Charged Particles in Electric and Magnetic Fields Learn

A charged particle feels a force in an electric field and a different, velocity-dependent force in a magnetic field. Together these are the Lorentz force, and they control the particle's path: electric fields speed it up or slow it down, while magnetic fields bend its path without changing its speed.

This physics steers electrons in old TV tubes, separates isotopes in mass spectrometers, focuses beams in accelerators and explains the aurora. Mastering how the two fields act differently is the key to all charged-particle optics.

Key Concepts

Lorentz force: total electromagnetic force on charge q. Electric force: along (or opposite) E, independent of velocity; it does work and changes kinetic energy. Magnetic force: perpendicular to both velocity v and field B; it does no work and changes only direction. Radius of circular path in B: r = mv/(qB). Velocity selector: crossed E and B where forces cancel for one speed, v = E/B.

Formulas

F = q(E + v × B), magnitudes: FE = qE and FB = qvB sinθ; forces in newton (N), E in V/m, B in tesla (T), v in m/s, q in coulomb (C). Circular motion in uniform B: qvB = mv²/r, so r = mv/(qB) and period T = 2πm/(qB), independent of speed. Assumption: non-relativistic speeds and uniform fields. Limit: at speeds near light, relativistic mass increase changes radius and period.

Worked Example

A proton (m = 1.67 × 10⁻²⁷ kg, q = 1.6 × 10⁻¹⁹ C) moves at 1.0 × 10⁶ m/s perpendicular to B = 0.50 T. Find its path radius.

1. r = mv/(qB) = (1.67 × 10⁻²⁷ × 1.0 × 10⁶)/(1.6 × 10⁻¹⁹ × 0.50).
2. Numerator = 1.67 × 10⁻²¹. Denominator = 8.0 × 10⁻²⁰.
3. r = 0.021 m ≈ 2.1 cm.

The proton circles with radius about 2.1 cm. Doubling its speed would double the radius but keep the revolution period unchanged in the non-relativistic limit.

Common Misconceptions

• Magnetic force speeds particles up: It is always perpendicular to motion, so speed and kinetic energy stay constant; only direction changes.
• Electric force depends on velocity: FE = qE works even for a stationary charge; the magnetic part vanishes at rest instead.
• Heavier and lighter particles circle with the same radius: Radius grows with momentum mv and shrinks with charge q, so mass matters.

Real-World Applications

• Mass spectrometers and velocity selectors in analytical instruments.
• Cathode-ray and electron microscopes, where fields focus electron beams.
• Particle accelerators, aurora formation and magnetic confinement in fusion research.

Electric fields change a charge's energy, magnetic fields change only its direction; the Lorentz force combines both and dictates every curved or straight path a charged particle takes.