing N identical turns:

|ε| = N|ΔΦ/Δt|

Therefore:

greater rate of change of magnetic flux → greater induced EMF

The full form is:

ε = −N ΔΦ/Δt

The negative sign represents Lenz's law.

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Flux Linkage

For a coil with N turns, the quantity:

NΦ

is called magnetic flux linkage.

Faraday's law can therefore be written as:

ε = −Δ(NΦ)/Δt

A coil with many turns can produce a larger induced EMF because each turn experiences the changing magnetic flux.

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Changing the Magnetic Field Strength

Suppose a loop remains stationary but the magnetic field passing through it becomes stronger.

Since:

Φ = BA cos θ

increasing B increases the magnetic flux.

Because the flux changes, an EMF is induced.

Similarly, if the magnetic field decreases:

• magnetic flux decreases
• an EMF is induced

The field does not have to disappear completely. It only has to change.

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Constant Magnetic Field

A constant magnetic field does not automatically produce an induced current.

Suppose a loop is stationary in a constant magnetic field.

If:

• B remains constant
• A remains constant
• θ remains constant

then:

ΔΦ = 0

Therefore:

ε = 0

Even though magnetic flux exists, no EMF is induced because the flux is not changing.

This is one of the most important ideas in electromagnetic induction.

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Moving a Loop into a Magnetic Field

Suppose a conducting loop moves from a region with no magnetic field into a region containing a uniform magnetic field.

Initially:

Φ = 0

As the loop enters the field, more of its area lies inside the magnetic field.

The flux increases.

Therefore:

an EMF is induced

Once the entire loop is inside a uniform field and continues moving without changing orientation:

• B is constant
• A inside the field is constant
• θ is constant

Therefore the flux becomes constant again and the induced EMF becomes zero.

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Moving a Loop Out of a Magnetic Field

When the loop begins leaving the field, the amount of its area inside the magnetic region decreases.

Therefore:

magnetic flux decreases

and an EMF is induced again.

The direction of the induced current is opposite to the direction produced while the loop was entering.

This follows from Lenz's law.

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Changing the Orientation of a Loop

Magnetic flux depends on:

cos θ

Therefore, rotating a loop changes the magnetic flux even if:

• B remains constant
• A remains constant

For example:

At θ = 0°:

Φ = BA

At θ = 60°:

Φ = BA cos 60°

Φ = 0.5BA

At θ = 90°:

Φ = 0

Changing orientation therefore provides another way to induce an EMF.

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Example: Flux at Different Orientations

A loop has:

A = 0.020 m²

and is placed in:

B = 0.50 T

At θ = 0°:

Φ = BA

Φ = (0.50)(0.020)

Φ = 0.010 Wb

At θ = 60°:

Φ = (0.50)(0.020)(0.50)

Φ = 0.0050 Wb

At θ = 90°:

Φ = 0 Wb

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Maximum Induced EMF

Maximum induced EMF occurs when the magnetic flux is changing at its greatest rate.

This is not necessarily when magnetic flux itself is greatest.

For a loop rotating steadily in a magnetic field:

• maximum flux occurs when the field is perpendicular to the loop
• maximum rate of flux change occurs when the field is parallel to the plane of the loop

Therefore, for a steadily rotating loop:

maximum EMF occurs when Φ = 0

and:

zero EMF occurs when |Φ| is maximum

This relationship is fundamental to AC generators.

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Why Maximum Flux Gives Zero EMF During Rotation

Imagine a rotating loop at the instant when:

θ = 0°

The flux is:

Φ = BA

which is maximum.

But at this position, the flux-versus-time curve is momentarily flat.

Its rate of change is zero.

Therefore:

ε = 0

A quarter-turn later:

θ = 90°

The flux is zero, but it is changing most rapidly.

Therefore:

|ε| is maximum

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Rotating Loops

Suppose a loop rotates continuously in a uniform magnetic field.

Its magnetic flux changes continuously.

For steady rotation:

Φ = BA cos(ωt)

where ω is angular speed.

For a coil with N turns:

NΦ = NBA cos(ωt)

The resulting induced EMF varies with time.

This produces an alternating EMF.

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Maximum EMF of a Rotating Coil

For a coil rotating steadily in a uniform magnetic field, the maximum EMF is:

ε_max = NBAω

where:

• N = number of turns
• B = magnetic field strength
• A = area of each turn
• ω = angular speed

Therefore, maximum generated EMF can be increased by increasing:

• number of turns
• magnetic field strength
• coil area
• rotational speed

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Effect of Loop Area

From:

Φ = BA cos θ

magnetic flux is directly proportional to loop area.

If B and θ remain unchanged:

Φ ∝ A

A larger loop can contain more magnetic flux.

However, a large constant flux does not by itself produce an EMF.

There must still be a change in flux.

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Example: Changing Loop Area

Suppose a flexible conducting loop is perpendicular to a uniform magnetic field of:

B = 0.40 T

Its area changes from:

A₁ = 0.020 m²

to:

A₂ = 0.050 m²

Initial flux:

Φ₁ = BA₁

Φ₁ = (0.40)(0.020)

Φ₁ = 0.0080 Wb

Final flux:

Φ₂ = BA₂

Φ₂ = (0.40)(0.050)

Φ₂ = 0.020 Wb

Change in flux:

ΔΦ = 0.020 − 0.0080

ΔΦ = 0.012 Wb

If this occurs in 0.20 s:

|ε| = ΔΦ/Δt

|ε| = 0.012/0.20

|ε| = 0.060 V

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Effect of Number of Turns

Suppose two coils experience the same change in magnetic flux per turn.

Coil A has:

N = 10

Coil B has:

N = 100

Since:

|ε| = N|ΔΦ/Δt|

Coil B produces:

10 times the EMF

of Coil A.

This is why practical generators and transformers commonly use coils containing many turns of wire.

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Worked Example: Changing Magnetic Field

A single loop has area:

A = 0.030 m²

The magnetic field perpendicular to the loop increases from:

0.20 T

to:

0.80 T

in:

0.15 s

Initial flux:

Φ₁ = B₁A

Φ₁ = (0.20)(0.030)

Φ₁ = 0.0060 Wb

Final flux:

Φ₂ = B₂A

Φ₂ = (0.80)(0.030)

Φ₂ = 0.024 Wb

Change:

ΔΦ = 0.018 Wb

Therefore:

|ε| = 0.018/0.15

|ε| = 0.12 V

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Worked Example: Multiple-Turn Coil

A coil contains:

N = 50 turns

Each loop has area:

A = 0.010 m²

A perpendicular magnetic field changes from:

0.20 T

to:

0.60 T

in:

0.10 s

Change in field:

ΔB = 0.40 T

Change in flux per turn:

ΔΦ = AΔB

ΔΦ = (0.010)(0.40)

ΔΦ = 0.0040 Wb

Induced EMF:

|ε| = NΔΦ/Δt

|ε| = 50(0.0040)/0.10

|ε| = 2.0 V

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Induced Current

If the conducting loop forms a complete circuit, the induced EMF can produce current.

Using Ohm's law:

I = ε/R

where:

• I = induced current
• ε = induced EMF
• R = total resistance

If the loop is broken or open:

• an EMF may still exist
• continuous current cannot flow

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Worked Example: Induced Current

A coil experiences an induced EMF of:

1.5 V

Its total resistance is:

6.0 Ω

Then:

I = ε/R

I = 1.5/6.0

I = 0.25 A

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Lenz's Law

Faraday's law tells us the magnitude of the induced EMF.

Lenz's law helps determine its direction.

Lenz's law states:

The induced current produces a magnetic effect that opposes the change in magnetic flux that caused it.

The induced current does not simply oppose the magnetic field.

It opposes the change in flux.

This distinction is essential.

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Increasing Magnetic Flux

Suppose the external magnetic field through a loop points:

into the page

and becomes stronger.

The inward magnetic flux is increasing.

The induced current produces a magnetic field that opposes this increase.

Therefore, the induced field points:

out of the page

The current direction can then be determined using the right-hand grip rule.

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Decreasing Magnetic Flux

Now suppose the external magnetic field points into the page but becomes weaker.

The inward flux is decreasing.

The induced current attempts to maintain it.

Therefore, the induced magnetic field points:

into the page

The direction of induced current is therefore opposite to the previous case.

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Right-Hand Grip Rule for a Loop

Once the direction of the induced magnetic field is known:

• curl the fingers of your right hand in the direction of conventional current
• your thumb points in the direction of the magnetic field through the loop

Equivalently:

counterclockwise current → field out of page

clockwise current → field into page

when viewed directly at the loop.

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A Reliable Method for Current Direction

Use this sequence:

1. Determine the direction of the external magnetic field.
2. Determine whether the magnetic flux is increasing or decreasing.
3. Use Lenz's law to determine the direction of the induced magnetic field.
4. Use the right-hand grip rule to determine the induced current direction.

For example:

External field:

into page

Flux:

increasing

Induced field must be:

out of page

Therefore induced current is:

counterclockwise

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Example: Magnet Approaching a Loop

Suppose the north pole of a bar magnet approaches a conducting loop.

The magnetic flux through the loop increases.

According to Lenz's law, the loop produces a magnetic field that opposes this increase.

The near side of the loop behaves magnetically in a way that resists the approaching magnet.

This creates a repulsive interaction.

The effect demonstrates that induced current follows conservation of energy.

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Magnet Moving Away

If the magnet moves away from the loop:

• magnetic flux decreases
• induced current reverses
• the loop's magnetic field attempts to maintain the original flux

The magnetic interaction now tends to oppose the magnet's departure.

Therefore, both approaching and departing motion experience effects that oppose the change producing the induction.

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Faster Change Produces Greater EMF

Faraday's law shows:

|ε| ∝ |ΔΦ/Δt|

Suppose the same flux change occurs in:

0.10 s

instead of:

1.0 s

The change occurs ten times faster.

Therefore:

the induced EMF is ten times greater

This is why moving a magnet quickly through a coil produces a larger voltage than moving it slowly.

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No Change Means No Induction

Consider a magnet held stationary beside a stationary loop.

The loop may experience a magnetic field.

It may even have substantial magnetic flux.

But if nothing changes:

ΔΦ = 0

Therefore:

ε = 0

This distinction can be summarized as:

magnetic flux does not cause induction by itself

changing magnetic flux causes induction

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Four Main Ways to Change Magnetic Flux

From:

Φ = BA cos θ

flux can change by changing:

B — magnetic field strength

A — loop area

θ — loop orientation

A fourth practical method is moving the loop into or out of a magnetic-field region, which changes the effective area exposed to the field.

These are the major mechanisms students should look for in induction problems.

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Practical Example: Electrical Generator

A generator rotates a coil through a magnetic field.

As the coil rotates:

• its orientation changes
• magnetic flux changes
• an EMF is induced
• current can flow through an external circuit

Mechanical energy is therefore converted into:

electrical energy

Sources of mechanical energy can include:

• wind
• falling water
• steam
• engines
• human motion

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Practical Example: Bicycle Generator

A bicycle generator uses mechanical rotation to create changing magnetic flux.

As the bicycle wheel turns, components inside the generator rotate relative to a magnetic field.

The changing flux induces an EMF.

The electrical energy can power lights or other small devices.

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Practical Example: Wireless Charging

Wireless charging uses changing magnetic fields.

An alternating current in a transmitting coil creates a changing magnetic field.

A receiving coil experiences changing magnetic flux.

Faraday's law then produces an induced EMF in the receiving coil.

The electrical energy can be used to charge a battery.

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Practical Example: Induction Cooking

An induction cooktop uses rapidly changing magnetic fields.

Alternating current flows through a coil beneath the cooking surface.

This produces a changing magnetic field.

The changing flux induces currents in suitable metal cookware.

Electrical resistance in the cookware converts electrical energy into thermal energy.

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Practical Example: Metal Detection

Metal detectors generate changing magnetic fields.

Conductive objects within these fields can develop induced currents.

Those currents create their own magnetic fields.

The detector senses these changes and can indicate the presence of metal.

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Loops and Motional EMF

Motional EMF and induction in loops are closely related.

In motional EMF, a conductor physically moves through a magnetic field.

In loop induction, an EMF occurs whenever magnetic flux changes.

A moving loop can therefore experience induction because its:

• position changes
• orientation changes
• exposed area changes

Both phenomena are part of the broader theory of electromagnetic induction.

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A Complete Flux Problem

A 20-turn coil has area:

A = 0.025 m²

A uniform magnetic field increases from:

B₁ = 0.10 T

to:

B₂ = 0.50 T

The field is perpendicular to the plane of the coil.

The change occurs in:

Δt = 0.20 s

Change in field:

ΔB = 0.50 − 0.10

ΔB = 0.40 T

Change in flux per turn:

ΔΦ = AΔB

ΔΦ = (0.025)(0.40)

ΔΦ = 0.010 Wb

Induced EMF:

|ε| = NΔΦ/Δt

|ε| = 20(0.010)/0.20

|ε| = 1.0 V

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A Complete Current Problem

Suppose the coil from the previous example has resistance:

R = 4.0 Ω

Then:

I = ε/R

I = 1.0/4.0

I = 0.25 A

If the external magnetic field into the page is increasing, the induced field must point out of the page.

Therefore, the induced conventional current is:

counterclockwise

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Comparing Two Loops

Loop A and Loop B experience the same change in magnetic field over the same time.

Loop A:

A = 0.020 m²

Loop B:

A = 0.060 m²

Since:

|ε| ∝ A

Loop B has three times the area.

Therefore:

Loop B produces three times the induced EMF

assuming their orientations and number of turns are identical.

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Comparing Two Coils

Coil A contains:

50 turns

Coil B contains:

200 turns

Both experience the same change in flux per turn over the same time.

Since:

|ε| ∝ N

then:

ε_B/ε_A = 200/50

ε_B/ε_A = 4

Coil B produces four times the induced EMF.

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A Reliable Strategy for Induction Problems

First determine what is changing.

Ask:

• Is B changing?
• Is A changing?
• Is θ changing?
• Is the loop moving into or out of the field?

Then calculate magnetic flux using:

Φ = BA cos θ

Find the change:

ΔΦ = Φ_final − Φ_initial

For N turns:

|ε| = N|ΔΦ|/Δt

If resistance is known:

I = |ε|/R

For current direction:

1. identify the external field
2. determine whether its flux is increasing or decreasing
3. use Lenz's law to find the induced field
4. use the right-hand grip rule to find current direction

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Common Mistakes

Thinking a magnetic field automatically creates current

A constant magnetic flux produces no induced EMF.

Forgetting that flux must change

Electromagnetic induction depends on the rate of change of flux.

Using the angle between the field and the surface

In Φ = BA cos θ, θ is measured between B and the normal to the surface.

Confusing maximum flux with maximum EMF

Maximum flux does not necessarily mean maximum induced EMF.

Assuming zero flux means zero EMF

A rotating loop can have zero instantaneous flux while its flux is changing most rapidly.

Ignoring the number of turns

For a coil:

|ε| = N|ΔΦ/Δt|

Ignoring loop area

A larger area generally produces greater flux and can produce greater EMF for the same rate of field change.

Using Lenz's law to oppose the field instead of the change

The induced field opposes the change in magnetic flux.

Confusing clockwise and counterclockwise current

Determine the induced magnetic-field direction first, then use the right-hand grip rule.

Calculating current without checking whether the circuit is complete

An induced EMF can exist without continuous current.

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Key Terms

Electromagnetic induction — production of an EMF by changing magnetic flux.

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

Weber (Wb) — SI unit of magnetic flux.

Loop — closed conducting path capable of carrying current.

Coil — conductor containing one or more loops or turns.

Normal — imaginary line perpendicular to a surface.

Flux linkage — magnetic flux multiplied by the number of turns, NΦ.

Faraday's law — relationship between induced EMF and rate of change of magnetic flux.

Lenz's law — principle stating that induced effects oppose the change in magnetic flux that produced them.

Induced EMF — potential difference produced through electromagnetic induction.

Induced current — current produced by an induced EMF in a complete circuit.

Natural orientation — orientation of a loop relative to a magnetic field, described using the angle between B and the loop's normal.

Maximum flux — largest magnitude of magnetic flux, occurring when the field is perpendicular to the loop.

Maximum induced EMF — largest induced voltage, occurring when flux changes most rapidly.

Alternating EMF — EMF that repeatedly reverses direction.

Generator — device that converts mechanical energy into electrical energy using electromagnetic induction.

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Key Takeaways

• Changing magnetic flux through a conducting loop produces an induced EMF.
• Magnetic flux for a uniform field is Φ = BA cos θ.
• The angle θ is measured between the magnetic field and the normal to the loop.
• Magnetic flux is maximum when the field is perpendicular to the plane of the loop.
• Magnetic flux is zero when the field is parallel to the plane of the loop.
• A constant magnetic flux produces no induced EMF.
• Flux can change because B, A, or θ changes.
• Moving a loop into or out of a magnetic field can also change its flux.
• Faraday's law relates induced EMF to the rate of change of magnetic flux.
• For N turns, |ε| = N|ΔΦ/Δt|.
• Faster flux changes produce larger induced EMFs.
• Increasing loop area can increase magnetic flux and induced EMF.
• Increasing the number of turns increases induced EMF.
• In a rotating coil, maximum flux and maximum EMF occur at different orientations.
• Maximum induced EMF occurs when magnetic flux is changing most rapidly.
• Lenz's law determines the direction of induced current.
• The induced magnetic effect opposes the change in flux that produced it.
• An induced EMF produces current only when a complete conducting path exists.
• Induced current can be calculated using I = ε/R.
• Generators use rotating loops and changing magnetic flux to produce electrical energy.
• Wireless charging, induction cooking, and many sensing technologies also use electromagnetic induction.

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Check Your Understanding

1. Explain why a conducting loop in a constant magnetic field does not necessarily have an induced current.
2. Define magnetic flux and state the equation used to calculate it for a flat loop in a uniform magnetic field.
3. A loop has area 0.040 m² and is perpendicular to a 0.50 T magnetic field. Calculate its magnetic flux.
4. The normal to a loop makes an angle of 60° with a 0.80 T magnetic field. The loop area is 0.020 m². Calculate the magnetic flux.
5. Explain why maximum magnetic flux and maximum induced EMF occur at different positions for a steadily rotating loop.
6. A loop experiences a change in magnetic flux from 0.010 Wb to 0.040 Wb in 0.20 s. Calculate the magnitude of the induced EMF.
7. A 40-turn coil experiences a change in magnetic flux of 0.0050 Wb per turn in 0.10 s. Calculate the induced EMF.
8. A coil produces an induced EMF of 3.0 V and has a total resistance of 12 Ω. Calculate the induced current.
9. Explain what happens to the induced EMF if the same magnetic-flux change occurs in half the time.
10. Two single loops experience the same change in magnetic field. Loop B has three times the area of Loop A. Compare their induced EMFs if the changes occur over the same time.
11. A magnetic field directed into the page through a loop becomes stronger. Determine the direction of the induced magnetic field and the direction of conventional current.
12. A magnetic field directed into the page becomes weaker. Determine the induced magnetic-field direction and conventional current direction.
13. A loop moves completely into a uniform magnetic field, continues moving entirely inside the field, and then leaves it. Describe when an EMF is induced during this motion and explain why.
14. A 100-turn coil has area 0.015 m². A perpendicular magnetic field changes from 0.20 T to 0.80 T in 0.30 s. Calculate the magnitude of the induced EMF.
15. Explain how changing magnetic flux is used in an electrical generator, and identify at least three changes to the generator design or operation that could increase its maximum induced EMF.