Self-Induction

Self-induction occurs when a changing current in a circuit produces a changing magnetic field, which then induces an EMF in the same circuit.

The process can be summarized as:

changing current → changing magnetic field → changing magnetic flux → induced EMF

According to Lenz's law, the induced EMF acts in a direction that opposes the change in current that produced it.

Self-induction is especially important in coils and inductors, where the magnetic field produced by the current is concentrated and linked through many turns of wire.

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Current Produces a Magnetic Field

Whenever electric current flows through a conductor, it produces a magnetic field.

For a straight wire, the magnetic field forms circular patterns around the wire.

For a coil, the fields from the individual turns combine, producing a stronger magnetic field through the coil.

Therefore:

current → magnetic field

If the current changes, the magnetic field also changes.

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Changing Current Produces Changing Magnetic Flux

Suppose the current through a coil increases.

The magnetic field produced by the coil becomes stronger.

Therefore, the magnetic flux through the turns of the coil increases.

If the current decreases:

• the magnetic field becomes weaker
• the magnetic flux decreases

Faraday's law tells us that a change in magnetic flux produces an induced EMF.

Because the coil's own current produces the changing flux, the process is called:

self-induction

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Why It Is Called Self-Induction

In earlier examples of electromagnetic induction, one object often caused an EMF in another.

For example:

moving magnet → induced EMF in a separate coil

In self-induction:

changing current in coil → changing magnetic field → induced EMF in the same coil

The circuit effectively induces an EMF in itself.

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The Self-Induced EMF

The magnitude of the self-induced EMF depends on how quickly the current changes.

The relationship is:

ε = −L ΔI/Δt

or, for continuously changing current:

ε = −L dI/dt

where:

• ε = self-induced EMF in volts (V)
• L = inductance in henries (H)
• ΔI = change in current in amperes (A)
• Δt = time interval in seconds (s)

The negative sign represents Lenz's law.

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Inductance

Inductance measures how strongly a circuit opposes changes in its current through self-induction.

Its symbol is:

L

Its SI unit is the:

henry (H)

A component has an inductance of 1 H if a current changing at 1 A/s produces a self-induced EMF of magnitude 1 V.

Therefore:

1 H = 1 V·s/A

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Rate of Change of Current

From:

|ε| = L|ΔI/Δt|

the induced EMF depends on the rate at which current changes, not simply on the size of the current.

For example, a current changing from:

1 A → 5 A

in 0.10 s produces a greater induced EMF than the same change occurring over 10 s.

Therefore:

faster current change → greater self-induced EMF

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Constant Current

Suppose a steady current flows through an inductor.

The current is not changing.

Therefore:

ΔI = 0

and:

ε = 0

The inductor still produces a magnetic field, but there is no self-induced EMF because the magnetic flux is no longer changing.

This distinction is important:

current produces a magnetic field

but:

changing current produces self-induced EMF

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Increasing Current

Suppose the current through a coil begins to increase.

The magnetic field becomes stronger.

The increasing magnetic flux produces a self-induced EMF.

According to Lenz's law, this induced EMF acts to oppose the increase in current.

Therefore:

current trying to increase → induced EMF acts against the increase

This is why current through an inductor cannot change instantaneously in an ideal circuit.

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Decreasing Current

Now suppose the current through the coil begins to decrease.

The magnetic field becomes weaker.

The magnetic flux decreases.

According to Lenz's law, the self-induced EMF acts to oppose this decrease.

The inductor therefore produces an EMF that attempts to keep the current flowing in its original direction.

So:

current trying to decrease → induced EMF acts to maintain the current

An inductor therefore opposes changes in current in either direction.

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The Meaning of the Negative Sign

In:

ε = −L ΔI/Δt

the negative sign does not mean that the magnitude of the EMF is negative.

It represents direction.

The induced EMF acts in a direction that opposes the change producing it.

This is another expression of Lenz's law.

When only the magnitude is required:

|ε| = L|ΔI/Δt|

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Self-Induction Does Not Always Oppose the Current

A common misunderstanding is that self-induced EMF always opposes the current.

More precisely, it opposes the change in current.

When current increases, the induced EMF opposes the increase.

When current decreases, the induced EMF acts to maintain the current.

Therefore:

an inductor resists changes in current, not current itself.

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Worked Example: Increasing Current

An inductor has:

L = 0.50 H

The current increases from:

2.0 A to 6.0 A

in:

0.20 s

First calculate:

ΔI = 6.0 − 2.0

ΔI = 4.0 A

Rate of change:

ΔI/Δt = 4.0/0.20

ΔI/Δt = 20 A/s

Magnitude of induced EMF:

|ε| = L|ΔI/Δt|

|ε| = (0.50)(20)

|ε| = 10 V

The induced EMF acts to oppose the increase in current.

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Worked Example: Decreasing Current

A 2.0 H inductor carries a current that decreases from:

5.0 A to 1.0 A

in:

0.50 s

Change in current:

ΔI = 1.0 − 5.0

ΔI = −4.0 A

Magnitude of rate of change:

|ΔI/Δt| = 4.0/0.50

|ΔI/Δt| = 8.0 A/s

Therefore:

|ε| = (2.0)(8.0)

|ε| = 16 V

The induced EMF acts in a direction that attempts to maintain the original current.

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Worked Example: Finding Inductance

A changing current produces an induced EMF of:

12 V

The current changes at:

30 A/s

Use:

|ε| = L|ΔI/Δt|

Rearrange:

L = |ε| / |ΔI/Δt|

L = 12/30

L = 0.40 H

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Worked Example: Finding the Rate of Current Change

An inductor has:

L = 0.25 H

and develops an induced EMF of:

5.0 V

Use:

|ε| = L|ΔI/Δt|

Therefore:

|ΔI/Δt| = |ε|/L

|ΔI/Δt| = 5.0/0.25

|ΔI/Δt| = 20 A/s

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Inductors

An inductor is an electrical component designed to have significant inductance.

A basic inductor consists of a coil of conducting wire.

When current flows through the coil:

• a magnetic field forms
• energy is stored in the magnetic field

When the current changes:

• the magnetic field changes
• magnetic flux changes
• a self-induced EMF is produced

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Factors Affecting Inductance

The inductance of a coil depends on its construction.

Important factors include:

• number of turns
• coil dimensions
• cross-sectional area
• coil length
• core material

Generally, inductance can be increased by:

• increasing the number of turns
• increasing the cross-sectional area
• using a suitable magnetic core

For a long ideal solenoid, inductance is approximately:

L = μN²A/l

where:

• μ = magnetic permeability of the core/material
• N = number of turns
• A = cross-sectional area
• l = length of the coil

This shows particularly clearly that inductance increases strongly with the number of turns.

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Number of Turns

More turns produce greater magnetic flux linkage.

Because the induced EMF depends on changing flux linkage, increasing the number of turns increases the inductance.

For an ideal solenoid:

L ∝ N²

Doubling the number of turns can therefore produce approximately four times the inductance if the other properties remain unchanged.

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Magnetic Core

Placing a suitable magnetic material inside a coil can increase its magnetic field and magnetic flux.

This can greatly increase the coil's inductance.

Common core materials include:

• iron-based magnetic materials
• ferrites

The best material depends on the operating conditions and frequency.

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Energy Stored in an Inductor

An inductor stores energy in its magnetic field.

The energy stored is:

E = ½LI²

where:

• E = stored energy in joules (J)
• L = inductance in henries (H)
• I = current in amperes (A)

This energy does not appear instantly.

Energy must be supplied while the magnetic field is being established.

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Worked Example: Stored Energy

An inductor has:

L = 0.80 H

and carries:

I = 3.0 A

The stored magnetic energy is:

E = ½LI²

E = ½(0.80)(3.0²)

E = (0.40)(9.0)

E = 3.6 J

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Why an Inductor Resists Increasing Current

When a circuit containing an inductor is switched on, the current attempts to increase.

The increasing current creates an increasing magnetic field.

This produces a self-induced EMF that opposes the increase.

As a result, the current rises gradually rather than instantaneously.

This behaviour is sometimes described as the inductor having electrical inertia.

The analogy is useful:

• mass resists changes in velocity
• inductance resists changes in current

However, inductance is an electromagnetic effect, not a mechanical one.

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Switching On an Inductive Circuit

Consider a battery, resistor, inductor, and switch connected in series.

Immediately after the switch closes:

• current begins increasing
• magnetic field begins increasing
• self-induced EMF is large
• the inductor strongly opposes the increase

As time passes:

• current increases more slowly
• the rate of current change decreases
• self-induced EMF decreases

Eventually, in an ideal DC circuit:

• current becomes steady
• ΔI/Δt = 0
• self-induced EMF becomes zero

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Current Growth in an RL Circuit

A circuit containing resistance R and inductance L is called an RL circuit.

After connection to a DC supply, the current does not immediately jump to its final value.

Instead, it gradually approaches that value.

The characteristic time is called the time constant:

τ = L/R

where:

• τ = time constant in seconds
• L = inductance in henries
• R = resistance in ohms

A larger inductance produces a slower change in current.

A larger resistance produces a shorter time constant.

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The Time Constant

After one time constant:

t = τ

the current during switch-on has reached about:

63% of its final value

After several time constants, the current is very close to its final value.

This gives a useful way of describing how quickly an inductive circuit responds.

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Switching Off an Inductive Circuit

When the circuit is switched off, the current attempts to decrease rapidly.

The magnetic field around the inductor begins to collapse.

The changing magnetic flux produces a self-induced EMF.

According to Lenz's law, this EMF attempts to keep the current flowing.

Therefore, an inductor can temporarily continue driving current even after the original supply is disconnected.

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Large Voltage When Current Changes Rapidly

Recall:

|ε| = L|ΔI/Δt|

If current changes extremely rapidly, the induced voltage can become large.

This is especially important when an inductive circuit is suddenly switched off.

A rapid current decrease means:

large |ΔI/Δt|

which can produce:

large induced voltage

This voltage may cause a spark across switch contacts.

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Sparks Across Switches

When a switch suddenly opens an inductive circuit:

• current falls rapidly
• magnetic field collapses
• a large self-induced EMF can develop

If the voltage becomes large enough, the electric field across the switch gap can ionize the air.

A spark may occur.

This is one practical effect of self-induction.

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Relays and Electromagnets

Relay coils and electromagnets are inductors.

When current through their coils is switched:

• their magnetic fields change
• self-induced EMFs are produced

This must be considered when designing circuits that control:

• relays
• solenoids
• motors
• electromagnets

Protective components are often added to prevent large voltage spikes from damaging other circuit components.

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Motors

Electric motors contain coils.

As current through these coils changes, self-induction occurs.

Motor circuits can therefore produce induced voltages when:

• switched on
• switched off
• electronically controlled
• current direction changes

Self-induction is one reason motor-control circuits require careful electrical design.

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Transformers and Self-Induction

Transformer coils also experience self-induction.

Alternating current in the primary coil continuously changes.

Therefore:

• the primary magnetic field changes
• flux through the primary changes
• a self-induced EMF occurs in the primary

At the same time, changing flux through the secondary coil produces mutual induction.

So a transformer involves both:

• self-induction
• mutual induction

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Self-Induction vs Mutual Induction

Both effects are based on:

changing magnetic flux and Faraday's law

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DC and Inductors

In a DC circuit, the current eventually becomes constant.

Once the current is steady:

ΔI/Δt = 0

so:

ε = 0

An ideal inductor then behaves like a conducting path with no resistance.

Real inductors contain wire resistance, so real devices still have some voltage drop and energy loss.

Self-induction is most significant while the current is changing.

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AC and Inductors

Alternating current continually changes in magnitude and direction.

Therefore, an inductor in an AC circuit continually experiences self-induction.

The induced EMF continuously opposes the changing current.

As the AC frequency increases, the current changes more rapidly.

Therefore, inductive effects generally become more significant.

This property makes inductors useful in:

• filters
• power supplies
• radio circuits
• signal processing
• energy-storage circuits

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Inductive Reactance

In an AC circuit, the opposition an inductor provides to alternating current is described by inductive reactance.

It is given by:

X_L = 2πfL

where:

• X_L = inductive reactance in ohms (Ω)
• f = frequency in hertz (Hz)
• L = inductance in henries (H)

Therefore:

higher frequency → greater inductive reactance

and:

greater inductance → greater inductive reactance

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Example: Inductive Reactance

An inductor has:

L = 0.20 H

and operates at:

f = 50 Hz

Then:

X_L = 2πfL

X_L = 2π(50)(0.20)

X_L ≈ 62.8 Ω

Therefore:

X_L ≈ 63 Ω

This means the inductor provides substantial opposition to the changing current at this frequency.

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Inductors in Filters

Because inductors oppose rapidly changing currents, they can help filter electrical signals.

An inductor can allow slowly changing or steady current more easily while providing greater opposition to rapidly changing current.

This makes inductors useful in circuits designed to:

• reduce unwanted electrical noise
• separate frequencies
• smooth electrical signals
• control current changes

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Inductors in Power Supplies

Inductors are commonly used in electronic power supplies.

They can:

• store magnetic energy
• release that energy later
• smooth changes in current
• help convert voltage levels
• reduce current ripple

Switching power supplies repeatedly store and release energy using inductors.

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Energy Transfer During Switch-On

When an inductor is connected to a source, the source supplies electrical energy.

Some of that energy becomes energy stored in the magnetic field:

E = ½LI²

The inductor therefore does not simply "block" the current.

Instead, it temporarily stores energy while the current increases.

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Energy Transfer During Switch-Off

When current decreases, the magnetic field collapses.

The energy stored in the field is released.

The inductor can transfer this energy back into the circuit.

Therefore:

electrical energy → magnetic-field energy → electrical energy

Some of the energy may ultimately become thermal energy because real circuits contain resistance.

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Why Current Cannot Change Instantly

Suppose current through an ideal inductor changed instantly.

Then:

Δt → 0

For a finite change in current:

ΔI/Δt → extremely large

From:

|ε| = L|ΔI/Δt|

the required induced EMF would become extremely large.

Therefore, an ideal inductor prevents an instantaneous change in current.

This gives an important circuit rule:

current through an ideal inductor is continuous.

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Comparing Inductors

Consider two inductors experiencing the same rate of current change.

Inductor A:

L = 0.20 H

Inductor B:

L = 1.0 H

Since:

|ε| = L|ΔI/Δt|

Inductor B has five times the inductance.

Therefore, it produces:

five times the self-induced EMF

for the same rate of current change.

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Complete Calculation Example

A 0.75 H inductor carries a current that changes from:

1.0 A to 5.0 A

in:

0.20 s

Calculate the change in current:

ΔI = 5.0 − 1.0

ΔI = 4.0 A

Calculate the rate:

ΔI/Δt = 4.0/0.20

ΔI/Δt = 20 A/s

Calculate the induced EMF:

|ε| = L|ΔI/Δt|

|ε| = (0.75)(20)

|ε| = 15 V

The induced EMF acts to oppose the increase in current.

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Complete Energy Example

A 0.50 H inductor carries a steady current of:

4.0 A

Calculate its stored energy:

E = ½LI²

E = ½(0.50)(4.0²)

E = (0.25)(16)

E = 4.0 J

If the current is switched off, this magnetic-field energy must be transferred elsewhere.

It may:

• drive current temporarily
• charge another component
• produce useful electrical output
• become thermal energy
• contribute to a voltage spike if the circuit is not properly controlled

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Identifying Self-Induction

When deciding whether self-induction occurs, ask:

Is current changing?

If no:

there is no self-induced EMF from that steady current.

If yes, ask:

Does the changing current produce changing magnetic flux linked with the same circuit?

If yes:

self-induction occurs

Common situations include:

• switching a coil on
• switching a coil off
• changing current through an electromagnet
• AC flowing through an inductor
• current changing in a motor winding
• current changing in a relay coil
• current changing in a transformer winding

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A Reliable Strategy for Self-Induction Problems

First identify whether the current is changing.

Then calculate:

ΔI = I_final − I_initial

Find the rate:

ΔI/Δt

Use:

|ε| = L|ΔI/Δt|

when only the magnitude is required.

For direction, use Lenz's law:

• increasing current → induced EMF opposes the increase
• decreasing current → induced EMF attempts to maintain the current

If energy is required:

E = ½LI²

If an RL time constant is required:

τ = L/R

Always connect the calculation back to the physical behaviour of the circuit.

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Common Mistakes

Thinking self-induction requires two coils

Self-induction occurs within the same circuit or coil.

Thinking a steady current continuously produces self-induced EMF

A steady current produces a magnetic field, but not a changing magnetic flux.

Saying an inductor opposes current

An inductor specifically opposes changes in current.

Ignoring the rate of current change

Induced EMF depends on ΔI/Δt, not simply ΔI.

Forgetting the negative sign represents direction

The negative sign in ε = −L ΔI/Δt represents Lenz's law.

Assuming the induced EMF always acts opposite to the current

When current decreases, the induced EMF acts to maintain the original current.

Confusing inductance with resistance

Resistance dissipates electrical energy mainly as heat. Inductance stores energy in a magnetic field and opposes changes in current.

Forgetting that an inductor stores energy

The stored energy is:

E = ½LI²

Assuming current through an ideal inductor can change instantly

An instantaneous change would require an unrealistically large voltage.

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Key Terms

Self-induction — production of an induced EMF in a circuit due to changes in its own current.

Self-induced EMF — voltage generated by a circuit's own changing magnetic flux.

Inductance — measure of a circuit's ability to produce self-induced EMF when current changes.

Henry (H) — SI unit of inductance.

Inductor — electrical component designed to have significant inductance.

Coil — conductor wound into turns, often used to produce magnetic fields and inductance.

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

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

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

Lenz's law — principle stating that an induced effect opposes the change producing it.

Back EMF — an induced EMF that acts against a change driving the circuit; the term is often used for opposing induced voltages.

RL circuit — circuit containing both resistance and inductance.

Time constant — characteristic time describing how rapidly current changes in an RL circuit; τ = L/R.

Magnetic energy — energy stored in the magnetic field of an inductor.

Inductive reactance — opposition an inductor provides to alternating current, X_L = 2πfL.

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Key Takeaways

• Electric current produces a magnetic field.
• Changing current produces a changing magnetic field.
• Changing magnetic flux can induce an EMF in the same circuit.
• This process is called self-induction.
• Self-induced EMF is described by ε = −L ΔI/Δt.
• The magnitude is |ε| = L|ΔI/Δt|.
• Inductance is measured in henries.
• Greater inductance produces a larger induced EMF for the same rate of current change.
• Faster current changes produce larger self-induced EMFs.
• A steady DC current produces no self-induced EMF once it becomes constant.
• Lenz's law causes self-induced EMF to oppose changes in current.
• When current increases, the induced EMF opposes the increase.
• When current decreases, the induced EMF acts to maintain the current.
• An inductor therefore resists changes in current rather than simply resisting current.
• Inductors store energy in magnetic fields.
• Stored energy is E = ½LI².
• Current through an ideal inductor cannot change instantaneously.
• Sudden switching can produce large induced voltages.
• Self-induction occurs in inductors, relays, electromagnets, motors, transformers, and many electronic circuits.
• In AC circuits, inductive effects become stronger as frequency increases.
• Inductors are widely used for energy storage, filtering, current control, and power conversion.

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Check Your Understanding

1. Explain what is meant by self-induction.
2. Describe the sequence of events that causes a changing current in a coil to produce a self-induced EMF.
3. Why does a constant current produce no self-induced EMF once the current has stabilized?
4. Explain why an inductor opposes a change in current rather than simply opposing current itself.
5. A 0.40 H inductor experiences a current change from 1.0 A to 5.0 A in 0.20 s. Calculate the magnitude of the self-induced EMF.
6. A 2.0 H inductor has a current that decreases from 8.0 A to 2.0 A in 0.50 s. Calculate the magnitude of the induced EMF and describe its effect on the current.
7. An inductor produces an EMF of 6.0 V when current changes at 15 A/s. Calculate its inductance.
8. A 0.25 H inductor produces an induced EMF of 10 V. Calculate the magnitude of the rate of change of current.
9. Explain what happens to the self-induced EMF when the rate of current change doubles.
10. Describe what happens to the magnetic field, induced EMF, and current immediately after an inductive DC circuit is switched on.
11. Explain why opening a switch in an inductive circuit can produce a large voltage or spark.
12. A 0.60 H inductor carries a current of 5.0 A. Calculate the energy stored in its magnetic field.
13. A circuit contains an inductance of 0.80 H and resistance of 4.0 Ω. Calculate its time constant.
14. Compare self-induction and mutual induction. Identify one important similarity and one important difference.
15. Explain how self-induction is useful in an inductor used in a power supply or filter, including how the inductor responds to changing current and how energy is stored in its magnetic field.