Magnetic force on a moving charge

A charged particle moving perpendicular to a uniform magnetic field experiences a magnetic force. The force acts sideways: it is perpendicular to both the particle’s velocity and the magnetic field.

The magnitude of the force is:

Here, $$F$$ is the force in newtons, $$B$$ is the magnetic flux density in teslas, $$|Q|$$ is the magnitude of the particle’s charge in coulombs, and $$v$$ is its speed in metres per second. Using $$|Q|$$ gives a positive force magnitude for either sign of charge.

Direction of the magnetic force

For a positive charge, the force is in the direction of the vector cross product $$\mathbf{v}\times\mathbf{B}$$, where $$\mathbf{v}$$ is the velocity and $$\mathbf{B}$$ is the magnetic field. For a negative charge, the force is in the opposite direction.

For example, if a positive particle moves to the right through a field directed into the page, the force acts upwards. A negative particle moving in the same direction through the same field experiences a force downwards.

Reversing any one of the charge sign, velocity direction or field direction reverses the force direction, provided the other two remain unchanged.

Effect on the particle’s motion

Because the magnetic force is perpendicular to the velocity, it does no work on the particle. It changes the direction of the velocity, but not the particle’s speed.

Circular motion of charged particles in a magnetic field

A charged particle moving perpendicular to a uniform magnetic field follows a circular path when the magnetic force is the only significant force acting on it. The magnetic force is always perpendicular to the particle’s velocity, so it changes the direction of motion without changing the particle’s speed.

As the particle turns, the force turns with it and continues to point towards the centre of the circle. It therefore provides the centripetal force needed for circular motion.

Radius of the circular path

For a particle moving perpendicular to the field, the magnetic force has magnitude $$B|Q|v$$. Equating this to the required centripetal force gives:

Rearranging gives the radius:

Here, $$m$$ is the particle’s mass, $$v$$ its speed, $$B$$ the magnetic field strength, $$|Q|$$ the magnitude of its charge, and $$r$$ the radius of its path.

For a fixed mass and charge magnitude, a greater speed gives a larger radius, while a stronger magnetic field gives a smaller radius. Particles with opposite charge signs curve in opposite directions when they have the same initial velocity in the same field.

Orbital period

The orbital period $$T$$ is the time taken to complete one circle. It is the distance around the circle, $$2\pi r$$, divided by the speed. Substituting the expression for $$r$$ gives, in the non-relativistic case:

The period is independent of speed: a faster particle travels farther around a larger circle, but the increase in distance exactly matches the increase in speed.

Cyclotron

A cyclotron accelerates charged particles by repeatedly passing them through an electric field. A magnetic field bends the particles back towards the accelerating region, so they can gain energy many times before they are extracted.

The cyclotron contains two hollow semicircular electrodes called dees, separated by a narrow gap. A uniform magnetic field acts perpendicular to the plane in which the particles move.

Motion inside the dees

Inside each dee, the magnetic force acts perpendicular to a particle’s motion. It provides the centripetal force that bends the particle’s path into a semicircle. Because the force is perpendicular to the motion, it changes the particle’s direction but does not increase its speed.

For a particle of mass $$m$$, charge magnitude $$|q|$$ and speed $$v$$ in a magnetic field of strength $$B$$, the radius of its path inside a dee is:

As the particle gains speed, the radius increases. Its successive semicircular paths therefore form an outward spiral.

Acceleration across the gap

An alternating potential difference between the dees produces an electric field across the gap. As the particle crosses the gap, the electric force accelerates it, increasing its kinetic energy and speed.

After crossing, the particle follows a semicircle inside the next dee and returns to the gap from the opposite direction. The potential difference reverses during this half-orbit, so the electric force accelerates the particle again. Repeated crossings increase its speed and move it outwards until it is extracted.

Timing of the alternating field

For a non-relativistic particle in a fixed magnetic field, the time for a complete orbit is:

The period $$T$$ does not depend on speed: although a faster particle travels along a larger path, it completes each orbit in the same time. The potential difference can therefore alternate at a fixed frequency, reversing every half-orbit to accelerate the particle at each successive crossing.