Faraday Induction & Plasma

1. Faraday's Law of Induction

Michael Faraday discovered in 1831 that a changing magnetic field through a loop induces an EMF. Maxwell expressed this as:

The negative sign is fundamental — it expresses Lenz's law. The flux Φ_B can change because:

• The magnitude of B changes with time.
• The loop moves relative to a static field.
• The loop area or orientation changes (as in an electric generator).

The motional EMF in a conductor of length L moving with velocity v perpendicular to field B is: EMF = BLv. This is the operating principle of every electric generator.

2. Lenz's Law & Energy Conservation

Lenz's law states that the induced current flows in a direction that opposes the change that created it. This is a direct consequence of energy conservation — if the induced current reinforced the change, energy would be created from nothing.

Eddy currents are induced in bulk conductors exposed to changing fields. They are used beneficially in induction cooktops and magnetic braking, and are minimised by laminating transformer cores to reduce losses.

3. Self-Inductance & Mutual Inductance

A coil carrying current creates a magnetic field that links its own turns — this is self-inductance:

When two coils are near each other, flux from coil 1 links coil 2 — this is mutual inductance M:

In LC circuits, energy oscillates between the inductor (magnetic field) and capacitor (electric field) at the resonant frequency ω₀ = 1/√(LC). Real circuits include resistance and the oscillations decay, just as mechanical pendulums do.

4. Transformers

An ideal transformer with N₁ primary and N₂ secondary turns obeys:

Real transformers lose energy through: core hysteresis losses (B–H curve area), eddy currents (reduced by lamination), resistive winding losses (I²R), and leakage flux. Modern power transformers achieve >99% efficiency.

High-voltage transmission lines operate at hundreds of kV because for fixed power P = VI, higher voltage means lower current, and resistive losses scale as I²R. Step-up and step-down transformers mediate between generation (10–25 kV) and long-distance transmission (100–765 kV) and local distribution (240 V).

5. Magnetohydrodynamics (MHD)

Magnetohydrodynamics treats plasma as a single conducting fluid coupled to electromagnetic fields. The MHD equations combine Navier–Stokes with Maxwell:

6. The Magnetic Pinch Effect

When current flows through a plasma column, the resulting azimuthal magnetic field B_φ creates a J × B force directed inward — squeezing or "pinching" the plasma:

The z-pinch is inherently unstable: the sausage instability (m=0 kink) and kink instability (m=1) destroy confinement in microseconds. The tokamak stabilises confinement geometry by adding a strong toroidal field and shaping the plasma into a torus — the approach used by ITER and commercial ventures like Commonwealth Fusion's SPARC.

The theta-pinch (current along θ, field along z) reverses the geometry and is also unstable. The field-reversed configuration (FRC) and spheromak are compact torus concepts pursued by private fusion companies.

7. JavaScript — MHD Flux Tube Simulation

A simplified 2D MHD simulation tracking field line advection under a prescribed velocity field (flux-freezing regime, Rₘ ≫ 1).

// 2D flux-frozen field line advection (ideal MHD, Rm >> 1)
// Field lines represented as chains of points; advected by fluid velocity

class FluxTube {
  constructor(points) {
    // points: [{x, y}, ...]
    this.pts = points.map(p => ({x: p.x, y: p.y}));
  }

  advect(velocityFn, dt) {
    for (const p of this.pts) {
      const v = velocityFn(p.x, p.y);
      p.x += v.vx * dt;
      p.y += v.vy * dt;
    }
  }

  draw(ctx, color = '#f87171') {
    ctx.beginPath();
    ctx.strokeStyle = color;
    ctx.lineWidth = 2;
    this.pts.forEach((p, i) =>
      i === 0 ? ctx.moveTo(p.x, p.y) : ctx.lineTo(p.x, p.y)
    );
    ctx.stroke();
  }
}

// Solar-convection-like velocity: rotating cells
function convectiveVelocity(x, y, t = 0) {
  const scale = 0.02;
  // Stream function ψ = cos(scale·x)·cos(scale·y)
  // vx = ∂ψ/∂y, vy = −∂ψ/∂x
  return {
    vx:  Math.cos(scale * x) * (-Math.sin(scale * y)) * scale * 50,
    vy:  Math.sin(scale * x) *   Math.cos(scale * y)  * scale * 50,
  };
}

// Faraday induction EMF integrator
function computeEMF(Bfield, loop, dt) {
  // loop: array of {x,y} vertices (closed polygon)
  // Φ = ∫∫ B · dA  (Shoelace-like area integration)
  let flux = 0;
  const N = loop.length;
  for (let i = 0; i < N; i++) {
    const x = (loop[i].x + loop[(i + 1) % N].x) / 2;
    const y = (loop[i].y + loop[(i + 1) % N].y) / 2;
    flux += Bfield(x, y); // B_z component
  }
  flux *= 1; // multiply by cell area for proper integral
  return -flux / dt; // EMF = −dΦ/dt (Faraday)
}

// Dipole magnetic field (2D cross-section)
function dipoleBfield(x, y, mx = 0, my = 0, m = 1e4) {
  const mu0 = 4e-7 * Math.PI;
  const dx = x - mx, dy = y - my;
  const r2 = dx ** 2 + dy ** 2;
  if (r2 < 1) return {bx: 0, by: 0};
  const r = Math.sqrt(r2);
  const pre = (mu0 * m) / (4 * Math.PI * r2 ** 2);
  return {
    bx: pre * (3 * dx * dy),       // ∝ 3 cos θ sin θ / r³
    by: pre * (2 * dy ** 2 - dx ** 2), // ∝ (3 cos²θ − 1) / r³
  };
}

// Bennett pinch: radial balance for a z-pinch current column
function bennettPressure(I, r, R) {
  // Uniform current density J = I/(πR²), pressure inside r < R
  const mu0 = 4e-7 * Math.PI;
  const J = I / (Math.PI * R ** 2);
  const p0 = (mu0 * J ** 2 * R ** 2) / 4; // peak pressure at r=0
  return r < R ? p0 * (1 - r ** 2 / R ** 2) : 0;
}

8. Applications

• Electric motors & generators: All rotating electrical machines operate on Faraday induction — motional EMF in conductors crossing a magnetic field.
• Wireless charging: Inductive coupling between coils transfers energy via mutual inductance at 100–200 kHz (Qi standard).
• MRI scanners: Rapidly switched gradient coils induce current in the patient (RF coils) and back-induce EMFs for detection.
• Tokamaks (ITER, SPARC): MHD confinement using toroidal + poloidal fields targeting plasma temperatures >150 million K to achieve D-T fusion ignition.
• Stellar dynamos: Differential rotation and convection stretch and amplify magnetic field lines (Ω and α effects), generating the solar magnetic cycle (~11 years).
• Geodynamo: Convective motion of Earth's liquid iron outer core sustains the global magnetic field via MHD induction.