# 7.8.5 Circular Path of Particles

### Motion of a Charged Particle in a Magnetic Field

• A charged particle in uniform magnetic field which is perpendicular to its direction of motion travels in a circular path
• This is because the magnetic force F will always be perpendicular to its velocity v
• F will always be directed towards the centre of the path in circular motion A charged particle moves travels in a circular path in a magnetic field

• The magnetic force F provides the centripetal force on the particle
• The equation for centripetal force is: • Where:
• F = centripetal force (N)
• m = mass of the particle (kg)
• v = linear velocity of the particle (m s-1)
• r = radius of the orbit (m)

• Equating this to the magnetic force on a moving charged particle gives the equation: • Rearranging for the radius r obtains the equation for the radius of the orbit of a charged particle in a perpendicular magnetic field: • This equation shows that:
• Faster moving particles with speed v move in larger circles (larger r): r v
• Particles with greater mass m move in larger circles: r m
• Particles with greater charge q move in smaller circles: r ∝ 1 / q
• Particles moving in a strong magnetic field B move in smaller circles: r ∝ 1 / B
• The centripetal acceleration is in the same direction as the centripetal (and magnetic) force
• This can be found using Newton’s second law:

F = ma

#### Cyclotrons

• Cyclotrons are a type of particle accelerator that accelerates charged particles (eg. protons) from their centre along a spiral path
• They are used for medical research such as:
• Producing medical isotopes (tracers)
• Creating high-energy beams of radiation for radiotherapy
• Cyclotrons make use of the circular trajectory of charged particles in a magnetic field to create the spiral path
• Cyclotrons are made up of:
• Two hollow semicircular electrodes called ‘dees
• A uniform magnetic field applied perpendicular to the electrodes
• An alternating potential difference is also applied between the electrodes, which creates an electric field between them

• The process of accelerating a particle in a cyclotron is:
• A source of charged particles is placed at the centre of the cyclotron and they are fired into one of the electrodes
• The magnetic field in the electrode makes them follow a semi-circular path, since it is perpendicular to their motion until they eventually leave the electrode
• The potential difference applied between the electrode accelerates the particles across the gap to the next electrode (since there is an electric field in the gap)
• Since the speed of the particles is now higher, they will follow a circular path with a larger radius (since r v) before leaving the electrode again
• The potential difference is then reversed so the particles accelerate towards the opposite electrode
• This process is repeated as the particles spiral outwards and eventually have a speed large enough to exit the cyclotron
• The alternating potential difference is needed to accelerate the particles across the gap between opposite electrodes
• Otherwise, the particles will only accelerate in one direction

#### Worked Example

An electron with charge-to-mass ratio of 1.8 × 1011 C kg-1 is travelling at right angles to a uniform magnetic field of flux density 6.2 mT. The speed of the electron is 3.0 × 106 m s-1.

Calculate the radius of the circle path of the electron. #### Exam Tip

Make sure you’re comfortable with deriving the equation for the radius of the path of a particle travelling in a magnetic field, as this is a common exam question.

Similar to orbits in a gravitational field, any object moving in circular motion will obey the equations of circular motion. Make sure to refresh your knowledge on these equations. They will be in the ‘Circular Motion’ section of the data sheet, and not under ‘Magnetic Fields’. ### Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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