Question

Two particles Y and Z emitted by a radioactive source at P made tracks in a chamber as illustrated in the figure. A magnetic field acts downward into the paper. Careful measurements showed that both tracks were circular, the radius of Y track being half that of the Z track. Which one of the following statements is certainly true?

A. Both particles Y and Z carried a positive charge
B. The mass of particle Z was one half that of particle Y
C. The mass of particle Z was twice that of particle Y
D. The charge of particle Z was twice that of particle Y

Answer 1

Unless explicitly specified, it is impossible to quantitatively distinguish between the properties of two particles subjected to a magnetic field. However, their collective response to the magnetic field can be used to arrive at suitable conclusions about the nature of their properties, if not the magnitude. In such a case, use Fleming’s Left-hand rule and the magnetic force parameters to determine the certainty with which the given properties of the particles can be deduced from the relation between the two track radii and arrive at the appropriate answer.
Formula used:
Centripetal force $F_{centripetal} =\dfrac{mv^2}{r}$
Magnetic force $F_{magnetic} = Bqv$

Complete step by step answer:
We know that magnetic fields deflect the motion of particles possessing a net electrical charge, and when such particles encounter a magnetic field, they experience a Lorentz force due to which their direction of motion gets altered.
We are given that the magnetic field is directed into the plane of the paper and the particles Y and Z are emitted in the chamber perpendicular to this magnetic field. As a consequence of this magnetic field, the two particles experience a deflecting force due to which they begin to trace out curved paths of motion. Thus, the particles experience a centripetal force equivalent to the magnetic force acting on them as they traverse through the magnetic field in the chamber, i.e.,
$F_{centripetal} = F_{magnetic}$
$\Rightarrow \dfrac{mv^2}{r} = Bqv$
Where r is the radius of the path traced by the particle, v is the velocity of the particle, m is the mass of the particle, B is the magnetic field strength and q is the charge of the particle.
The radius of the path traced by the particle can thus be given as:
$\Rightarrow r = \dfrac{mv}{Bq}$
For the same magnetic field, $r \propto \dfrac{mv}{q}$
If $r_y$ is the radius of path traced by Y and $r_z$ is the radius of path traced by Z, we are given that:
$r_y = \dfrac{1}{2} \times r_z$
But we have $r \propto \dfrac{mv}{q}$, which means that the radius of the path traced by the particle depends on three parameters, namely mass, velocity and charge. Thus, the relation between the radius of tracks Y and Z cannot be accurately extrapolated to determine the relation between the two particles’ charge or mass since the difference in radius could arise from a variation in any of the three parameters or a combination of two or more of them.
However, we know that the magnetic field is directed (into the plane of the paper) perpendicular to the initial path of the particles and this magnetic field exerts a force perpendicular to the motion of the particles, making both the particles deflect in the same directional sense, as seen in the figure. This uniformity in the direction of their deflection can only mean that the two particles are like-charged.

So, the correct answer is “Option A”.

Note: Remember that for particles traversing through an electromagnetic field, they get kinetic energy for their accelerative motion from the potential energy stored due to the potential difference in the electric field, and the particles exert a centripetal force in response to the magnetic force acting on them due to which they execute circular motion. This means that the magnetic force acting on the particles will always be perpendicular to the plane containing the magnetic field and the velocity of the particles, illustrated by the expression:
$\vec{F}_{magnetic} = q(\vec{v} \times \vec{B})$