Mutual Induction

Mutual induction occurs when a changing current in one coil produces a changing magnetic field that induces an EMF in a second coil.

The process can be summarized as:

changing current in Coil 1 → changing magnetic field → changing magnetic flux through Coil 2 → induced EMF in Coil 2

Unlike self-induction, where a changing current induces an EMF in the same coil, mutual induction involves two electrically separate circuits linked by a changing magnetic field.

Mutual induction is the fundamental principle behind transformers and is also used in wireless charging, induction sensors, ignition systems, and many other technologies.

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Two Coils

Consider two coils placed close together.

The first coil is connected to a power source. It is called the primary coil.

The second coil is connected to another circuit or measuring device. It is called the secondary coil.

When current flows through the primary coil, it produces a magnetic field.

If that current changes, the magnetic field also changes.

If some of this changing magnetic field passes through the secondary coil, its magnetic flux changes.

Faraday's law then predicts an induced EMF in the secondary coil.

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The Basic Process

Mutual induction occurs through a sequence of events:

1. Current changes in the primary coil.

2. The primary coil's magnetic field changes.

3. Magnetic flux through the secondary coil changes.

4. An EMF is induced in the secondary coil.

5. If the secondary circuit is complete, an induced current flows.

The two coils do not need to be electrically connected.

Energy can be transferred through their interacting magnetic fields.

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Conditions Required for Mutual Induction

For mutual induction to occur effectively:

• current in the primary must change
• the changing current must produce a changing magnetic field
• some of this changing field must pass through the secondary coil
• magnetic flux through the secondary must change

For an induced current rather than simply an induced EMF, the secondary must also form a complete circuit.

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Constant Current Does Not Produce Continuous Mutual Induction

Suppose the primary coil carries a steady DC current.

The current produces a magnetic field.

However, once the current becomes constant:

ΔI/Δt = 0

The magnetic field becomes constant.

Therefore, the magnetic flux through the secondary becomes constant.

Faraday's law then gives:

ε = 0

So a steady current in the primary does not produce a continuous induced EMF in the secondary.

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Switching a DC Circuit On

Suppose the primary coil is connected to a DC battery through a switch.

When the switch is closed:

• primary current rises
• primary magnetic field increases
• flux through the secondary increases
• an EMF is temporarily induced in the secondary

Once the primary current becomes steady:

• magnetic field becomes steady
• flux becomes constant
• induced EMF falls to zero

Therefore, with DC, mutual induction occurs mainly while the current is changing.

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Switching the Primary Off

When the primary circuit is opened:

• primary current decreases rapidly
• magnetic field collapses
• flux through the secondary decreases
• another EMF is induced

The induced EMF is in the opposite direction from the EMF produced when the primary current was increasing.

This follows from Lenz's law.

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Alternating Current

Alternating current continually changes in:

• magnitude
• direction

Therefore, an AC current in the primary produces a continuously changing magnetic field.

This produces continuously changing magnetic flux through the secondary.

As a result:

AC in primary → continuously induced AC voltage in secondary

This is why transformers normally operate using alternating current.

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Faraday's Law and the Secondary Coil

Faraday's law gives the induced EMF:

ε = −N ΔΦ/Δt

where:

• ε = induced EMF
• N = number of turns
• ΔΦ = change in magnetic flux through each turn
• Δt = time interval

Therefore, a larger induced EMF can result from:

• more turns in the secondary
• greater magnetic flux
• faster changes in magnetic flux

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Mutual Inductance

The strength of the interaction between two circuits can be described by mutual inductance.

Its symbol is:

M

For two coupled coils:

ε₂ = −M ΔI₁/Δt

where:

• ε₂ = induced EMF in Coil 2
• M = mutual inductance
• ΔI₁/Δt = rate of change of current in Coil 1

The SI unit of mutual inductance is the:

henry (H)

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What Mutual Inductance Means

A large mutual inductance means that a change in current in one coil produces a relatively large induced EMF in the other.

A small mutual inductance means the coils interact only weakly.

Mutual inductance depends on factors including:

• number of turns
• distance between coils
• orientation of coils
• coil size
• overlap
• core material
• how much magnetic flux from one coil links the other

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Worked Example: Mutual Inductance

Two coils have mutual inductance:

M = 0.40 H

The current in Coil 1 changes at:

15 A/s

The magnitude of the induced EMF in Coil 2 is:

|ε₂| = M|ΔI₁/Δt|

|ε₂| = (0.40)(15)

|ε₂| = 6.0 V

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Worked Example: Finding Mutual Inductance

A changing current of:

25 A/s

in a primary coil produces:

5.0 V

in a secondary coil.

Use:

M = |ε₂| / |ΔI₁/Δt|

M = 5.0/25

M = 0.20 H

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Coil Position and Mutual Induction

The position of the coils strongly affects mutual induction.

The effect is greatest when a large fraction of the magnetic field produced by the primary passes through the secondary.

This is called strong magnetic coupling.

If very little primary magnetic flux passes through the secondary, the coupling is weak.

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Distance Between Coils

Moving the secondary coil farther from the primary generally reduces mutual induction.

As distance increases:

• less of the primary magnetic field reaches the secondary
• magnetic flux linkage decreases
• induced EMF becomes smaller

Therefore:

closer coils generally produce stronger mutual induction

assuming the other factors remain unchanged.

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Coil Alignment

Mutual induction also depends on orientation.

If the coils are positioned so that much of the primary magnetic field passes through the secondary:

coupling is strong

If the secondary is rotated so that less magnetic field passes through it:

coupling becomes weaker

At some orientations, the magnetic flux through the secondary can become very small.

Therefore:

coil orientation affects induced EMF

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Why Orientation Matters

Magnetic flux depends on:

Φ = BA cos θ

where θ is the angle between the magnetic field and the normal to the coil.

If:

θ = 0°

then:

Φ = BA

and flux is maximum.

If:

θ = 90°

then:

Φ = 0

Therefore, coil alignment can strongly affect how much of the primary magnetic field links the secondary.

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Overlapping Coils

If two coils are arranged so that their areas overlap strongly and their axes are aligned, magnetic coupling is usually greater.

If the coils:

• barely overlap
• are widely separated
• are poorly aligned

less flux from the primary links the secondary.

This reduces mutual induction.

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Increasing the Number of Turns

More turns can increase magnetic flux linkage.

Increasing turns in the primary can strengthen its magnetic effect.

Increasing turns in the secondary increases the total flux linkage experienced by the secondary.

Therefore, coil turns are an important design factor in devices based on mutual induction.

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Magnetic Cores

Placing both coils around a common magnetic core can greatly improve magnetic coupling.

The core provides an efficient path for magnetic flux.

More of the magnetic field produced by the primary is directed through the secondary.

This is why transformers commonly use magnetic cores.

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Air-Core vs Magnetic-Core Coupling

If two coils are separated by air, some of the primary magnetic field may spread outward and fail to pass through the secondary.

A suitable magnetic core can:

• concentrate magnetic flux
• guide flux through both coils
• reduce magnetic leakage
• increase mutual inductance

This makes energy transfer more effective.

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Flux Leakage

Not all magnetic flux produced by the primary necessarily passes through the secondary.

Flux that fails to link both coils is called leakage flux.

Greater leakage generally means:

• weaker coupling
• smaller mutual inductance
• less effective energy transfer

Transformer designs therefore attempt to keep the primary and secondary strongly magnetically coupled.

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Lenz's Law and Mutual Induction

The induced EMF in the secondary follows Lenz's law.

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

Suppose the primary field through the secondary is increasing.

The induced current produces a field opposing that increase.

If the primary flux begins decreasing, the secondary current reverses so that its magnetic effect attempts to maintain the original flux.

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Direction of Induced Current

To determine the direction of current in the secondary:

1. identify the direction of the primary magnetic field through the secondary
2. determine whether that flux is increasing or decreasing
3. use Lenz's law to determine the required secondary magnetic field
4. use the right-hand grip rule to determine secondary current direction

The induced current always acts so that its magnetic effect opposes the change in flux.

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Self-Induction vs Mutual Induction

Self-induction and mutual induction are closely related.

Both are explained using:

• magnetic fields
• magnetic flux
• Faraday's law
• Lenz's law

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Transformers

A transformer is one of the most important applications of mutual induction.

A simple transformer contains:

• a primary coil
• a secondary coil
• a magnetic core

An alternating voltage applied to the primary produces alternating current.

This creates a changing magnetic field in the core.

The changing magnetic flux passes through the secondary coil.

Faraday's law then produces an alternating EMF in the secondary.

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Transformer Operation

The sequence is:

AC voltage applied to primary

↓

alternating current in primary

↓

changing magnetic field

↓

changing magnetic flux through core

↓

changing flux through secondary

↓

induced AC voltage in secondary

No direct electrical connection between the two coils is required.

The coils are coupled magnetically.

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Why Transformers Need Changing Current

If a transformer primary were connected to an ideal constant DC current:

• current would become constant
• magnetic field would become constant
• magnetic flux would become constant
• secondary EMF would become zero

Therefore, continuous transformer operation requires a changing current, normally AC.

A brief induced voltage can occur when a DC circuit is switched on or off because the current changes during those moments.

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Transformer Turns Ratio

For an ideal transformer:

V_s/V_p = N_s/N_p

where:

• V_s = secondary voltage
• V_p = primary voltage
• N_s = number of secondary turns
• N_p = number of primary turns

The voltage ratio is therefore determined by the turns ratio.

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Step-Up Transformer

A step-up transformer increases voltage.

For a step-up transformer:

N_s > N_p

Therefore:

V_s > V_p

The secondary contains more turns than the primary.

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Step-Down Transformer

A step-down transformer decreases voltage.

For a step-down transformer:

N_s < N_p

Therefore:

V_s < V_p

The secondary contains fewer turns than the primary.

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Worked Example: Step-Up Transformer

A transformer has:

N_p = 200 turns

N_s = 1000 turns

The primary voltage is:

V_p = 24 V

Use:

V_s/V_p = N_s/N_p

V_s/24 = 1000/200

V_s/24 = 5

Therefore:

V_s = 120 V

This is a step-up transformer.

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Worked Example: Step-Down Transformer

A transformer has:

N_p = 1200 turns

N_s = 120 turns

The primary voltage is:

240 V

Then:

V_s/240 = 120/1200

V_s/240 = 0.10

Therefore:

V_s = 24 V

This is a step-down transformer.

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Transformer Power

For an ideal transformer, energy is conserved.

Therefore:

P_p = P_s

Since:

P = VI

then:

V_pI_p = V_sI_s

This means that increasing voltage decreases current, while decreasing voltage increases current, assuming an ideal transformer.

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Voltage and Current Ratios

For an ideal transformer:

V_s/V_p = N_s/N_p

and:

I_s/I_p = N_p/N_s

Voltage and current ratios therefore behave oppositely.

A step-up transformer:

• increases voltage
• decreases current

A step-down transformer:

• decreases voltage
• increases current

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Worked Example: Transformer Current

An ideal transformer has:

V_p = 120 V

I_p = 2.0 A

and:

V_s = 24 V

Input power:

P_p = V_pI_p

P_p = (120)(2.0)

P_p = 240 W

For an ideal transformer:

P_s = 240 W

Therefore:

I_s = P_s/V_s

I_s = 240/24

I_s = 10 A

The lower secondary voltage corresponds to a larger secondary current.

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Real Transformers

Real transformers are not perfectly efficient.

Some energy is lost through:

• resistance in the windings
• eddy currents in the core
• magnetic hysteresis
• magnetic flux leakage

Therefore:

P_output < P_input

Efficiency can be calculated using:

efficiency = P_output/P_input × 100%

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Laminated Transformer Cores

Transformer cores are often laminated.

Thin insulated layers reduce large circulating eddy currents in the core.

This reduces:

• heating
• energy losses

and increases transformer efficiency.

This connects mutual induction directly with the earlier study of eddy currents.

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Wireless Charging

Wireless charging uses mutual induction without a solid magnetic connection between the coils.

A transmitter coil carries alternating current.

This creates a changing magnetic field.

A nearby receiver coil experiences changing magnetic flux.

An EMF is induced in the receiver.

The electrical energy can then be used to charge a battery.

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Why Wireless Charger Alignment Matters

Mutual induction depends on magnetic coupling.

If the transmitting and receiving coils are well aligned:

• more magnetic flux links the receiver
• mutual inductance is larger
• energy transfer is more effective

If the device is badly positioned:

• less flux links the receiver
• induced voltage decreases
• charging may become slower or less efficient

This is a practical example of the effect of coil position.

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Electric Toothbrush Chargers

Some rechargeable electric toothbrushes use inductive charging.

A coil in the charging base produces a changing magnetic field.

A second coil inside the toothbrush experiences changing magnetic flux.

An induced voltage provides energy for charging.

Because electrical contact between exposed metal connectors is unnecessary, this approach can be useful in devices used around water.

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Inductive Sensors

Mutual induction can also be used in sensors.

A changing current in one coil creates a magnetic field.

Nearby conductive or magnetic objects can alter:

• magnetic flux
• coupling
• inductance

Electronic circuits detect these changes.

Applications include:

• proximity sensing
• position measurement
• industrial automation
• metal detection

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Ignition Systems

Traditional vehicle ignition systems use rapidly changing current and electromagnetic induction to produce high voltages.

Current changes in one winding create changing magnetic flux.

This changing flux induces voltage in another winding containing many more turns.

The large turns ratio allows a high voltage to be produced.

The system therefore operates using transformer-like mutual induction.

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Audio and Signal Transformers

Transformers can also transfer changing electrical signals between circuits.

Applications include:

• audio equipment
• communication systems
• impedance matching
• electrical isolation

Again, energy is transferred through changing magnetic flux rather than a direct conducting connection between the two windings.

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Electrical Isolation

Because transformer windings can be electrically separate, transformers can provide galvanic isolation.

There may be:

• no direct conducting path between primary and secondary
• magnetic coupling between them

This can be useful for:

• safety
• noise reduction
• circuit protection
• separating different electrical systems

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Effect of Faster Current Changes

From:

|ε₂| = M|ΔI₁/Δt|

a faster change in primary current produces a larger induced EMF.

Suppose the current changes by:

4 A

Case A:

Δt = 2 s

Rate:

2 A/s

Case B:

Δt = 0.20 s

Rate:

20 A/s

The current changes ten times faster in Case B.

Therefore, for the same mutual inductance:

the induced EMF is ten times larger

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Effect of Increasing Mutual Inductance

Suppose two coil systems experience the same primary current change.

System A:

M = 0.10 H

System B:

M = 0.50 H

Since:

|ε₂| = M|ΔI₁/Δt|

System B has five times the mutual inductance.

Therefore:

System B produces five times the induced EMF

for the same rate of current change.

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Improving Magnetic Coupling

Mutual induction can often be increased by:

• moving coils closer together
• aligning their axes
• increasing overlap
• increasing the number of turns
• using a suitable common magnetic core
• reducing leakage flux

These changes increase the amount of magnetic flux linking the two coils.

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Weakening Magnetic Coupling

Mutual induction can be reduced by:

• increasing the distance between coils
• rotating the coils out of alignment
• reducing overlap
• removing a magnetic core
• placing the coils so little magnetic flux links them

Sometimes weak coupling is unwanted.

In other situations, reducing unwanted mutual induction helps prevent electrical interference between nearby circuits.

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Mutual Induction and Energy Transfer

Mutual induction allows energy to move between circuits without direct electrical contact.

The primary circuit supplies energy.

Its changing magnetic field transfers energy electromagnetically.

The secondary circuit receives electrical energy.

Therefore:

electrical energy in primary → magnetic-field interaction → electrical energy in secondary

Energy is conserved; the secondary does not generate energy from nothing.

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A Complete Mutual-Induction Problem

Two coils have mutual inductance:

M = 0.30 H

The primary current increases from:

1.0 A to 7.0 A

in:

0.20 s

Change in current:

ΔI = 7.0 − 1.0

ΔI = 6.0 A

Rate of change:

ΔI/Δt = 6.0/0.20

ΔI/Δt = 30 A/s

Magnitude of secondary EMF:

|ε₂| = M|ΔI/Δt|

|ε₂| = (0.30)(30)

|ε₂| = 9.0 V

The direction of the induced EMF is such that its magnetic effect opposes the change in flux.

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A Complete Transformer Problem

An ideal transformer has:

N_p = 500

N_s = 100

V_p = 230 V

Find the secondary voltage.

V_s/V_p = N_s/N_p

V_s/230 = 100/500

V_s = 230(0.20)

V_s = 46 V

If the primary current is:

I_p = 1.0 A

then input power is:

P_p = V_pI_p

P_p = 230(1.0)

P_p = 230 W

For an ideal transformer:

P_s = 230 W

Therefore:

I_s = 230/46

I_s = 5.0 A

The transformer decreases voltage while increasing current.

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Comparing Self-Induction and Mutual Induction in a Transformer

The primary winding experiences self-induction because its changing current produces changing flux that induces an EMF in the primary itself.

The secondary experiences mutual induction because the changing current in the primary creates changing flux that induces an EMF in the secondary.

Therefore, a transformer demonstrates both effects simultaneously.

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A Reliable Strategy for Mutual-Induction Problems

First identify the two circuits:

primary → produces changing magnetic field

secondary → experiences changing flux

Then ask:

• Is the primary current changing?
• Does the primary field pass through the secondary?
• Are the coils well coupled?
• Is there a magnetic core?
• Is the secondary circuit complete?

For mutual-inductance calculations:

|ε₂| = M|ΔI₁/Δt|

For transformer calculations:

V_s/V_p = N_s/N_p

For ideal power calculations:

V_pI_p = V_sI_s

Finally, use Lenz's law if the direction of the induced current is required.

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Common Mistakes

Thinking a constant current continuously induces voltage in the secondary

Mutual induction requires changing magnetic flux.

Thinking the coils must touch

They need magnetic coupling, not physical contact.

Thinking the coils must be electrically connected

Mutual induction can occur between electrically isolated circuits.

Ignoring coil position

Distance and orientation affect how much magnetic flux links the secondary.

Assuming all primary flux reaches the secondary

Real systems have leakage flux.

Confusing mutual induction with self-induction

Self-induction occurs in the same circuit; mutual induction occurs between circuits.

Thinking transformers operate normally from steady DC

Continuous transformer operation requires changing current.

Reversing the transformer turns ratio

Remember:

V_s/V_p = N_s/N_p

Assuming a step-up transformer increases both voltage and current

For an ideal transformer, increasing voltage corresponds to decreasing current.

Assuming transformers create energy

They transfer energy. Real transformers actually deliver slightly less output power than their input power because of losses.

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Key Terms

Mutual induction — production of an EMF in one circuit because current changes in another magnetically coupled circuit.

Primary coil — coil in which the changing current produces the changing magnetic field.

Secondary coil — coil in which changing magnetic flux produces an induced EMF.

Mutual inductance — measure of how strongly changing current in one circuit induces EMF in another.

Henry (H) — SI unit of mutual inductance and self-inductance.

Magnetic coupling — degree to which magnetic flux from one coil links another coil.

Flux linkage — magnetic flux linked with the turns of a coil.

Leakage flux — magnetic flux produced by one coil that does not link effectively with the other coil.

Faraday's law — principle relating induced EMF to the rate of change of magnetic flux.

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

Transformer — device that uses mutual induction to transfer electrical energy between circuits.

Step-up transformer — transformer with a secondary voltage greater than its primary voltage.

Step-down transformer — transformer with a secondary voltage smaller than its primary voltage.

Turns ratio — ratio between the numbers of turns in transformer windings.

Magnetic core — magnetic material used to guide and concentrate magnetic flux.

Galvanic isolation — electrical separation between circuits with no direct conductive connection.

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Key Takeaways

• Mutual induction occurs between two magnetically coupled circuits.
• Changing current in the primary produces a changing magnetic field.
• Changing magnetic flux through the secondary produces an induced EMF.
• A constant primary current does not produce continuous mutual induction.
• Switching DC on or off can briefly produce mutual induction.
• Alternating current produces continuous changing flux and therefore continuous induction.
• The magnitude of induced EMF depends on how rapidly the primary current changes.
• Mutual inductance is represented by M and measured in henries.
• The relationship |ε₂| = M|ΔI₁/Δt| describes simple mutual-induction situations.
• Closer and better-aligned coils generally have stronger magnetic coupling.
• Increasing the number of turns can increase induced voltage.
• A common magnetic core can greatly improve coupling.
• Leakage flux reduces effective magnetic coupling.
• Lenz's law determines the direction of the induced effect.
• Mutual induction transfers energy without requiring a direct electrical connection.
• Transformers operate using mutual induction.
• An AC primary produces changing magnetic flux that induces AC voltage in the secondary.
• Transformer voltage ratio depends on the turns ratio.
• Step-up transformers increase voltage and reduce current.
• Step-down transformers reduce voltage and increase current.
• Wireless charging is a practical application of mutual induction.
• Self-induction and mutual induction are based on the same principles of changing magnetic flux and electromagnetic induction.

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Check Your Understanding

1. Explain what is meant by mutual induction.
2. Describe the sequence by which changing current in a primary coil produces an EMF in a secondary coil.
3. Why does a steady DC current in the primary not continuously induce an EMF in the secondary?
4. Explain what happens in the secondary when a DC primary circuit is first switched on and then switched off.
5. State four factors that can affect the strength of mutual induction between two coils.
6. Explain why moving two coils farther apart usually decreases the induced EMF.
7. Explain how rotating the secondary coil can affect mutual induction.
8. A pair of coils has mutual inductance 0.50 H. The primary current changes at 12 A/s. Calculate the magnitude of the induced EMF in the secondary.
9. A secondary coil experiences an induced EMF of 8.0 V when the primary current changes at 20 A/s. Calculate the mutual inductance.
10. Explain why a magnetic core can increase mutual induction between two coils.
11. A transformer has 400 primary turns and 1200 secondary turns. The primary voltage is 50 V. Calculate the secondary voltage and identify whether the transformer is step-up or step-down.
12. An ideal transformer changes 240 V to 24 V. If the primary current is 1.5 A, calculate the secondary current.
13. Explain why transformers normally operate using AC rather than steady DC.
14. Describe how coil alignment affects the operation and efficiency of a wireless charging system.
15. Compare self-induction and mutual induction, and explain how both processes can occur within a transformer.