Charged Particle in a Magnetic Field Simulator

How a charged particle moves in a magnetic field

A charged particle moving across a uniform magnetic field goes round a circle of radius r = mv/(qB), because the magnetic force F = qvB sin θ always acts at right angles to its velocity. If the velocity makes an angle with the field, the particle spirals along the field on a helix instead, and if it moves along the field it is not deflected at all. The time for one turn, T = 2πm/(qB), is the same at every speed. This simulator draws each path exactly, from the solution of the Lorentz force law, and adds a velocity selector, where crossed electric and magnetic fields let only the speed v = E/B through undeflected, and a mass spectrometer that separates isotopes by the radius of their paths.

The magnetic force never has a part along the velocity, so it does no work: the particle’s speed and kinetic energy stay exactly the same while its direction turns. That is the whole reason the path closes into a circle, with the magnetic force supplying the centripetal force, qvB = mv²/r. The Centripetal Force Calculator works out the same force for any circular motion.

How to use the simulator

Choose a set-up, then the particle, its speed and the field. The scene is drawn to scale, and the scale bar in its corner says what a length on screen is. For the circle, the helix and the spectrometer the scale changes only in steps of 1, 2 and 5, so a stronger field visibly shrinks the path. At the opening settings a proton turns in well under a microsecond, so the picture is slowed by the factor printed in the other corner. With the scene focused, the arrow keys change the field and the speed.

• Circle. The magnetic field points into the page, drawn as crosses, and the particle starts moving to the right. A positive charge turns anticlockwise and a negative one clockwise: the electron and the positron have the same mass and trace mirror-image circles of the same size in opposite directions. The plot is the radius against the speed, a straight line.
• Helix. The field runs to the right and the velocity makes the angle you set with it. The part of the velocity across the field, v sin θ, sets the radius; the part along it, v cos θ, carries the particle a pitch p = v cos θ × T along the field in every turn. The far half of each turn is dashed.
• Velocity selector. Two plates make an electric field pointing down, the upper plate positive, across the magnetic field. The electric force qE and the magnetic force qvB are drawn on the particle as arrows; they cancel only at v = E/B. The plates are drawn one r₀ long, where r₀ is the radius the selected speed would have in the magnetic field alone, 0.2 r₀ apart, with an exit slit 0.04 r₀ wide, so the picture is the same for every particle. Plates this short have a limit, as every selector does: a particle 24 or more times faster than E/B crosses them too quickly to be turned far aside and reaches the slit too, on a path that still curves.
• Mass spectrometer. Singly charged ions of two isotopes are accelerated through the voltage you set, enter the field, which points out of the page, through a slit, and land on a detector after half a circle. The taller peak is the more abundant isotope.

Worked example: a proton in a 0.10 T field

What is the radius of the path of a proton moving at 1.0 × 10⁶ m/s at right angles to a 0.10 T field, and how long does one turn take? These are the simulator’s opening settings, with the proton’s mass m = 1.673 × 10⁻²⁷ kg and charge q = 1.602 × 10⁻¹⁹ C.

• Radius: r = mv/(qB) = (1.673 × 10⁻²⁷ × 1.0 × 10⁶) ÷ (1.602 × 10⁻¹⁹ × 0.10) = 0.104 m, or 10.4 cm.
• Period: T = 2πm/(qB) = 6.56 × 10⁻⁷ s, or 0.656 µs, so the proton goes round 1.52 million times a second, the cyclotron frequency of 1.52 MHz.
• The force on it is qvB = 1.60 × 10⁻¹⁴ N, and its kinetic energy is 5.22 keV throughout.
• At twice the speed, 2.0 × 10⁶ m/s, the radius doubles to 20.9 cm and the period is still 0.656 µs: the circle is twice as long and the proton covers it twice as fast.

NCERT’s Class 12 physics textbook, in its chapter on moving charges and magnetism, works Example 4.3 for an electron at 3 × 10⁷ m/s in a field of 6 × 10⁻⁴ T, with m = 9 × 10⁻³¹ kg and e = 1.6 × 10⁻¹⁹ C, and gets r = 28 cm, a frequency of 17 MHz and an energy of 2.5 keV. Set the particle to Electron, the speed to 30000 km/s and the field to 0.6 mT: the exact constants give 28.4 cm, 16.8 MHz and 2.56 keV.

Worked example: the helix at 60°

If the same proton moves at 60° to the field, what are the radius and pitch of its helix? The speed across the field is 1.0 × 10⁶ × sin 60° = 8.66 × 10⁵ m/s and along it 1.0 × 10⁶ × cos 60° = 5.0 × 10⁵ m/s. So r = mv sin θ/(qB) = 9.04 cm, and in each turn of 0.656 µs the proton moves p = 5.0 × 10⁵ × 6.56 × 10⁻⁷ = 0.328 m along the field, a pitch of 32.8 cm.

Worked example: a velocity selector

What speed passes straight through plates with an electric field of 100 kV/m across a 0.10 T magnetic field? The two forces cancel when qE = qvB, so v = E/B = 1.0 × 10⁵ ÷ 0.10 = 1.0 × 10⁶ m/s, whatever the charge or mass, because both forces are proportional to the charge. A proton at 1.1 × 10⁶ m/s feels a magnetic force 10 percent larger than the electric force, curves up and misses the slit; one at 1.5 × 10⁶ m/s strikes the upper plate. An electron at the same speeds curves the other way.

Worked example: separating neon-20 from neon-22

Where do singly charged neon ions accelerated through 2.0 kV land after half a circle in a 0.10 T field? An ion accelerated from rest gains kinetic energy qV, so its speed is v = √(2qV/m) and it lands a distance 2r = (2/B)√(2mV/q) from the slit.

• Neon-20, with an ion mass of 3.320 × 10⁻²⁶ kg: v = 1.389 × 10⁵ m/s, and it lands 57.58 cm from the slit.
• Neon-22, with 3.652 × 10⁻²⁶ kg: v = 1.325 × 10⁵ m/s, landing at 60.39 cm.
• The two peaks are 2.81 cm apart. The ratio of their distances is √(21.991/19.992) = 1.049, the square root of the mass ratio.

Neon was the first stable element found to have isotopes. J. J. Thomson saw a second kind of neon atom, of mass 22, among positive rays bent by electric and magnetic fields in 1913, and Aston confirmed it with his mass spectrograph in 1919. Each pair offered here, at those settings:

The masses and abundances are from NIST’s Atomic Weights and Isotopic Compositions. The uranium ions land 1.3 cm apart after a half circle 2 m across. That narrow margin is one reason separating uranium-235 this way, as the calutrons at Oak Ridge did in the 1940s, was so hard.

Why the period does not depend on the speed

A faster particle needs a larger centripetal force to turn it on the same circle, and the magnetic force grows with its speed by exactly the right amount, so it goes round a larger circle instead: r is proportional to v, and the time for one turn, 2πr/v = 2πm/(qB), does not change. That is what makes a cyclotron work. Its particles are pushed by an alternating voltage at the fixed frequency qB/(2πm), and they stay in step with it as they speed up and spiral outwards. It holds only while the particle is much slower than light, because at higher speeds the radius grows as the momentum γmv and the turns take longer.

The radius depends on the momentum and the charge, not on the speed alone, so an alpha particle, with about four times the proton’s mass and twice its charge, goes round a circle almost exactly twice as large at the same speed. A uniform magnetic field like the one here is what a current in a long coil makes: the Solenoid Magnetic Field Simulator shows how uniform the field inside a real coil is, and the Electric Field Simulator draws the fields of point charges, the other half of the Lorentz force.

What this simulation leaves out

• Relativity. The motion is classical. The top of the speed range is a tenth of the speed of light, where the true radius, γmv/(qB), is 0.5 percent larger than the one shown; at 1.0 × 10⁶ m/s it is 6 parts in a million larger. The radius readout’s hint gives the figure for the speed set.
• Radiation. An accelerated charge radiates, and a charge going round a circle is always accelerating, so a real electron slowly spirals inwards. At these speeds and fields the energy lost in one turn is far too small to see.
• Real fields. Every field here is perfectly uniform and stops sharply at the edges drawn. Real magnets and plates have fringe fields at their edges, which bend a beam a little before it enters and after it leaves.
• Other particles. Each particle moves alone. In a real beam the particles repel each other and collide with the gas left in the vacuum, which spreads the beam and the spectrometer’s peaks.
• Gravity. The weight of a proton is about 10⁻²⁶ N, a trillion times smaller than the magnetic force on it at the opening settings, so it is left out.

Common mistakes

• Getting the direction wrong for an electron. The right-hand rule for v × B gives the force on a positive charge. For an electron the force, and the sense of rotation, is the opposite.
• Thinking the magnetic force speeds the particle up. It is always at right angles to the velocity, so it changes only the direction, and the kinetic energy stays the same.
• Using the whole speed in r = mv/(qB) for a helix. Only the part across the field, v sin θ, goes round the circle; the part along it sets the pitch.
• Expecting a faster particle to go round more often. It goes round a larger circle in the same time. The period depends only on the mass, the charge and the field.
• Forgetting the units. B must be in tesla and v in m/s for r to come out in metres: 100 mT is 0.10 T, and 1 gauss is 10⁻⁴ T.
• Thinking a velocity selector depends on the particle. Both forces are proportional to the charge, so v = E/B is the same for every particle. What differs is how sharply a particle at the wrong speed is thrown aside.