Question

Two particles X and Y having equal charge, after being accelerated through the same potential difference enter a region of uniform magnetic field and describe circular paths of radii R₁ and R₂ respectively. The ratio of the mass of X to that of Y is
(a) (R₁/R₂)^1/2
(b) R₁/R₂
(c) (R₁/R₂)²
(d) R₁R₂.

Answer 1

(c)$\frac{{R_1}^2}{{R_2}^2}$
Particles X and Y of respective masses m₁ and m₂ are carrying charge q describing circular paths with respective radii R₁ and R₂ such that
$$\begin{gathered} R_1=\frac{m_1{v}_1}{qB} \\ {R}_2=\frac{m_2{v}_2}{qB} \end{gathered}$$
Since both the particles are accelerated through the same potential difference, both will have the same kinetic energy.
$$\begin{gathered} \therefore \frac{1}{2}{m}_1{v_1}^2=\frac{1}{2}{m}_2{v_2}^2 \\ \because R_1=\frac{m_1{v}_1}{qB}\Rightarrow v_1=\frac{R_{1}qB}{m_1} \\ \mathrm{And}, \\ {R}_2=\frac{m_2{v}_2}{qB}\Rightarrow v_2=\frac{R_{2}qB}{m_2} \\ \therefore m_1{\left(\frac{R_{1}qB}{m_1}\right)}^2=m_2{\left(\frac{R_{2}qB}{m_2}\right)}^2 \\ \Rightarrow \frac{m_1}{m_2}=\frac{{R_1}^2}{{R_2}^2} \end{gathered}$$