Question

Two particles X and Y having equal charges after being accelerated through same potential difference enter a region of uniform magnetic field and describe a circular paths of radii '$r_1$‘ and ’$r_2$' respectively. The ratio of the mass of X to that of Y is:

Answer 1

Step 1: Calculation of the velocity of the particles

The magnetic force acting on the particles is equal to the centripetal force acting on each particle.

$qVB=m{V}^{2}r$

Here, $q$ is the charge of the particle, $V$ is the velocity, $m$ is the mass of the particle, and $r$ is the radii of the circular path.

$V=\frac{qB}{mr}$

Step 2: Calculation of the ratio of the masses

Since the work done on each particle is equal, the kinetic energy of the particles will be the same.

$$\begin{gathered} \frac{1}{2}{m}_X{V}_X^2=\frac{1}{2}{m}_Y{V}_Y^2 \\ {m}_X{\left(\frac{qB}{m_X{r}_1}\right)}^2=m_Y{\left(\frac{qB}{m_Y{r}_2}\right)}^2 \\ \frac{m_X}{m_Y}=\frac{r_2^2}{r_1^2} \\ \frac{m_X}{m_Y}={\left(\frac{r_2}{r_1}\right)}^2 \end{gathered}$$

Hence, the ratio of the mass of X to that of Y is ${\left(\frac{r_2}{r_1}\right)}^2$.