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).
Inside the solenoid the electron's velocity is parallel to B, so v×B=0 and the magnetic force vanishes - the electron keeps moving with uniform velocity along the axis, matching stem option (d).
Setting up the field and velocity
A long current-carrying solenoid produces a magnetic field that is (very nearly) uniform inside and directed along its axis. The electron is projected with velocity v=v0i^also along the axis, so v∥B.
Applying the Lorentz force law
F=q(v×B)
When two vectors are parallel (or antiparallel), their cross product is zero:
v×B=0⇒F=0.
Consequence
With no magnetic force (and no other force stated - gravity is negligible for an electron), Newton's first law applies: the electron continues at constant velocity, unchanged in magnitude and direction, straight along the axis.
Watch out
This is a common trap: a magnetic field does not automatically deflect every charge that enters it. It deflects only the component of velocity perpendicular to B. Motion purely along the field line feels no force at all.
Method: Classifying Charged-Particle Motion by the Angle Between v and B
Before writing any force equation, decide which of the three qualitatively different motions applies -- straight line, circle, or helix -- purely from the angle between the particle's velocity and the field.
Steps
Step 1: Resolve the velocity into components along and perpendicular to B
Write v=v∥B^+v⊥, where B^ is the field direction. This single decomposition is all you need, because the Lorentz force F=q(v×B) only ever acts on the perpendicular component -- v∥B^×B=0 always.
Step 2: Read off the case from which component is present
v⊥=0 (velocity purely along B): F=0 identically -- the particle is force-free and travels in a straight line at constant velocity (Newton's first law takes over).
v∥=0 (velocity purely perpendicular to B): the force is constant in magnitude and always centripetal -- the particle moves in a circle.
Both nonzero: the perpendicular part circles while the parallel part drifts steadily -- the particle traces a helix.