Magnetic Field

Lorentz force

Force on a charge q moving at velocity v in a magnetic field B:

F = q · v × B

The force is:

• Perpendicular to BOTH v and B (right-hand rule).
• Magnitude q·v·B·sin θ (θ = angle between v and B).
• Zero if v parallel to B (no force on charges moving along field).
• Maximum when v perpendicular to B.

Combined with electric force (F_total = q·E + q·v×B), this is the full Lorentz force.

Circular motion in B field

Charge q moving perpendicular to uniform B feels constant centripetal force:

q·v·B = m·v² / r
r = m·v / (q·B)

And period:

T = 2π·m / (q·B)  (cyclotron period; independent of v!)

Field magnitudes

Source	B
Earth's surface (avg)	50 µT
Refrigerator magnet	5 mT (50 G)
Bar magnet (near pole)	10 mT
Inside an MRI	1.5-7 T
Strongest commercial magnets	~10 T (electromagnets), ~1.5 T (permanent)
High-field research	30-45 T (steady), ~100 T (pulsed)
Neodymium magnet (small)	1-1.5 T at surface
Pulsar magnetic field	~10⁸ T (extreme)
Magnetar (neutron star)	~10¹¹ T (most extreme known)

JavaScript — magnetic field calculations

// Lorentz force on a charge
function lorentzForce(q, v_vec, B_vec) {
  // F = q · (v × B)
  const cross = [
    v_vec[1]*B_vec[2] - v_vec[2]*B_vec[1],
    v_vec[2]*B_vec[0] - v_vec[0]*B_vec[2],
    v_vec[0]*B_vec[1] - v_vec[1]*B_vec[0]
  ];
  return cross.map(c => q * c);
}

// Electron moving 1e6 m/s in 1 T field
const F = lorentzForce(-1.602e-19, [1e6, 0, 0], [0, 0, 1]);
console.log(F);  // [0, -1.6e-13, 0] N (downward y)

// Cyclotron radius
function cyclotronRadius(mass, velocity, charge, B) {
  return mass * velocity / (Math.abs(charge) * B);
}

// Electron in 1 T field, 1e6 m/s
const r_e = cyclotronRadius(9.11e-31, 1e6, 1.602e-19, 1);
console.log(`Electron radius: ${(r_e * 1e6).toFixed(2)} µm`);  // ~5.7 µm

// Cyclotron frequency (independent of speed!)
function cyclotronFrequency(charge, B, mass) {
  return Math.abs(charge) * B / (2 * Math.PI * mass);
}

// Proton in 1 T
const f_p = cyclotronFrequency(1.602e-19, 1, 1.673e-27);
console.log(`Proton cyclotron at 1T: ${(f_p / 1e6).toFixed(1)} MHz`);  // 15.2

// Magnetic field from a long straight wire
const mu_0 = 4 * Math.PI * 1e-7;  // T·m/A
function fieldFromWire(I, distance) {
  // B = μ_0·I/(2π·r)
  return mu_0 * I / (2 * Math.PI * distance);
}

// 10 A wire, 1 cm away
console.log(`B from 10A wire at 1cm: ${(fieldFromWire(10, 0.01) * 1e6).toFixed(0)} µT`); // ~200

// Solenoid field (interior)
function solenoidField(n_turns_per_meter, current) {
  return mu_0 * n_turns_per_meter * current;
}

// 1000 turns/m, 5 A
console.log(`Solenoid: ${(solenoidField(1000, 5) * 1000).toFixed(2)} mT`);  // 6.28

// Earth's field deflection
function earthDeflection(speed, charge, mass, B = 50e-6) {
  // Cyclotron radius
  return cyclotronRadius(mass, speed, charge, B);
}

// Cosmic ray proton (1e8 m/s) in Earth's field
console.log(`Cosmic proton radius in Earth's field: ${earthDeflection(1e8, 1.602e-19, 1.673e-27).toFixed(0)} m`);
// Massive radius — barely deflected (more deflection at higher latitudes due to angle)

// Tesla in different units
function teslaToGauss(T) { return T * 1e4; }
function gaussToTesla(G) { return G * 1e-4; }

console.log(`1 T = ${teslaToGauss(1)} G`); // 10000
console.log(`Earth ~ 0.5 G = ${gaussToTesla(0.5).toExponential(2)} T`); // 5e-5

Where magnetic fields matter

• Motors and generators. Convert between electrical and mechanical via magnetic forces and induction.
• Particle accelerators. Bend charged particle beams in circular paths.
• MRI imaging. Strong B fields align nuclear spins; modulate to image tissues.
• Compasses and navigation. Earth's field; magnetometers in phones.
• Plasma confinement. Tokamaks contain hot fusion plasmas with strong B fields.
• Mass spectrometry. B field separates ions by mass-to-charge ratio.
• Astrophysics. Magnetic fields in stars, galaxies, jets, accretion disks. Earth's magnetosphere protects from solar wind.

Common mistakes

• Confusing magnetic and electric fields. They're related but distinct (different units, sources, force laws). Maxwell unifies them but for most applications, treat separately.
• Thinking magnetic force does work. It doesn't — F is always perpendicular to v, so dW/dt = F·v = 0. Only changes direction; energy comes from work done by electric fields or other agents.
• Treating magnets as monopoles. Magnetic monopoles haven't been observed (despite searches). Magnets always have N AND S poles paired.
• Wrong direction with right-hand rule. Use right hand for positive charges; for negative (electrons), reverse direction.
• Confusing T and G. Earth's field is ~50 µT = 0.5 G. Watch units in tables and equations.
• Forgetting field can vary. Earth's field varies with location and time. Magnets have non-uniform fields. For complex situations, careful 3D analysis needed.