Helical MotionJEE Advanced

Concept

When a charge enters a magnetic field at an angle, the perpendicular velocity component is bent into a circle by the Lorentz force while the parallel component is untouched. The two together trace a helix whose radius, period and pitch follow directly from the cyclotron motion.

Key formula

Derivation

Split $\vec v$ into $v_\parallel = v\cos\alpha$ (along $\vec B$) and $v_\perp = v\sin\alpha$. The magnetic force $q\vec v\times\vec B$ has no component along $\vec B$, so $v_\parallel$ is constant.

The perpendicular part feels a central force $qv_\perp B = mv_\perp^2/r$, giving radius $r = mv_\perp/(qB)$ and period $T = 2\pi m/(qB)$ — note $T$ is independent of speed.

During one revolution the particle advances $p = v_\parallel T$ along the axis: the pitch of the helix.

Scenarios to explore

• Helical Motion — Charged particle spiralling in a magnetic field.
• Strong field, tight coil — A larger B shrinks the radius and speeds up the circling.

Real-world applications

• Mass spectrometers and cyclotron accelerators.
• Charged particles trapped in Earth's magnetic field (auroras).
• Magnetic confinement in fusion devices.

JEE exam tips

• At α = 90° the path is a pure circle (zero pitch); at α = 0° it is a straight line.
• The magnetic force does no work, so the speed — and kinetic energy — stay constant.

Common mistakes

• Using the full speed in the radius formula instead of $v_\perp$ only.
• Thinking the period depends on speed — it does not for non-relativistic motion.
• Forgetting that the parallel velocity is completely unaffected by B.

Exam traps to avoid

• Pitch is the axial advance per full turn, not per radian.
• Reversing the charge sign reverses the sense of circulation, not the radius.