Q.Two protons of equal kinetic energies enter a region of uniform magnetic field. The first proton enters normal to the field direction while the second enters at 30∘ to the field direction. Name the trajectories followed by them.
Assam AhsecCBSE Class XII Board 2018Subjective· 1mImportance★★★★★
Concept understanding — Charged Particle in Magnetic Field
Charged Particle in a Magnetic Field
When a charged particle moves through a magnetic field, the field grabs it sideways. Unlike an electric field, which can speed a charge up or slow it down, a magnetic field only bends the path — it never changes the particle's speed. Understanding why leads directly to circular and helical motion, the basis of cyclotrons, mass spectrometers and the aurora.
The force: always sideways
A particle of charge q moving with velocity v in a magnetic field B feels the magnetic (Lorentz) force:
F=q(v×B)
Because of the cross product, F is perpendicular to bothv and B. Its magnitude is
F=∣q∣vBsinθ
where θ is the angle between v and B.
Important
Since F⊥v, the force does no work: F⋅v=0. Therefore the kinetic energy and the speed stay constant — the field only changes the direction of motion, never the magnitude.
Case 1: velocity perpendicular to the field → a circle
If v⊥B (θ=90∘), the force F=qvB stays constant in size and always points toward one central point. That is exactly the condition for uniform circular motion, with the magnetic force acting as the centripetal force:
qvB=rmv2
Solving for the radius:
r=qBmv
The time period of one revolution is
T=v2πr=qB2πm
Note
The period T (and the frequency f=qB/2πm, the cyclotron frequency) does not depend on the speed or the radius. A faster particle traces a bigger circle but takes exactly the same time per loop. This speed-independence is what makes the cyclotron work.
Case 2: velocity at an angle → a helix
If v makes an angle θ with B, split it into two parts:
Perpendicular componentv⊥=vsinθ — feels the magnetic force and drives circular motion of radius r=qBmv⊥.
Parallel componentv∥=vcosθ — feels no force (since v∥×B=0) and carries the particle steadily along the field line.
Combining a circle with a steady drift gives a helix. The distance advanced along the field in one full turn is the pitch:
p=v∥T=vcosθ⋅qB2πm
A quick example
An electron (m=9.1×10−31kg, q=1.6×10−19C) enters a 0.02T field at 106m/s, perpendicular to B:
Charged Particle in a Magnetic Field — Why the Key Formulas Hold
Let's build this from first principles. The core idea is that a magnetic field exerts a force only on a moving charge, and that force is always perpendicular to both the velocity and the field.
1. The Fundamental Force Law: Lorentz Force
The starting point is the Lorentz force for a charge q moving with velocity v in a magnetic field B:
Fm=q(v×B)
Why this form?
Cross productv×B means the force is perpendicular to both v and B.
Magnitude: Fm=∣q∣vBsinθ, where θ is the angle between v and B.
Direction: given by the right-hand rule (for positive q).
Key insight: Because Fm⊥v, the magnetic force does no work — it changes only the direction of velocity, not its speed.
2. Circular Motion in a Uniform Magnetic Field
Consider a charge q moving with speed v perpendicular to a uniform B (so θ=90∘, sinθ=1).
Step 1: Force provides centripetal acceleration
The magnetic force is the only radial force:
Fm=qvB
This must equal the centripetal force required for circular motion:
Fc=rmv2
Step 2: Equate and solve for r
qvB=rmv2
Cancel one v (assuming v=0):
qB=rmv
Thus:
r=qBmv
This is the radius of the circular path (cyclotron radius).
Why this makes sense:
Larger mass m → harder to turn → larger r
Larger charge q or stronger B → stronger force → tighter turn → smaller r
Faster speed v → more momentum → larger r
3. Angular Frequency (Cyclotron Frequency)
From the circular motion relation:
ω=rv
Substitute r=qBmv:
ω=qBmvv=mqB
Thus:
ωc=mqB
Why this is remarkable:
ωc is independent of speed v — all particles with same q/m have the same angular frequency, regardless of how fast they move.
This is the principle behind cyclotrons (particle accelerators).
The magnetic force qv×B acts only on the velocity component perpendicular to B; the parallel component is unaffected. A proton entering normal (90∘) has no parallel component, while one at 30∘ has bo …
Concept. The Lorentz force F=qv×B is always perpendicular to v, so it changes direction but not speed. Only the velocity component ⊥B feels a force; the component ∥B moves uniformly.
Why this trajectory.
For the first proton (θ=90∘): v∥=vcos90∘=0, so the whole velocity is perpendicular. The force provides centripetal force and the path is a circle of radius r=qBmv. …
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
AHSEC Higher Secondary (HS) Final Examination 2025Set ANNUAL1 mark
Q.In case of an electron moving along a helical path what determines the pitch of motion?
›Reveal solutionSolution
Pitch = (velocity component along B) × (time period of one revolution).
When a charged particle (electron) enters a magnetic field B⃗ with velocity at an angle to B, its velocity can be resolved into two components: v⊥ (perpendicular to B, responsible for the circular motion) and v∥ (parallel to B, unaffected by the magnetic force since the force qv×B is zero along B).
The perpendicular component v⊥ makes the electron move in a circle of radius r = mv⊥/(qB), with time period T = 2πm/(qB) — this period is independent of speed, depending only on q, m, and B.
Meanwhile the parallel component v∥ is unaffected by the field and carries the electron steadily along the direction of B, so in one revolution (time T) the electron advances a distance called the pitch:
AHSEC Higher Secondary (HS) Final Examination 2020Set ANNUAL1 mark
Q.Name the beautiful natural phenomenon that occurs in the sky of polar regions of earth due to helical motion of charged particles.
›Reveal solutionSolution
The aurora is caused by charged particles from the solar wind spiralling (helical motion) along Earth's magnetic field lines into the polar upper atmosphere.
Charged particles (mainly electrons and protons) ejected from the Sun as the solar wind get trapped by Earth's magnetic field. Because a charged particle entering a magnetic field at an angle other than 0° or 90° experiences a force F=qv×B, it moves along a helical path, spiralling along the magnetic field lines. Near the poles, the field lines converge and dip steeply into the atmosphere, funnelling these fast charged particles down into the upper atmosphere (ionosphere), where they collide with and excite oxygen and nitrogen atoms/molecules. As these excited atoms return to their ground state …