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

Potential of an anode, V = 100 V

Magnetic field experienced by the electrons, B = 2.83 x 10^-4 T

Radius of the circular orbit r = 12.0 cm = 12.0 x 10^-2 m

Mass of each electron = m and

charge on each electron = e

Velocity of each electron = v

The energy of each electron is equal to its kinetic energy, i.e.,

1/2 mv² = eV

v²= 2eV/m————- Eq-1

It is the magnetic field, due to its bending nature, that provides the centripetal force (F = mv²/r) Centripetal force = Magnetic force

mv²/r  = evB

eB = mv/r

v = eBr/m ——Eq-2

Putting the value of v in equation (1), we get:

2eV/m = e²B²r²/m²

e/m  = 2V/B²r²

= 2 x 100 / (2.83 x 10^-4 ) x (12.0 x 10^-2) = 1.73 x 10¹¹Ckg^-1

Therefore, the specific charge ratio (e/m) is 1.73 x 10¹¹ Ckg^-1.