A magnetic field set up using Helmholtz coils (described in Exercise 4.16) is uniform in a small region and has a magnitude of 0.75 T. In the same region, a uniform electrostatic field is maintained in a direction normal to the common axis of the coils. A narrow beam of (single species) charged particles all accelerated through 15 kV enters this region in a direction perpendicular to both the axis of the coils and the electrostatic field. If the beam remains undeflected when the electrostatic field is 9.0 × 10⁻⁵ V m⁻¹, make a simple guess as to what the beam contains. Why is the answer not unique?

Magnetic field, B = 0.75 T

Accelerating voltage, V = 15 kV = 15 x 10³ V

Electrostatic field, E = 9 x 10⁵ V m⁻¹

Mass of the electron = m,

Charge of the electron = e,

Velocity of the electron = v

Kinetic energy of the electron = eV

=>1/2 x mv² =eV

Therefore e/m = v²/2V———————- Eq-1

Since the particle remains un-deflected by electric and magnetic fields, we can infer that the electric field is balancing the magnetic field.

Therefore eE =evB

v = E/B—————————–Eq-2

Putting Eq-2  in equation 1, we get

e/m = 1/2  x (E/B)² /V = E²/2VB²

=  (9 x 10⁵ )²/[2 x 15000 x (0.75)²]

=4.8 x 10⁷ C/Kg

This value of specific charge e/m is equal to the value of deuteron or deuterium ions. This is not a unique answer. Other possible answers are He⁺⁺ Li⁺⁺, etc