(a) A monoenergetic electron beam with electron speed of 5.20 × 106 m s–1 is subject to a magnetic field of 1.30 × 10–4 T normal to the beam velocity. What is the radius of the circle traced by the beam, given e/m for electron equals 1.76 × 1011C kg–1.
(b) Is the formula you employ in (a) valid for calculating the radius of the path of a 20 MeV electron beam? If not, in what way is it modified?
[Note: Exercises 11.20(b) and 11.21(b) take you to relativistic mechanics which is beyond the scope of this book. They have been inserted here simply to emphasize the point that the formulas you use in part (a) of the exercises are not valid at very high speeds or energies. See answers at the end to know what ‘very high speed or energy’ means.]
Given:
Speed of electron, v = 5.20 × 106 ms-1
Magnetic field strength normal to the beam = 1.30 × 10–4 T
Charge to mass ratio (e/m) = 1.76 × 1011 C Kg-1
(a) Force applied by magnetic field,
F = e|v × B|
F = e × v × B × sinθ …(1)
Where,
e = Charge on electron
v = velocity of particle
B = magnetic field strength
θ = angle between Magnetic field and velocity
Since the electron traces a circular path, we can use the following equation of centrifugal force:
F = 
…(2)
Where,
m = mass of particle
v = velocity of particle
r = radius of circle traced
By equating (1) and (2), we get,
evB × sin90° = 
r = 
→ r = 
…(3)
→ r = 
→ r = 22.7 6 cm
(b) Energy of the electron beam, E = 20 MeV
E = 20 × 106 × 1.6 × 10-19J
The energy of an electron is given by:
E = 
From the above equation, we can write,
v = 
…(4)
putting the value in equation (4)
v = 
v = 2.6 × 109 ms-1
This result is incorrect because the speed of any massive object can’t exceed the speed of light (i.e. 3 × 108 ms-1). We can’t use equation (4) in the case where the speed is relativistic:
At relativistic speeds, mass is given by,
m = 
Where,
m = relativistic mass
m0 = rest mass
v = velocity of particle
c = speed of light
the radius of the path traced is given by,
r = 
…(3)
By substituting the value of relativistic mass in equation (3) we get,
r = 
…(4)
By using equation (4) we can find the radius traced by electrons moving at relativistic speed.
By substituting value of relativistic mass in equation