Energy in Magnetic Fields

An inductor stores energy in the magnetic field produced by the current flowing through it.

When current through an inductor increases, the magnetic field becomes stronger. Because the changing magnetic field produces a self-induced EMF that opposes the increase in current, the power supply must do work to establish the current.

That work is stored as magnetic field energy.

When the current later decreases, the magnetic field collapses and the stored energy can be transferred back into the circuit.

The basic energy transfer is:

electrical energy → magnetic field energy → electrical energy

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Energy Storage in an Inductor

Consider an inductor connected to a power supply.

When the circuit is first switched on:

• current begins to increase
• the magnetic field becomes stronger
• magnetic flux through the coil increases
• a self-induced EMF is produced
• the self-induced EMF opposes the increase in current

The power supply must therefore transfer energy into the system.

Some of this energy becomes energy stored in the magnetic field surrounding and inside the inductor.

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Why Energy Is Required

An inductor opposes a change in current because of Lenz's law.

As current increases, the self-induced EMF acts against that increase.

The source must therefore do work to move charge against the induced EMF.

The work done becomes stored magnetic energy.

This is similar to doing work to compress a spring. The work is not destroyed; it is stored and can later be released.

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Energy Stored in an Inductor

The energy stored in an ideal inductor is:

E = ½LI²

where:

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

This is one of the most important equations for magnetic energy storage.

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What the Equation Tells Us

From:

E = ½LI²

the stored energy depends on:

• inductance
• current

For a fixed current:

E ∝ L

For a fixed inductance:

E ∝ I²

Therefore, current has a particularly strong effect on the amount of stored energy.

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Effect of Inductance

Suppose the current remains constant.

If inductance doubles:

L → 2L

then:

E → 2E

Therefore:

doubling inductance doubles the stored energy

for the same current.

A larger inductance means the circuit can store more magnetic energy at a given current.

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Effect of Current

Stored energy is proportional to the square of current:

E ∝ I²

If current doubles:

I → 2I

then:

E → ½L(2I)²

E → 4(½LI²)

Therefore:

doubling current produces four times the stored energy

If current triples:

stored energy becomes nine times larger

assuming inductance remains constant.

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Comparing Current and Inductance

Suppose an inductor initially stores energy E.

If its inductance doubles:

energy doubles

If its current doubles:

energy quadruples

This means that increasing current can have a much greater effect on stored energy than an equal proportional increase in inductance.

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Worked Example: Stored Energy

An inductor has:

L = 0.50 H

and carries:

I = 4.0 A

Use:

E = ½LI²

E = ½(0.50)(4.0²)

E = (0.25)(16)

E = 4.0 J

The magnetic field stores:

4.0 J

of energy.

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Worked Example: Smaller Inductor

An inductor has:

L = 25 mH

and carries:

I = 2.0 A

First convert:

25 mH = 0.025 H

Then:

E = ½LI²

E = ½(0.025)(2.0²)

E = (0.0125)(4.0)

E = 0.050 J

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Worked Example: Finding Current

An inductor stores:

E = 9.0 J

and has:

L = 0.50 H

Start with:

E = ½LI²

Rearrange:

I² = 2E/L

I² = 2(9.0)/0.50

I² = 36

Therefore:

I = 6.0 A

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Worked Example: Finding Inductance

An inductor stores:

E = 2.0 J

when carrying:

I = 4.0 A

Use:

E = ½LI²

Rearrange:

L = 2E/I²

L = 2(2.0)/(4.0²)

L = 4.0/16

L = 0.25 H

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Why the Factor of One-Half Appears

The induced EMF is not constant while current increases from zero.

For an ideal linear inductor:

ε = L ΔI/Δt

As current builds, energy is progressively transferred into the magnetic field.

The total work required to increase current from zero to I gives:

E = ½LI²

The factor ½ appears because the energy is accumulated as the current and magnetic field build from zero to their final values.

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Power During Energy Storage

Power is the rate of energy transfer:

P = VI

For an inductor:

V = L ΔI/Δt

Therefore:

P = LI ΔI/Δt

When current increases, energy is being transferred into the magnetic field.

When current decreases, stored magnetic energy can be transferred out of the inductor.

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Energy Is Stored in the Field

It is common to say that energy is "stored in the inductor."

More precisely, the energy is associated with the magnetic field produced by the current.

For a solenoid, much of the magnetic field is concentrated:

• inside the coil
• and, if present, within its magnetic core

The stronger the field, the greater the magnetic energy that can be associated with the system.

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Magnetic Energy Density

Energy can also be described in terms of the amount stored per unit volume of magnetic field.

For a simple linear magnetic medium:

u = B²/(2μ)

where:

• u = magnetic energy density in J/m³
• B = magnetic field strength in T
• μ = magnetic permeability

For free space:

u = B²/(2μ₀)

This equation shows that stronger magnetic fields contain greater energy density.

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Energy in a Solenoid

For an ideal solenoid:

L = μN²A/l

and:

E = ½LI²

Combining these ideas shows that changing the geometry of a solenoid can affect its ability to store magnetic energy.

For the same current:

• more turns generally increase stored energy
• greater cross-sectional area increases stored energy
• suitable magnetic cores can increase stored energy
• changes in coil length affect inductance and therefore stored energy

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Building the Magnetic Field

When an inductor is first connected to a DC supply, current does not immediately reach its final value.

Instead:

1. current begins increasing
2. magnetic field begins building
3. self-induced EMF opposes the increase
4. energy is transferred into the magnetic field
5. current gradually approaches its steady value

Once the current becomes constant, the magnetic field becomes constant.

The stored energy remains:

E = ½LI²

as long as the current continues.

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Releasing the Stored Energy

Suppose the current through an inductor begins to decrease.

The magnetic field begins collapsing.

According to Lenz's law, the inductor produces an EMF that attempts to maintain the current.

The magnetic field therefore transfers its stored energy back into the electrical circuit.

The sequence becomes:

magnetic field energy → electrical energy

This is why an inductor can continue driving current temporarily even after the original source is disconnected.

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Energy Conservation

Inductors provide a clear example of conservation of energy.

During current growth:

electrical energy supplied → magnetic field energy

During current decay:

magnetic field energy → electrical energy

In a real circuit, some energy also becomes thermal energy because wires and other components have resistance.

Therefore, a real energy pathway might be:

electrical energy → magnetic field energy → electrical energy + thermal energy

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Ideal and Real Inductors

An ideal inductor has no electrical resistance and no energy losses.

It can theoretically store and return magnetic energy without loss.

A real inductor has:

• wire resistance
• magnetic-core losses
• eddy-current losses
• hysteresis losses
• magnetic flux leakage

Therefore, some energy is converted into unwanted thermal energy.

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Resistance vs Inductance

Resistance and inductance affect electrical energy differently.

A resistor mainly converts electrical energy into thermal energy:

electrical energy → thermal energy

An ideal inductor temporarily stores energy:

electrical energy → magnetic field energy → electrical energy

This distinction is important.

An inductor does not simply "use up" energy in the same way as a resistor.

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Electrical Energy Storage

Electrical systems can store energy in different forms.

Two important components are:

• capacitors
• inductors

A capacitor stores energy in an electric field.

An inductor stores energy in a magnetic field.

These two devices play complementary roles in many electrical circuits.

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Capacitors and Inductors

This comparison is extremely useful when studying electrical energy storage.

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Similarity Between the Energy Equations

For a capacitor:

E_C = ½CV²

For an inductor:

E_L = ½LI²

Notice the similar structure.

For capacitors:

• C describes the component
• V describes its electrical state

For inductors:

• L describes the component
• I describes its electrical state

In both cases, stored energy depends on the square of the electrical quantity.

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Comparing Energy Storage

Consider:

Capacitor: E = ½CV²

Inductor: E = ½LI²

If voltage across a capacitor doubles:

capacitor energy becomes 4 times larger

If current through an inductor doubles:

inductor energy becomes 4 times larger

Both systems therefore show a squared dependence.

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Batteries Are Different

A battery also stores energy, but primarily as chemical energy, not as electric or magnetic field energy.

Therefore:

battery → chemical energy

capacitor → electric field energy

inductor → magnetic field energy

These devices can all supply electrical energy to circuits, but they store energy in fundamentally different ways.

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Inductor and Capacitor Energy Exchange

A capacitor and inductor can exchange energy with each other.

Suppose a charged capacitor is connected to an inductor.

Initially, much of the energy may be stored in the capacitor's electric field.

As the capacitor discharges:

• current increases
• the inductor's magnetic field strengthens
• energy moves into the magnetic field

Later, the magnetic field collapses and can recharge the capacitor.

In an ideal system:

electric field energy ↔ magnetic field energy

This behaviour is the basis of LC oscillations.

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Energy in an LC Circuit

At one moment, the capacitor may have maximum voltage and the current may be zero.

Energy is mainly:

electric field energy

Later, the capacitor voltage decreases while current increases.

Energy moves into the inductor.

When current reaches its maximum, energy is mainly:

magnetic field energy

The process can then reverse.

Real circuits gradually lose energy because of resistance.

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Practical Use: Switching Power Supplies

Modern electronic power supplies frequently use inductors to temporarily store energy.

A switching circuit rapidly changes the current through an inductor.

During part of the switching cycle:

energy is stored in the magnetic field

During another part:

energy is released to the circuit

This allows efficient conversion between different voltage and current levels.

Inductors are therefore important in devices such as:

• phone chargers
• computer power supplies
• LED drivers
• battery-powered electronics

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Buck Converters

A buck converter reduces DC voltage.

An inductor helps maintain relatively smooth current while electronic switches repeatedly turn on and off.

During part of the cycle, the inductor stores magnetic energy.

During another part, it releases energy to the load.

This helps convert electrical power efficiently.

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Boost Converters

A boost converter can produce an output voltage greater than its input voltage.

Current first builds through an inductor.

The inductor stores energy in its magnetic field.

When the switching configuration changes, the collapsing magnetic field produces an induced voltage.

The stored energy is transferred to the output.

This demonstrates a direct practical use of:

E = ½LI²

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Practical Use: Electrical Filters

Inductors are also used to smooth current.

Because an inductor opposes rapid current changes, it can reduce unwanted fluctuations.

Energy is alternately stored and released as the current changes.

This is useful in:

• power supplies
• audio circuits
• communication systems
• electronic filters

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Practical Use: Ignition Systems

Inductive energy storage can be used to produce large voltages.

Current is first established through a coil.

Energy is stored in its magnetic field.

When the current is interrupted rapidly:

• magnetic field collapses
• flux changes rapidly
• a large induced EMF can be produced

This principle is used in some ignition and pulsed electrical systems.

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Practical Use: Electromagnets

An electromagnet stores energy in its magnetic field while current flows.

Examples include:

• relays
• solenoids
• electromagnetic actuators
• lifting magnets
• magnetic valves

When the current is switched off, the stored magnetic energy must go somewhere.

Circuit designers therefore need to manage the released energy safely.

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Voltage Spikes

The energy stored in an inductor cannot disappear instantly.

If a current-carrying inductor is suddenly disconnected, the current attempts to decrease very rapidly.

Since:

|ε| = L|ΔI/Δt|

a very rapid change can produce a large voltage.

This voltage may cause:

• sparks
• damage to electronic components
• electrical interference

Protective circuits are often used to provide a safe path for the stored energy.

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Flyback Protection

A coil controlled electronically may have a protective path added across it.

When the switch opens:

• the magnetic field collapses
• the inductor attempts to maintain current
• stored energy continues circulating through the protective path
• the current gradually decreases
• the energy is safely dissipated

This helps protect sensitive switching components from large induced voltages.

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Energy Storage in Electric Vehicles

Inductors are important components in the power electronics used in electric vehicles.

They help:

• convert voltages
• control motor currents
• smooth electrical power
• temporarily store magnetic energy
• manage energy transfer between batteries, motors, and electronic systems

The energy stored in an individual inductor may be temporary, but repeated storage and release can occur thousands of times per second.

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Renewable Energy Systems

Power converters used with:

• solar panels
• wind turbines
• battery storage

often contain inductors.

The inductors temporarily store and release magnetic energy as electronic switches control the flow of electrical power.

This helps convert variable electrical inputs into forms suitable for:

• batteries
• electrical grids
• motors
• electronic equipment

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Magnetic Energy and Current Changes

The relationship between energy and current helps explain why inductors resist changes in current.

If current increases:

magnetic energy must increase

That requires an energy input.

If current decreases:

magnetic energy decreases

That energy must be transferred elsewhere.

Therefore, changing current involves changing the amount of energy stored in the magnetic field.

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Worked Comparison

Inductor A:

L = 0.20 H

I = 4.0 A

Energy:

E_A = ½(0.20)(4.0²)

E_A = 1.6 J

Inductor B:

L = 0.40 H

I = 4.0 A

Energy:

E_B = ½(0.40)(4.0²)

E_B = 3.2 J

Doubling inductance has doubled the stored energy.

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Current Comparison

An inductor has:

L = 0.50 H

At:

I₁ = 2.0 A

E₁ = ½(0.50)(2.0²)

E₁ = 1.0 J

At:

I₂ = 4.0 A

E₂ = ½(0.50)(4.0²)

E₂ = 4.0 J

The current doubled, but the stored energy increased by a factor of four.

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Multiple-Change Example

Suppose an inductor initially has:

L₁ = L

I₁ = I

Its inductance doubles and its current triples.

Since:

E = ½LI²

the new energy factor is:

2 × 3²

= 2 × 9

= 18

Therefore:

the new magnetic energy is 18 times the original energy.

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A Complete Calculation

An inductor has:

L = 0.80 H

The current increases from zero to:

I = 5.0 A

Calculate the final stored energy.

E = ½LI²

E = ½(0.80)(5.0²)

E = (0.40)(25)

E = 10 J

If the current later decreases to:

I = 3.0 A

the remaining energy is:

E = ½(0.80)(3.0²)

E = (0.40)(9)

E = 3.6 J

Energy released:

10 − 3.6

6.4 J

Therefore, 6.4 J of magnetic energy has been transferred out of the inductor.

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Comparing an Inductor and Capacitor

An inductor has:

L = 2.0 H

I = 3.0 A

Its energy is:

E_L = ½(2.0)(3.0²)

E_L = 9.0 J

A capacitor has:

C = 0.020 F

V = 30 V

Its energy is:

E_C = ½(0.020)(30²)

E_C = (0.010)(900)

E_C = 9.0 J

Both devices store the same amount of energy, but:

• the inductor stores it in a magnetic field
• the capacitor stores it in an electric field

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A Reliable Strategy for Energy Problems

First identify:

• inductance L
• current I

Convert units if necessary.

Then use:

E = ½LI²

If current is unknown:

I = √(2E/L)

If inductance is unknown:

L = 2E/I²

For comparisons, remember:

E ∝ L

and:

E ∝ I²

If the problem asks how much energy is released as current changes:

energy released = E_initial − E_final

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Unit Conversions

Inductance must usually be converted into henries.

Remember:

1 mH = 10⁻³ H

1 μH = 10⁻⁶ H

For example:

50 mH = 0.050 H

250 mH = 0.250 H

20 μH = 2.0 × 10⁻⁵ H

Current should normally be expressed in amperes.

Energy is then obtained in joules.

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Common Mistakes

Forgetting to square the current

The equation is:

E = ½LI²

not ½LI.

Forgetting the factor ½

The factor one-half is essential.

Using millihenries as though they were henries

Always convert units before calculating.

Thinking the energy is stored in the electric current itself

The energy is associated with the magnetic field produced by the current.

Thinking an ideal inductor permanently consumes energy

An ideal inductor can store energy and later return it.

Confusing an inductor with a resistor

A resistor dissipates energy mainly as heat; an ideal inductor stores magnetic energy.

Confusing magnetic and electric energy storage

Inductors store magnetic field energy. Capacitors store electric field energy.

Assuming the energy disappears when current falls

The stored energy must be transferred somewhere else.

Ignoring voltage spikes when interrupting current

A rapid decrease in current can produce a large induced EMF.

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Key Terms

Magnetic field energy — energy associated with a magnetic field.

Inductor — circuit component that stores energy in a magnetic field and opposes changes in current.

Inductance — measure of a circuit's self-inductive behaviour, represented by L.

Henry (H) — SI unit of inductance.

Self-induction — production of an EMF caused by changes in a circuit's own current.

Self-induced EMF — voltage produced by changing magnetic flux caused by changing current in the same circuit.

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

Energy density — energy stored per unit volume.

Capacitor — component that stores energy in an electric field.

Electric field energy — energy associated with an electric field, such as that between capacitor plates.

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

RL circuit — circuit containing resistance and inductance.

LC circuit — circuit containing inductance and capacitance that can exchange energy between magnetic and electric fields.

Flyback protection — circuit arrangement used to safely manage energy released when current through an inductive load is interrupted.

Energy conservation — principle that energy cannot be created or destroyed, only transferred or transformed.

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Key Takeaways

• Inductors store energy in magnetic fields.
• Energy must be supplied to establish the magnetic field around an inductor.
• Self-induction opposes the increase in current while the magnetic field is being established.
• The energy stored in an inductor is E = ½LI².
• Stored energy is directly proportional to inductance.
• Stored energy is proportional to the square of current.
• Doubling inductance doubles the stored energy for the same current.
• Doubling current quadruples the stored energy.
• When current decreases, the magnetic field collapses and stored energy is released.
• Magnetic energy can be transferred back into the electrical circuit.
• Energy conservation applies throughout the storage and release process.
• Real inductors lose some energy through resistance and magnetic-core effects.
• Capacitors store energy in electric fields, while inductors store energy in magnetic fields.
• Capacitor energy is E = ½CV².
• Inductor energy is E = ½LI².
• Inductors oppose rapid changes in current, while capacitors oppose rapid changes in voltage.
• LC circuits can exchange energy between electric and magnetic fields.
• Sudden interruption of current can release stored magnetic energy as a large induced voltage.
• Inductors are important in power converters, filters, motor controllers, renewable-energy systems, and many other electronic devices.
• Magnetic energy storage allows circuits to control and transfer electrical energy efficiently.