• The velocity of an electron beam can be found using a deflection tube
• This consists of electrons accelerated from a cathode (negative terminal) to an anode (positive terminal)
• They are then passed between two oppositely charged parallel horizontal plates and a magnetic field is applied perpendicular to them
• This means an electron beam is passed through a combined electric (E) and magnetic (B) field

• By adjusting the strengths of the E and B fields, the two electric and magnetic forces can be made to balance the path of the electrons and keep the beam horizontal
• When the electron beam remains straight, the electric and magnetic forces on them are equal in magnitude but opposite in direction
• The magnetic field is provided by two coils (Helmhol coils) which provide a uniform field between them
• The force due to an electric field is given by:

F = eE

• The force due to a magnetic field is given by:

F = Bev

• Equating the E and B forces gives:

eE = Bev

• Where:
  • e = charge of an electron (C)
  • E = electric field strength (N C^-1)
  • B = magnetic flux density (T)
  • v = velocity of the electron beam (m s^-1)

• Rearranging for the velocity v, and using the definition of electric field strength between two plates, the equation becomes:

• Where:
  • V = voltage between the plates (V)
  • d = distance between the plates (m)

Measuring the charge-to-mass ratio of the electrons

• When electrons travel through a magnetic field that is perpendicular to its velocity, they travel in a circular path
• This means the magnetic force, which is perpendicular to its velocity, is equivalent to the centripetal force:

• Where:
  • r = radius of the circular path of the electrons (m)
  • m = mass of an electron (kg)
• Rearranging this for the charge to mass ratio e/m gives the equation:

• This is equivalent to 1.76 × 10¹¹ C kg^-1
• A common analytical tool used to measure the charge-to-mass ratio of particles is a mass spectrometer