# 25.3.6 Charge to Mass Ratio of Electrons

### Charge To Mass Ratio of Electrons

• 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

### Author: Katie

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.
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