EM-19 · Magnetostatics field

Magnetic Energy

The energy stored in a coil or a magnetic field in fact resides in the magnetic field itself. See the energy density grow as B² as you strengthen the field, and learn that a coil's energy is the field's energy (u = B²/2μ).

Strengthen the field and watch the energy

Use the slider to strengthen the magnetic field inside the coil. How does the glow filling the coil (the energy density) brighten? When you double the field, by how much does the energy grow?

Field strength BB = 4.0

Drag to orbit. The slider sets the field strength B.

The energy density in the field

u = B²/2μ · energy lives in the magnetic field

U = ½LI² = ∫u dV · the coil energy is the sum of field energy

Energy density (vs maximum) · 16%

Middle

Where the energy is

Building up the current in a coil stores energy (EM-17). But where is that energy? It seems to sit on the current, yet the deeper answer is that it resides in the magnetic field the coil makes. Cut the current and, as long as a field lingers, the energy lingers there too. Magnetic energy is spread through every part of space the field fills.

Energy density u=B²/2μ

The energy per unit volume at a point where a field exists is the energy density u: u = B²/(2μ). The key is that it scales with the square of the field. Double the field and the energy density quadruples, so even a modest increase makes the energy climb steeply. It closely mirrors the electrostatic u = ½εE². The glow inside the coil brightening as B² on the first screen is exactly this relation.

Observeu = B²(2μ)

Energy density is B squared over 2μ.

Chooseu ∝ ?

Double the field and the energy density quadruples.

It concentrates where the field is strong

Since the density goes as B², energy concentrates where the field is strong. In a solenoid the field is uniform inside and almost all the energy is trapped within the coil; outside the field is weak and so is the energy. That is why electromagnets and transformers confine the field in a narrow iron core to gather energy efficiently. Given just the field map, you can read off where and how much energy is stored.

Coil energy = field energy

Two views meet. The circuit view’s coil energy U = ½LI² and the field view’s ∫u dV (the density integrated over volume) are exactly equal. The solenoid confirms it directly: putting B = μnI and the volume Al into u = B²/(2μ) reproduces ½LI². Counting the same energy through the current or through the field gives one answer. This view that energy lives in the field, as we saw for electrostatic energy, leads to why electromagnetic waves can carry energy through empty space.

Fill inU = u ?

Total energy is density times volume.

On your ownU = ?

The energy stored in the coil equals ½LI².

Back to the first screen

On the first screen, strengthening the field made the glow inside the coil brighten steeply: double the field, four times the brightness. That is because the energy density follows the square of the field, u = B²/(2μ). The glow was the energy spread through space — the field, not the current, held it. The ½LI² gathered in the coil was exactly the sum of B²/(2μ) over every bit of volume.

Magnetic energy resides not in the current but in the magnetic field itself. The energy density per unit volume is u = B²/(2μ), so energy concentrates where the field is strong (it scales with the square of the field). The total energy is this density integrated over volume, U = ∫u dV, exactly equal to ½LI² for a coil. Doubling the field B quadruples the energy density.

What comes next

With this, the seven pieces of magnetostatics are complete — from Biot–Savart through Ampère, the Lorentz force, inductance, magnetic materials, and magnetic energy: the world of steady currents. Until now electrostatics and magnetostatics stood apart. But once fields vary in time, the two begin to interweave. The next block opens with Faraday's law (EM-20), where a changing magnetic field makes an electric field. The view that energy lives in fields reaches its peak with the Poynting vector of electromagnetic waves (EM-24).