7.8.5 Circular Path of Particles

• 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

A cyclotron is a type of particle accelerator that creates a high energy proton beam by accelerating protons in circular motion

• 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