(a) Estimate the speed with which electrons emitted from a heated emitter of an evacuated tube impinge on the collector maintained at a potential difference of 500 V with respect to the emitter.
Ignore the small initial speeds of the electrons. The specific charge of the electron, i.e., its e/m is given to be 1.76 × 1011 C kg–1.
(b) Use the same formula you employ in (a) to obtain electron speed for a collector potential of 10 MV. Do you see what is wrong? In what way is the formula to be modified?
Given:
Potential difference between collector and emitter = 500V
Specific charge of electron (charge per unit mass e/m) = 1.76 × 1011 C
Kinetic energy of an electron is given by:
…(1)
Where,
M = mass of electron
v = velocity of electron
e = charge of electron
V = potential difference (accelerating potential)
(a) From equation (1), we can write
…(2)
By putting the values in equation (2) we can find electron velocity.
v = 1.327 × 107 ms-1
(b) Accelerating potential, V = 10MV = 106V
Let speed of electron be v1
Again putting the values in equation (2),
v1 = 
v1 = 1.8 × 109ms-1
This result is wrong as we understand that speed of light
(i.e. 3 × 108 ms-1) is the theoretical limit of the speed.
Such problems can be dealt using relativistic mechanics,
Relativistic mass is given by:
m = 
Where,
m = relativistic mass
m0 = rest mass
v = velocity of particle
c = speed of light
At relativistic speeds, kinetic energy is given by,
KE = mc2-m0c2