JEE/NEET Physics · Moving Charges & Magnetism series · Part 2 of 8 · All parts →
- Charge moving ⊥ to B: uniform circular motion — B supplies the centripetal force
- Radius: r = mv/(qB) — fast or heavy means wide; strong field or big charge means tight
- Period: T = 2πm/(qB) — independent of SPEED (the cyclotron’s secret)
- Velocity with a component along B: helix (circle + slide)
- Work done by B is always zero: speed never changes
Fire a charge across a magnetic field and it can’t escape — the sideways force bends it into a perfect circle. The field is the ultimate invisible roundabout, and the radius of the circle tells you the particle’s momentum. Part 2 of the Moving Charges & Magnetism series.
- The magnetic roundabout
- Radius and period
- Speed-blind timing
- The helix
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
The Magnetic Roundabout
Magnetic force ⊥ velocity is exactly the job description of centripetal force (the Circular Motion card): set qvB = mv²/r and the circle appears by itself. Unlike friction or gravity, the field never tires and never changes the speed — steering only, forever.
Radius and Period
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| r | orbit radius | m |
| T | period of one lap | s — no v inside! |
| m, q | particle’s mass and charge | kg, C |
Speed-Blind Timing
The miracle: fast particles travel bigger circles in the SAME time — T depends only on m, q, B. A slow particle’s small circle and a fast one’s big circle complete together, like runners on concentric tracks finishing in a tie.
The Helix
Velocity partly along B: the parallel component slides freely (no force along B), the perpendicular part circles — the path is a helix, a spiral around the field line. This is how charged particles spiral along Earth’s field lines to the poles.
Solved Examples
r = mv/qB = (9.1×10⁻³¹ × 10⁶)/(1.6×10⁻¹⁹ × 0.01) ≈ 5.7×10⁻⁴ m — about half a millimetre.
✔
Answer: ≈0.57 mm
T = 2πm/(qB) = 2π(9.1×10⁻³¹)/(1.6×10⁻¹⁹ × 0.01) ≈ 3.6 ns — a billion laps per second-ish.
✔
Answer: ≈3.6 ns
r ∝ m (same q, v): r₂₃₈/r₂₃₅ = 238/235 ≈ 1.013 — a 2 mm separation on a 15 cm track.
Enough to separate uranium isotopes: the Manhattan Project’s calutrons were this exact card.
✔
Answer: 238 : 235
- Changing speed inside B. Never — magnetic forces steer only; if speed changes, something electric is also present.
- Period depending on v. It doesn’t — T = 2πm/qB: fast particles simply ride wider circles.
- Helix forgotten for angled entry. Component along B slides unchecked — the particle escapes along the field line while circling.
- Using the wrong m for electrons vs protons. Same formula, 1836× different mass: radii and periods differ correspondingly.
This Physics in Your Daily Life
- Mass spectrometers — r = mv/qB separates molecules by mass: drug tests, forensics, and doping labs all measure circle sizes.
- Van Allen belts — charged particles helixing along Earth’s field lines: radiation belts discovered by this geometry.
- Auroras spiral along field lines to the poles — the helix in action at planetary scale, glowing where they meet the atmosphere.
- Cyclotrons in hospitals — proton therapy beams circle up to cancer-killing energies: r = mv/qB written into medical hardware.
- Bubble chambers of particle physics history — trails of tiny circles, each radius a particle’s identity card: discovery by curvature.
Tie a ball to a string and swing: the string’s inward pull makes a circle. A magnetic field is an invisible string that never shortens, never snaps, never lets go — and never changes the speed. The result can only be perfect uniform circular motion.
Double the speed: the circle doubles its radius, but the lap time (2πm/qB) hasn’t changed — the fast particle’s longer path exactly cancels its speed. Triple it: same again. It’s the only force law with this property, and it’s why cyclotrons work at all.
Draw velocity arrows around a circle, each ⊥ to its radius, with the magnetic force arrow repeatedly turning the velocity’s tip: the arrows trace the circle like a compass drawing it. Add a slide-along-B arrow and the drawing becomes a spring-shape: the helix.
Practice set (answers hidden — try first)
(NEET-level) Doubling v in fixed B: r
(JEE Main-level) Doubling B at fixed v: r
(NEET-level) T depends on:
(Concept) Charge with velocity component along B moves in a
(JEE Main-level) Proton vs electron, same v, B: r_p/r_e =
- qvB = mv²/r → circles
- r = mv/(qB) — a momentum meter
- T = 2πm/(qB), speed-independent
- angled entry → helix
- speed never changes in B alone
- 🔁 circular motion condition
- 🔁 radius and period formulas
- 🔁 speed independence
- 🧠 Chant: ‘fast rides wide, arrives on time’.
- 🧠 Radius reads momentum — mass spectrometry in one line.
- 🏠 Daily: auroras spiral on field lines.
- 🏠 Daily: hospital cyclotrons circle protons to therapy energy.
Quick revision
- Charge moving ⊥ to B: uniform circular motion — B supplies the centripetal force
- Radius: r = mv/(qB) — fast or heavy means wide; strong field or big charge means tight
- Period: T = 2πm/(qB) — independent of SPEED (the cyclotron’s secret)
- Velocity with a component along B: helix (circle + slide)
- Work done by B is always zero: speed never changes
- This physics in your daily life
Have a doubt on this topic?
