Final answer:
The ratio of the mass of particle X to that of Y, when both are charged particles that have been accelerated through the same potential difference and move in a magnetic field, is equal to the ratio of their circular path radii, R1/R2.
Step-by-step explanation:
When two particles X and Y with equal charges and accelerated through the same potential difference, enter a region of uniform magnetic field and move perpendicular to it, they are subjected to a magnetic force that causes them to move in circular paths.
The radius of the path, r, for each particle is determined by the mass m, charge q, and velocity v of the particle, as well as the magnetic field strength B. The circle's radius is given by the formula: r = mv/(qB). Since X and Y have equal charges and are accelerated through the same potential difference, they will have the same velocity.
Now if the radii of their circular paths are R1 and R2 respectively, then the ratio of their masses can be found by taking the ratio of the radii since q and B are constant for both particles which means we can express the ratio of R1 to R2 as m1/m2. Thus, the ratio of the mass of X to that of Y is R1/R2.