Question

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

Consider two parallel co-axial circular coils of equal radius $R$, and number of turns $N$, carrying equal currents in the same direction, and separated by a distance $R$. Show that the field on the axis around the mid-point between the coils is uniform over a distance that is small as compared to $R$, and is given by, $B=0.72\frac{{\mu }_{0}NI}{R}$.
[Such an arrangement to produce a nearly uniform magnetic field over a small region is known as Helmholtz coils].

Answer 1

Kinetic energy of the electron =eV

under equilibrium conditions, $\frac{1}{2}m{v}^2=eV-\left(1\right)$

Since the particle remains undeflected by electric and magnetic field, we can infer that the electric field is balancing the magnetic field.

so, $eE=evB-\left(2\right)$

From $\left(1\right)and\left(2\right),e/m=E^2/2V{B}^2=4.8\times {10}^{7}C/kg$

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 $H{e}^{++},L{i}^{++}$, etc.