ENERGY IN MAGNETIC FIELDS

Wire up a battery, a resistor, and an inductor in a loop. Close the switch. The current doesn't leap to its final value — it ramps, slowly, climbing an exponential curve with time constant τ = L/R. During that ramp the battery is pushing charges against something: the self-induced EMF the coil is generating in opposition to its own rising current. Joseph Henry named it back-EMF and the name stuck.

Pushing against a force costs work. W = ∫V·I dt is the shape of that work, where V = L·dI/dt is the voltage the inductor presents back to the source. The battery pays it out current-second by current-second, and the energy has to go somewhere. It's not heat — ideal inductors are lossless; take the resistor out and nothing warms up. It's not kinetic energy — the wire is bolted to the bench. So where is it?

It's in the magnetic field the current builds.

The answer to "how much?" falls out of the integral cleanly. Start with zero current and ramp to I. At each instant the source delivers power P = V·I = L·I·(dI/dt). Integrate from zero to the final current:

That ∫ is just the running total. Read it in words: "add up, across all the intermediate currents between 0 and I, the little bit of work done to push each extra ampere against the back-EMF." The total comes out to the elegant half-product ½ L I².

Watch the amber trace approach its asymptote and the magenta one approach the square of that asymptote. The green-cyan back-EMF arrow fades as dI/dt decays — at steady state the coil is a short circuit and the energy reservoir is full.

½ L I² tells us how much energy a specific inductor holds — a quantity that depends on the coil's geometry through L. The deeper question is: how much energy is there per unit volume of magnetic field, regardless of what's making the field?

The cleanest place to pin it down is a long solenoid. For that geometry we already know two things. The self-inductance is L = μ₀ N² A / ℓ (from the self-inductance topic). The interior field is B = μ₀ N I / ℓ — uniform, axial, zero outside. The energy stored is ½ L I². The volume where the field lives is A · ℓ. Divide energy by volume and simplify:

Three substitutions and the coil geometry falls out. What's left is a purely local statement: every cubic metre of space where the magnetic field has magnitude B holds B²/(2μ₀) joules, and nothing about how that field got there matters. It is a property of the field itself.

Slide the |B| slider. The left bar (the field magnitude) moves linearly; the right bar (the energy density) moves as the square. At B = 1 T the density is about 398 kJ/m³ — a single cubic metre of one-tesla field carries enough to run a hundred-watt lamp for over an hour. MRI scanners sit at 1.5–3 T; their cryogenic coils store megajoules in the field that images your knee.

There's a pattern here you've seen before. In §01 we found that charging a capacitor stores U = ½CV² — and that every cubic metre between the plates holds u = ½ε₀E². Now the same arithmetic has landed on its magnetic twin.

Capacitance and inductance are geometric reservoirs; voltage and current are the charging variables; ½ε₀E² and B²/(2μ₀) are the field densities. Michael Faraday's instinct — that fields are physical things with locatable properties, not bookkeeping devices — is what makes this symmetry more than algebra. Both densities integrate to the total energy. Both formulas say the same deep thing: the field is a reservoir.

Inside ordinary matter, the clean vacuum form needs a small amendment. For a linear material (most of them), the correct expression is

where H is the auxiliary field. In the linear regime B = μH with μ = μ₀(1 + χ_m), and this reduces to B²/(2μ) — the vacuum formula with μ₀ replaced by the absolute permeability of the material. A ferrite core with μ_r = 2000 stores two thousand times more energy at the same B than vacuum would. That's why transformer and inductor designers chase high-μ cores.

For nonlinear materials — the ferromagnets that walk a hysteresis loop — the single-valued formula breaks down and the true energy stored is the area enclosed as H sweeps from 0 to its working point. That enclosed area, each time round the loop, is the energy lost as heat: the hysteresis-loss story from §04.3.

Every switching power supply on your desk parks a few microjoules in an inductor's field thousands of times per second, handing them on to a capacitor and around the loop — that's how a laptop brick converts 120 V AC to 19 V DC without wasting half as heat. Superconducting magnetic energy storage (SMES) plants hold megajoules in ring-shaped coils at liquid-helium temperatures, releasing them in milliseconds to smooth grid transients. Tokamak fusion reactors pump gigajoules into the toroidal field that confines their plasma; the field is literally holding a star in a jar. MRI coils need kilowatt-minutes just to ramp up — the magnet doesn't light up; it fills.

And the hook for §07 Maxwell: this energy density is what lets electromagnetic waves exist. Light is an oscillation of E and B in free space, carrying ½ε₀E² + B²/(2μ₀) joules per cubic metre as it propagates — no wires, no coils, no battery. The Sun warms your face because a field reservoir built eight minutes ago in a plasma 150 million kilometres away arrives at your skin and is reabsorbed. Start with ½LI² and you have already glimpsed the radiant edge of electromagnetism.