(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 =

v_{1 =} 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