Energy Stored in an Inductor

In one sentence: A coil with current flowing through it stores energy in its magnetic field, and doubling the current makes that stored energy four times bigger.

$E = \frac{1}{2} L I^2$

Read it out loud: “E equals one half times L times I squared. E is the energy stored in the coil, measured in joules. L is the inductance, measured in henries. I is the current, measured in amperes. I squared means I times I.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Magnetic field and the energy in a coil: A magnetic field is an invisible field around a magnet or a current, where magnetic force acts. A current flowing in a wire makes a circular magnetic field around the wire, and when the wire is wound into a coil the fields add up inside it. The field holds energy in the same way a stretched spring holds energy. The energy goes into the field while the current builds up, and it comes back out when the current falls.
Current, the ampere (A) and the milliamp (mA): Current is the flow of electric charge through a wire, measured in amperes (A). One ampere is one coulomb of charge passing a point every second. The prefix milli, written m, means one thousandth, so 20 mA is 0.02 A. A current is always measured as a flow past one point, and the same current flows through every part of a single loop.
The henry (H) and the millihenry (mH): The henry (H) is the unit of inductance, which measures how strongly a coil opposes changes in its current. One henry is one volt-second per ampere. A millihenry, written mH, is one thousandth of a henry, so 10 mH is 0.01 H. Inductance is the number that tells you how much energy a coil stores for a given current.
Squares and powers of ten, like $I^2$ and $10^{-3}$: A square, written $I^{2}$ , means the number multiplied by itself, so $2^{2} = 2 \times 2 = 4$ and $4^{2} = 16$ . Squaring makes numbers grow quickly. A power of ten is 10 with a small raised exponent that says how many times to multiply by ten. $10^{3}$ is 1,000, and $10^{-3}$ means one divided by 1,000, which is 0.001. When two powers of ten are multiplied, add their exponents.
Energy, the joule (J) and the watt (W): Energy is the ability to make something happen, and its unit is the joule, written J. One joule is roughly the energy needed to lift a small apple, about 100 grams, one meter straight up. A watt (W) is one joule per second, so a 1 W bulb uses 1 joule every second. Power is the rate at which energy is used or stored, and it is measured in watts.

What each symbol means

Symbol	Meaning
$E$	The energy stored in the magnetic field of the coil, measured in joules (J). It is the answer the formula gives. It stays in the coil as long as the current keeps flowing, and it is released when the current falls.
$L$	The inductance of the coil, measured in henries (H). A bigger inductance stores more energy at the same current. The worked example uses a 10 millihenry coil, which is 0.01 H.
$I$	The current through the coil, measured in amperes (A). The formula uses I squared, so doubling the current makes the stored energy four times bigger. The worked example uses 2 A, and then 4 A.

The formula piece by piece

$I$: The current through the coil, in amperes. Current is the flow of charge, and one ampere is one coulomb per second. The stored energy depends on how much current flows, so this is the starting point.
$I^2$: The current multiplied by itself. Squaring means that doubling the current makes the square four times bigger, so the stored energy rises quickly as the current rises.
$L$: The inductance of the coil, in henries. It measures how strongly the coil opposes changes in its current, and it sets how much energy a given current stores. Twice the inductance stores twice the energy at the same current.
$L I^2$: Inductance times current squared. This product is twice the stored energy, because the one half in the full formula comes from building the current up from zero, as explained below.
$\frac{1}{2}$: One half. The current starts at zero and climbs to its final value, so the average power during the rise is only half of the power at the end of it.
$E = \frac{1}{2} L I^2$: The whole formula in one sentence: the energy stored in a coil equals one half times its inductance times the square of the current flowing through it.

What the formula is doing, and why

A coil with current flowing through it stores energy in its magnetic field, and the energy stays there as long as the current keeps flowing. The inputs are the inductance $L$ and the current $I$ . The output is the energy $E$ , in joules. Building the current up from zero takes work, because the coil resists each change in current, and that work is what gets stored as magnetic energy.

Why square the current? The voltage across the coil while the current rises is $L\frac{dI}{dt}$ , and the power is that voltage times the current. Power grows in step with the current. A bigger final current also takes a longer rise at the same rate of change. Energy is power multiplied by time, so the current appears twice: once in the power and once in the time it takes to reach. Doubling the current makes the stored energy four times bigger, from 0.02 J at 2 A to 0.08 J at 4 A.

Why one half? Take a current that rises in a straight line from zero to $I$ . At the start the power is zero, and at the end it reaches its full value. The average power over the rise is therefore half of the final power. The energy is that average power multiplied by the time of the rise, and working this through gives $\frac{1}{2} L I^2$ . It is the same half that appears in the capacitor formula, for the same reason.

Bigger inductance stores more. At the same current, a coil with more inductance stores more energy, in a straight line: twice the inductance stores twice the energy. A bigger inductance means the coil opposes changes in current more strongly, so more work is needed to build up the same current.

Where the energy goes. When the current is cut, the stored energy has to go somewhere. It turns into heat in a resistor or a flyback diode, or into a spark across the switch, and the coil produces a large voltage to push the current through that path. This is the reason switching off a coil causes a voltage spike.

An everyday picture

Think of a moving car. Its kinetic energy, the energy of its motion, is $\frac{1}{2} m v^2$ , where $m$ is its mass and $v$ is its speed. Going twice as fast takes four times the energy, which is the same squared rule. Map the pieces onto the coil: the inductance $L$ plays the part of the mass, because it sets how hard the current is to change, and the current $I$ plays the part of the speed. The magnetic field is where the coil keeps this stored motion while the current flows.

Worked example: Energy in a 10 millihenry coil carrying 2 amps

The same 10 mH coil from the inductor voltage example now carries a steady current of $2\ \text{A}$ . These are made-up values, chosen for easy arithmetic. How much energy is stored in its magnetic field? Then find the energy if the current is doubled to 4 A.

Write the formula and list the inputs. The formula is $E = \frac{1}{2} L I^2$ . We have $L = 10\ \text{mH}$ and $I = 2\ \text{A}$ . The current is already in amperes, but the inductance is in millihenries, so it must be converted to henries before we multiply.
Convert the inductance to henries. Milli means one thousandth, $10^{-3}$ . So $10\ \text{mH} = 10 \times 10^{-3}\ \text{H} = 0.01\ \text{H}$ . This is the same conversion used in the inductor voltage example, and it keeps the final unit in joules.
Square the current. Squaring means multiplying the current by itself: $I^2 = (2\ \text{A})^2 = 2\ \text{A} \times 2\ \text{A} = 4\ \text{A}^2$ . We multiply the numbers 2 times 2 to get 4, and the units A times A to get A squared. This squaring is the reason the energy depends so strongly on the current.
Multiply by one half and by the inductance. Now $E = \frac{1}{2} \times 0.01\ \text{H} \times 4\ \text{A}^2 = 0.5 \times 0.01 \times 4 = 0.02$ . Half of 4 is 2, and 2 times 0.01 is 0.02. The number is complete here, and the units are checked in the next step.
Check the units. A henry is a volt-second per ampere, so $\text{H} \cdot \text{A}^2 = \frac{\text{V s}}{\text{A}} \cdot \text{A}^2 = \text{V} \cdot \text{s} \cdot \text{A}$ . A volt times an ampere is a watt, and a watt times a second is a joule. So the answer is 0.02 J, which is the correct unit for energy.
Double the current to 4 A. Now $I^2 = (4\ \text{A})^2 = 16\ \text{A}^2$ , so $E = \frac{1}{2} \times 0.01\ \text{H} \times 16\ \text{A}^2 = 0.08\ \text{J}$ . The current doubled, but the energy grew by $2^2 = 4$ times, from 0.02 J to 0.08 J, because the energy depends on the square of the current.
Convert to millijoules. Milli means one thousandth, so 1 joule is 1000 millijoules. Multiplying $0.02\ \text{J}$ by 1000 gives $20\ \text{mJ}$ . The energy at 2 A is 20 millijoules, which is a small amount compared with a whole joule.

Answer: The coil stores 0.02 J (20 millijoules) at 2 A. At 4 A it stores 0.08 J, four times as much. Doubling the current quadruples the stored energy.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Forgetting the square. Energy goes with $I^2$ , so doubling the current quadruples the energy rather than doubling it.
Forgetting the one half. Without it the answer is twice the true energy.
Mixing up the coil and capacitor formulas. The coil formula uses inductance and current, $E = \frac{1}{2} L I^2$ , while the capacitor formula uses capacitance and voltage, $E = \frac{1}{2} C V^2$ .
Thinking the stored energy is zero when there is no voltage. For a steady current, an ideal inductor has no voltage across it, but its magnetic field still holds energy.

Where you will meet it

Switching power supplies, where an inductor stores energy in its magnetic field during one part of each switching cycle and releases it during another.
Car ignition coils, which store energy in a magnetic field and release it as a high-voltage spark.
Relay and solenoid coils, whose stored energy must be removed safely with a diode when the coil is switched off.
Tuned radio circuits, where energy swings back and forth between a coil and a capacitor.

Questions people ask

Why does the energy depend on the current squared?

Power is voltage times current, and while the current rises the power grows in step with the current. A bigger current also takes a longer rise to reach, so the energy, which is power multiplied by time, picks up a second factor of $I$ . Together those two factors give $I^2$ .

Is the energy in an inductor stored the same way as in a capacitor?

The two formulas have the same shape. A capacitor stores $\frac{1}{2} C V^2$ in its electric field, and an inductor stores $\frac{1}{2} L I^2$ in its magnetic field. The capacitor uses voltage and the inductor uses current, but the one half and the square have the same origin.

Where does the energy go when the current stops?

It is released into the circuit. It often ends up as heat in a resistor or a diode, or as a spark at a switch. A flyback diode gives it a gentle path to dissipate, which keeps the voltage spike small.

Energy Stored in a Capacitor: A charged capacitor holds energy in the electric field between its two plates, and doubling the voltage across it makes the stored energy four times bigger.
Inductor Voltage: A coil of wire makes a voltage across itself only while its current is changing, and the faster the current changes, the bigger that voltage is.
RL Time Constant: The RL time constant tells you how long a coil takes to let its current build up through a resistor: divide the coil’s inductance by the resistor’s resistance.