A magnetic field set up using Helmholtz coils (described in Exercise $4.16$ ) is uniform in a small region and has a magnitude of $0.75 \mathrm{~T}$. 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 $15 \mathrm{kV}$ 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 $9.0 \times 10^{-5} \mathrm{~V} \mathrm{~m}^{-1}$, make a simple guess as to what the beam contains. Why is the answer not unique?

Magnetic field, $B=0.75 \mathrm{~T}$

Accelerating voltage, $\mathrm{V}=15 \mathrm{kV}=15 \times 10^{3} \mathrm{~V}$

Electrostatic field, $E=9.0 \times 10^{-5} \mathrm{~V} \mathrm{~m}^{-1}$

Kinetic energy of the electron, $E=(1 / 2) \mathrm{mv}^{2}$

$\mathrm{eV}=(1 / 2) \mathrm{mv}^{2}$

Therefore, $(e / m)=\left(v^{2} / 2 V\right)$

Here,

Mass of the electron $=m$

Charge of the electron $=\mathrm{e}$

Velocity of the electron $=v$

The electric and magnetic fields have no effect on the particles. As a result, the force exerted on the particle by the electric field is balanced by the force exerted by the magnetic field.

$e E=e v B$

$\Rightarrow \mathrm{v}=\mathrm{E} / \mathrm{B}$

Therefore, $(1 / 2) \mathrm{m}(\mathrm{E} / \mathrm{B})^{2}=\mathrm{eV}$

$\mathrm{e} / \mathrm{m}=\mathrm{E}^{2} / 2 \mathrm{VB}^{2}$

$=\frac{\left(9.0 \times 10^{5}\right)^{2}}{2 \times 15000 \times(0.75)^{2}}=4.8 \times 10^{7} C / \mathrm{kg}$

The beam contains Deuterium ions or deuterons

Because just the charge-to-mass ratio is established, the response is not unique. Other probable responses are He++, Lit+, and so on.