An electron gun with its collector at a potential of 100 V fires out electrons in a spherical bulb containing hydrogen gas at low pressure (∼10^–² mm of Hg). A magnetic field of 2.83 × 10^–⁴ T curves the path of the electrons in a circular orbit of radius 12.0 cm. (The path can be viewed because the gas ions in the path focus the beam by attracting electrons, and emitting light by electron capture; this method is known as the ‘fine beam tube’ method.) Determine e/m from the data.

Question

Philoid · Accepted Answer

Given:

Potential at collector, V = 100V

Magnetic field strength, B = 2.83 × 10^-4 T

Radius of orbit traced by electron, r = 12.0 cm

Let specific charge (i.e. e/m) = K

Kinetic energy of an electron is given by:

KE = = eV

→ v = …(1)

Where,

M = mass of electron

v = velocity of electron

e = charge of electron

V = potential difference (accelerating potential)

We also know that the bending is caused by the magnetic field,

So we can write,

Magnetic force = centrifugal force

e × v × B =

→ e × B =

→ v = …(2)

Where,

e = Charge on electron

v = velocity of particle

B = magnetic field strength

m = mass of electron

r = radius of trajectory

By equating (1) and (2) we can write,

.

→

By putting the values in above equation we get,

Hence the specific charge is 1.7 × 10^-11 C Kg^-1.