MC-A2 · Inductance & magnetic energy

Inductance and Magnetic Energy

Building current in a coil takes work against the rising flux, and that work does not vanish — it pools in the magnetic field. Sweep the current, watch the stored energy grow, and read off W = ½LI².

How fast does the stored energy grow with current?

The area under the line λ = LI is the magnetic energy stored in the coil. Push the current slider and watch the triangle grow. If you double the current, how many times larger is the area?

Current II = 1.5 A

Slide for continuous current. Triangle area = stored energy.

Stored magnetic energy

W = ½ L I² = 0.56 J

Half the product of current and flux linkage. It grows as the square of current, not linearly, so doubling the current quadruples the energy.

Double the current — W becomes?

Misaligned

Inductance is the flux-linkage efficiency

Inductance L is the flux linkage produced per unit current: L = λ/I = NΦ/I. Through the magnetic circuit Φ = NI/R, so λ = NΦ = N²I/R and therefore L = N²/R. It scales with the square of the turns and inversely with reluctance.

ObserveL = λI

Inductance is flux linkage per unit current.

ChooseL = ?

Insert the magnetic circuit and L becomes N²/R.

Why energy grows as the square

Raising the current by di costs work i·dλ = Li·di. The larger the current, the larger the flux linkage to push against, so the same di costs more. Summing from 0 to I gives the triangular area under the line, ½LI². The one-half is because the current averaged its way from 0 up to I.

Fill indW = i ? = Li di

The tiny work to raise the current further.

On your ownW = ?

The triangle summed from 0 to I.

The same energy in current and inductance

Put λ = LI into ½λI to get W = ½LI², and put I = λ/L back in to get W = ½λ²/L. The three forms are the same area written with different pairs of current, flux linkage and inductance. A larger inductance stores more energy at the same current.

Back to the first screen

When you doubled the current the triangle’s area quadrupled because the area scales both base (current) and height (flux linkage = LI) together. Both are proportional to current, so their product grows as the square. Hence the stored energy is W = ½LI², and the one-half marks the current averaging from 0 up to I.

The magnetic energy: a coil of inductance L carrying current I stores W = ½LI² in its field. The inductance L = λ/I = N²/R is flux linkage per unit current. The same quantity is also W = ½λI = ½λ²/L. The work done raising the current pools in the field and flows back out as the current falls.

Once you hold this energy

½LI² is the bridge between circuits and machines. A transformer’s leakage inductance, a motor’s torque production, an inverter’s switching energy — all are the giving and taking of this energy pooled in coils. The next unit’s ideal transformer shares one flux between two coils to move energy with almost no loss (MC-A3).