Magnetic Plasma Confinement: How Fusion Reactors Trap Plasma

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Larmor radius ≈ 4.6 mm — particles spiral tightly around field lines at 3 T.

At 3 T and 10 keV, deuterium ions have a Larmor radius of about 4.6 mm and a cyclotron frequency of ~23 MHz. The magnetic mirror with ratio 2 confines particles with pitch angles greater than 45°.

Formula

r_L = m × v_perp / (q × B) — Larmor radius
ω_c = q × B / m — cyclotron frequency
θ_loss = arcsin(√(B_min / B_max)) — loss cone angle

Trapping a Star on Earth

The core challenge of fusion energy is confinement: how do you hold a plasma at 150 million degrees — ten times hotter than the Sun's core — without it touching any material wall? The answer is magnetic fields. Charged particles spiral around magnetic field lines, effectively trapped in helical orbits. By arranging these field lines into closed loops (as in a tokamak) or complex 3D shapes (as in a stellarator), physicists can suspend a burning plasma in mid-air for seconds or even minutes.

Larmor Orbits and Cyclotron Motion

When a charged particle enters a uniform magnetic field, it traces a helix: circular motion perpendicular to the field combined with free motion along it. The radius of this circular orbit — the Larmor radius — is proportional to the particle's perpendicular momentum and inversely proportional to the field strength. For a 10 keV deuterium ion in a 5-tesla field, the Larmor radius is only a few millimeters, far smaller than the meter-scale confinement vessel. This tight spiraling is what makes magnetic confinement possible.

Magnetic Mirrors and Loss Cones

A magnetic mirror exploits the conservation of the magnetic moment μ = mv²⊥/(2B). As a particle moves into a region of stronger field, v⊥ must increase to keep μ constant, and v∥ decreases. If the field is strong enough, v∥ drops to zero and the particle reflects back. However, particles with small v⊥ relative to v∥ (pitch angles within the 'loss cone') escape through the mirror. The loss-cone angle depends on the mirror ratio R = Bmax/Bmin; higher ratios trap more particles.

From Mirrors to Tokamaks

Simple magnetic mirrors suffer from large loss-cone losses, which is why modern fusion research focuses on toroidal confinement. In a tokamak, the magnetic field lines wrap around a doughnut-shaped vacuum vessel, so particles following field lines never reach an end. The toroidal field provides the primary confinement, while a poloidal field (generated by plasma current) twists the field lines to prevent drift-driven losses. ITER, the world's largest tokamak, will use superconducting magnets producing 11.8 T to confine a plasma of 150 million degrees.

FAQ

How does magnetic confinement work?

Charged particles in a magnetic field experience the Lorentz force, which causes them to spiral around field lines. By shaping the magnetic field into a closed geometry (like a torus), particles are trapped and confined away from material walls. This is the principle behind tokamaks and stellarators.

What is the Larmor radius?

The Larmor radius (gyroradius) is the radius of the circular orbit a charged particle traces perpendicular to a magnetic field. It equals mv⊥/(qB), where m is the particle mass, v⊥ is the perpendicular velocity, q is the charge, and B is the field strength. Smaller Larmor radii mean tighter confinement.

What is a magnetic mirror?

A magnetic mirror is a region of increased magnetic field strength that reflects charged particles back toward weaker-field regions. Particles with sufficiently large perpendicular velocity (pitch angle above the loss-cone angle) are reflected. Those with too much parallel velocity escape through the 'loss cone'.

Why is plasma confinement so difficult?

Plasma confinement is challenging because plasmas are inherently unstable. Instabilities like interchange, ballooning, and kink modes can rapidly degrade confinement. Additionally, collisions scatter particles into the loss cone, and turbulent transport moves energy outward much faster than classical diffusion predicts.

Sources

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