Charged Particle in a Uniform Magnetic Field

A magnetic field does not speed a charged particle up — it only turns it. The Lorentz force F = |q|vB is always perpendicular to the velocity, so the particle moves on a circle of radius r = mv/(|q|B), and the time of one turn, T = 2πm/(|q|B), does not depend on the speed.
Drag the arrow v₀, change the charge, the mass and the field — and watch the circle rebuild itself. Three experiments: a circular orbit, entering a strip of field, and a mass spectrometer.

The idea in 30 seconds

A magnet “feels” only a moving charge

A charge at rest is invisible to a magnetic field. As soon as the charge moves, the Lorentz force starts to act on it: F = |q|·v·B (when v ⟂ B). The bigger the charge, the speed and the field, the stronger the force.

The force always acts sideways to the velocity

It neither pushes forward nor brakes — it only turns the particle. So the speed and the kinetic energy do not change, and the path bends into a circle. The work of this force is zero.

Faster and heavier — wider; stronger field — tighter

The radius is r = mv/(|q|B). But the time of one turn, T = 2πm/(|q|B), does not depend on the speed: a fast particle runs round a big circle, a slow one round a small circle — in the same time.

What to try

Press “v ×2” and look at the charts below: the radius doubled, while the dot on the T(v) chart stayed on the same horizontal line.
Flip the charge sign or the field direction on the fly: the particle turns around and the centre of the circle jumps to the other side of the v arrow.
Choose “Entering the field” and increase B: at first the particle leaves deflected, and when the radius becomes smaller than the strip width, the field turns it back.
Choose “Mass spectrometer”: three particles of different masses landed at different places on the plate. Which one landed the farthest, and why?

Charts: what changes and what does not

Top left: the particle’s coordinates — two waves shifted by a quarter of a period. Top right: the energy — flat in a magnetic field (the dashed line shows how it would grow in an electric one). Below is the key result: the radius r grows with the speed, the period T does not.

Coordinates x(t) and y(t): two waves shifted by ¼ of a period

Energy Ek(t): the magnetic field does not change it

Radius r(v): grows with the speed

Period T(v): does not depend on the speed

Experiment

|q| charge magnitude 1,0 C

m mass 1,0 kg

|B| field strength 1,0 T

v speed 3,00 m/s

α entry angle 0°

d field width 3,5 m

What if…

Slow motion / playback rate

Drag left to slow everything down (the physics does not change). Double-click resets to 1×

1×

0,05×1×4×

Show

velocity arrow v

Lorentz force arrow F

radius r

particle trail

metre grid

Challenge

The drawing is scaled to fit the scene, but the numbers in the panel are exact — they come straight from the formulas. The values here are classroom-sized: a real electron has a charge of 1.6·10⁻¹⁹ C and a mass of 9.1·10⁻³¹ kg.

In short: the field makes the particle turn, but it changes neither its speed nor its energy. Example: m = 1 kg, |q| = 1 C, B = 1 T, v = 3 m/s — the radius is r = 3 m, the period is T = 2π ≈ 6.28 s, the force is F = 3 N. Set these numbers and compare with the readout. Double the speed with “v ×2”: the radius becomes 6 m, while the period stays 6.28 s.

KEY FORMULAS

Lorentz force

$$\vec{F} = q\,\vec{v}\times\vec{B},\qquad F = qvB\sin\alpha$$

The force with which a magnetic field acts on a moving charge; for v ⟂ B its magnitude is F = |q|vB

▶ Detailed Explanation

The Lorentz force is a vector product: F = q·v×B. Its magnitude is F = |q|·v·B·sin α, where α is the angle between v and B. At α = 90° (the particle flies across the field, as in the simulator) the force is greatest and equals |q|vB; at α = 0° (along the field lines) it is zero — the field does not “notice” such a particle and it flies straight. Direction: F ⊥ v and F ⊥ B at the same time. For a positive charge it is found with the left-hand rule: the field lines enter the palm, the four fingers point along the velocity, and the thumb shows the force. For a negative charge the force points the other way — in the simulator this is the only thing the “+ / −” switch changes. Since F ⊥ v, the force can neither speed the particle up nor slow it down — it can only turn it.

Orbit radius (Larmor radius)

$$r = \frac{mv}{|q|B}$$

The radius of the circle the particle travels when v ⟂ B — from mv²/r = |q|vB

▶ Detailed Explanation

The Lorentz force plays the role of the centripetal force: m·v²/r = |q|·v·B. Cancelling v gives r = mv/(|q|B). The formula is easy to read: the radius is larger for a faster and heavier particle (it is harder to turn: v and m are in the numerator) and smaller for a strong field and a large charge (B and |q| are in the denominator). Check it with the buttons “v ×2”, “m ×2”, “B ×2”, “q ×2”. Example: m = 1 kg, |q| = 1 C, B = 1 T, v = 3 m/s gives r = 3 m — exactly the circle shown by default. If v → 0, then r → 0: the particle stays put, because the Lorentz force does not act on a charge at rest.

Period, frequency and cyclotron frequency

$$T = \frac{2\pi m}{|q|B},\qquad f = \frac{1}{T},\qquad \omega = \frac{|q|B}{m}$$

The period of revolution, the frequency and the angular frequency — none of them depends on the speed v

▶ Detailed Explanation

The period is the circumference divided by the speed: T = 2πr/v. Substitute r = mv/(|q|B): the speed v cancels between the numerator and the denominator, leaving T = 2πm/(|q|B). This is the main “miracle” of the formula: a fast particle traces a big circle, a slow one a small circle, but a turn takes the same time. Check it: press “v ×2” — the radius doubles, while the dot on the T(v) chart stays on the horizontal line. The frequency f = 1/T is the number of turns per second, and the angular (cyclotron) frequency ω = 2π/T = |q|B/m shows the angle through which the velocity turns each second. This is how the cyclotron works: the particle circles the ring in the same time, so the accelerating electric field can be switched at the constant frequency f = |q|B/(2πm), although the speed grows. Example: m = 1 kg, |q| = 1 C, B = 1 T gives T = 2π ≈ 6.28 s at any speed.

Work of the magnetic force and kinetic energy

$$A = 0,\qquad E_k = \frac{mv^2}{2} = \mathrm{const}$$

The Lorentz force is perpendicular to the displacement, so its work is zero and the particle’s kinetic energy stays constant

▶ Detailed Explanation

The work of a force is A = F·s·cos β, where β is the angle between the force and the displacement. For the Lorentz force β = 90° at every point of the path, cos 90° = 0, so A = 0. By the work–energy theorem ΔEk = A = 0 — so the speed and Ek = mv²/2 do not change. On the Ek(t) graph the line is horizontal, while the orange dashed line shows, for comparison, how the energy would grow if a force of the same size acted along the velocity (this is how an electric field behaves: F = qE speeds the charge up). A magnetic field can only turn a particle. The momentum is not constant as a vector: its magnitude mv is the same, but its direction keeps changing. A typical exam question: “Can a magnetic field change the kinetic energy of a charged particle?” — no, it cannot.

Entering a strip of field: the deflection angle

$$\sin\varphi = \frac{d}{r},\qquad r = \frac{mv}{|q|B}$$

A particle enters a field of width d perpendicular to its edge and leaves deflected by an angle φ; if r ≤ d the field turns it back

▶ Detailed Explanation

Outside the field the particle flies straight, inside it follows an arc of radius r. If it enters perpendicular to the edge of a strip of width d, the centre of the arc lies on the line of the entry edge, and the right triangle with hypotenuse r and leg d gives sin φ = d/r. The stronger the field (the smaller r), the larger the angle. When r ≤ d, the particle does not reach the far edge: it makes a half-circle and comes back through the same edge at a distance 2r from the entry point. Example: v = 5 m/s, m = |q| = B = 1, d = 3.5 m gives r = 5 m, sin φ = 0.7, φ ≈ 44.4°. Problems about a magnetic field of finite width are common in exams.

Mass spectrometer

$$x = 2r = \frac{2mv}{|q|B},\qquad m = \frac{|q|Bx}{2v}$$

The particle traces a half-circle and lands on the plate at a distance x = 2r from the slit — the mass is found from this distance

▶ Detailed Explanation

Particles with the same speed v enter through a slit in a plate perpendicular to the field. Each traces a half-circle of diameter 2r and lands on the plate at a distance x = 2r = 2mv/(|q|B) from the slit. For equal q, v and B the distance is proportional to the mass: a particle twice as heavy lands twice as far. By measuring x, the mass is found as m = |q|Bx/(2v). This is how isotopes (for example ¹²C and ¹³C) are separated and the composition of a substance is determined in chemistry, medicine and forensics. In the simulator, with v = 3 m/s and all other parameters equal to 1, particles of masses m, 1.5m and 2m land at 6, 9 and 12 m.

What is going on. A charged particle flies through a magnetic field. While the charge is at rest, the field does not “notice” it; as soon as the charge moves, the Lorentz force F = q·v×B acts on it. This force is strictly perpendicular to the velocity. Such a force neither pushes forward nor brakes — it only turns the velocity vector, just as a string turns a stone swung over your head. So the path is a circle while the speed stays the same.

Radius and period. The Lorentz force plays the role of the centripetal force: mv²/r = |q|vB, hence r = mv/(|q|B) — the Larmor (cyclotron) radius. The faster and heavier the particle, the wider the circle; the stronger the field and the larger the charge, the tighter it is. The big surprise is that the period T = 2πr/v = 2πm/(|q|B) does not depend on the speed: a fast particle runs round a big circle, a slow one round a small circle, yet a turn takes the same time. This is how the cyclotron works: the particle circles in the same time, so the accelerating electric field can be switched at a constant frequency.

A magnetic field does no work. At every point the Lorentz force is perpendicular to the displacement, so A = 0 and the kinetic energy E_k = mv²/2 is constant. This is the key difference from an electric field, where the force F = qE points along the field, speeds the particle up and E_k grows. On the E_k(t) graph the dashed line shows how the energy would grow if a force of the same size acted along the velocity.

Which way will it turn? The sense of rotation is set by the sign of the charge and the direction of the field. For a field pointing at you (⊙) and a positive charge, the particle goes clockwise (left-hand rule: the field lines enter the palm, the four fingers point along the velocity, the thumb shows the force). A negative charge or a field pointing away (⊗) reverses the rotation. Flip the sign or the field while it runs — the particle immediately turns the other way, and the centre of the circle jumps to the other side of the v arrow.

Three experiments. “Circular orbit” — the field is everywhere and the particle goes round a circle; the metre grid and the scale bar let you measure the radius right on the screen. “Entering the field” — the field exists only in a strip of width d: outside it the particle flies straight, inside it follows an arc, then goes straight again at an angle φ, with sin φ = d/r. If the radius is smaller than the width, the field turns the particle back. “Mass spectrometer” — three particles with the same charge and speed but different masses land on a plate at a distance 2r = 2mv/(|q|B): the heavier, the farther. This is how mass is read from the mark and isotopes are separated.

Where you meet it. Mass spectrometers in chemistry and forensics, particle accelerators and cyclotrons, bubble chambers, cathode-ray tubes, and the Earth’s magnetic field, which deflects charged particles of the solar wind and creates the auroras. The numbers in the experiment are classroom-sized: a real electron has a charge of 1.6·10⁻¹⁹ C and a mass of 9.1·10⁻³¹ kg.

How to use the simulator. Press “Start” and drag from the particle with the mouse to set the velocity v₀. Use the sliders to change |q|, m, |B| and v; the buttons “v ×2”, “m ×2”, “B ×2”, “q ×2” show at once what changes (the dashed circle is the previous orbit, and the readout on the right shows r, F, Ek). In “Measurements” every number is computed from the formulas and cross-checked. Switch on “Challenge” and tune the parameters to a given radius, angle or mark on the plate.