• 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