Physics/Lessons/Magnetic Fields/Charged Particles in Magnetic Fields
Physics · Magnetic Fields · A2 Level

Charged Particles in Magnetic Fields

A current is just a stream of moving charge, so the same force that pushes a current-carrying wire sideways pushes a single moving charged particle too — but a free particle isn't clamped in place. Left free to move, that sideways force bends it into a circle, and this lesson is about that circle: how fast it turns, how big it is, and how it's used to pick out particles of a chosen speed.

8 min read

Force on a moving charge

A current is a flow of charge, so it's no surprise that the motor-effect force on a current-carrying wire, $F=BIL\sin\theta$, has an equivalent for a single moving charge. A charge $Q$ moving at speed $v$ takes time $t=L/v$ to cross a length $L$, carrying an equivalent current $I=Q/t=Qv/L$. Substituting into $F=BIL\sin\theta$:

$F = B\left(\dfrac{Qv}{L}\right)L\sin\theta = BQv\sin\theta$

where $v$ is the particle's speed and $\theta$ is the angle between its velocity and the field. As before, the force is maximum when velocity and field are perpendicular ($F=BQv$) and zero when they're parallel.

Direction still comes from Fleming's left-hand rule, with velocity taking the place of current — First finger = Field, seCond finger = velocity (conventional current direction), thuMb = Force. For a negative charge, the force is reversed, since its motion corresponds to a conventional current in the opposite direction.

Circular motion in a magnetic field

This force has a property that makes it very different from the force on a wire clamped in place: it always acts perpendicular to the particle's velocity, whatever direction that velocity happens to be pointing. A force perpendicular to velocity does no work — it can't speed the particle up or slow it down — but it does change the direction of motion. That's exactly what a centripetal force does, and a magnetic force perpendicular to $v$ is precisely that: the particle moves at constant speed on a circular path, with the magnetic force supplying the centripetal force.

A circle representing the path of a positive charge moving through a uniform magnetic field directed into the page, shown by a grid of small crosses. At a point on the circle, an arrow labelled v is drawn tangent to the circle, showing the direction of velocity, and a shorter arrow labelled F equals Bqv points from that same point toward the centre of the circle, showing the centripetal force. A line labelled r is drawn from the centre to the point on the circle, showing the radius.
A charged particle moving through a uniform field traces a circle: the magnetic force $F=Bqv$ always points toward the centre, perpendicular to the velocity $v$, supplying exactly the centripetal force needed for circular motion at radius $r$. The ×'s mark a field pointing into the page — the standard convention for a field you can't draw as an arrow lying flat in the plane of the diagram (a dot, ⊙, would mean out of the page instead).
DerivationRadius of the circular path, r = mv / BQ
1
The magnetic force provides the centripetal force for circular motion: $BQv = \dfrac{mv^2}{r}$.
$mv^2/r$ is the standard centripetal force on a mass $m$ moving at speed $v$ on a circle of radius $r$.
2
Cancel one factor of $v$ from both sides: $BQ = \dfrac{mv}{r}$.
3
Rearrange for $r$: $\displaystyle r = \frac{mv}{BQ}$.
▪ derived

Faster particles, or more massive ones, sweep out larger circles; a stronger field or a larger charge pulls the circle tighter. This is the working principle behind bending and focusing charged-particle beams in accelerators and detectors — the radius of curvature in a known field reveals the particle's momentum.

Period and frequency of the circular motion

The time for one full revolution is the circumference divided by the speed, $T=\dfrac{2\pi r}{v}$. Substituting $r=mv/(BQ)$:

$T = \dfrac{2\pi}{v}\cdot\dfrac{mv}{BQ} = \dfrac{2\pi m}{BQ} \qquad\qquad f = \dfrac{1}{T} = \dfrac{BQ}{2\pi m}$
The period doesn't depend on speed

Notice $v$ cancels out of $T$ entirely — every particle of the same charge-to-mass ratio takes the same time to complete a full circle, however fast it's moving. Faster particles just trace bigger circles in that same time. This is the principle behind the cyclotron: a particle accelerator that can use a fixed-frequency alternating field to keep accelerating a particle, since the particle's orbital frequency stays constant even as its speed (and orbit radius) grows.

Velocity selector: crossed electric and magnetic fields

A charged particle moving through both an electric field and a magnetic field, arranged perpendicular to each other and to its velocity, feels two forces at once: an electric force $QE$ and a magnetic force $BQv$. If these two forces are arranged to point in opposite directions, they cancel exactly for one particular speed, letting that speed pass through undeflected while all others are deflected away:

$QE = BQv \quad\Rightarrow\quad v = \dfrac{E}{B}$

A narrow slit placed after the fields then only lets through particles travelling at $v=E/B$, whatever their mass or charge — this velocity selector is a standard way of producing a beam of particles with a single, known speed, for use in further experiments.

Key takeaways

  • Force on a moving charge: $F=BQv\sin\theta$, derived from $F=BIL\sin\theta$ by treating the moving charge as an equivalent current; direction from Fleming's left-hand rule (velocity in place of current), reversed for negative charges.
  • Since this force is always perpendicular to velocity, it does no work and acts as a centripetal force, giving circular motion at constant speed.
  • Radius of the circular path: $r=\dfrac{mv}{BQ}$ — larger for faster or more massive particles, smaller for a stronger field or larger charge.
  • Period and frequency: $T=\dfrac{2\pi m}{BQ}$, $f=\dfrac{BQ}{2\pi m}$ — independent of speed, the principle behind the fixed-frequency cyclotron.
  • A velocity selector uses balanced electric and magnetic forces ($QE=BQv$) to let through only particles travelling at $v=E/B$, whatever their mass or charge.
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