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Two particles $A$ and $B$ having equal charges $+6\,C$, after being accelerated through the same potential difference, enter in a region of uniform magnetic field and describe circular paths of radii $2\,cm$ and $3\,cm$ respectively. The ratio of mass of $A$ to that of $B$ is
$\frac{4}{9}$
$\frac{9}{5}$
$\frac{1}{2}$
$\frac{1}{3}$
Solution
Let $\mathrm{v}$ be velocity ecquired by the charged particle when accelerated through the potential difference $\mathrm{V}$
$\therefore$ $\frac{1}{2} \mathrm{mv}^{2}=\mathrm{qV}$
or $\quad \mathrm{v}=\sqrt{\frac{2 \mathrm{qV}}{\mathrm{m}}}$
As the charged particle describes a circular path of radius $\mathrm{R}$ in the uniform magnetic field.
$\therefore \quad \frac{\mathrm{mv}^{2}}{\mathrm{R}}=\mathrm{qvB}$
or $\quad \mathrm{R}=\frac{\mathrm{mv}}{\mathrm{qB}}=\frac{\mathrm{m}}{\mathrm{qB}} \sqrt{\frac{2 \mathrm{qV}}{\mathrm{m}}}=\frac{\sqrt{\mathrm{m}}}{\mathrm{B}} \cdot \sqrt{\frac{2 \mathrm{V}}{\mathrm{q}}}$
As $\mathrm{q}, \mathrm{B}$ and $\mathrm{V}$ remain the same.
$\therefore \quad \mathrm{R} \propto \sqrt{\mathrm{m}}$
$\frac{\mathrm{R}_{\mathrm{A}}}{\mathrm{R}_{\mathrm{B}}}=\sqrt{\frac{\mathrm{m}_{\mathrm{A}}}{\mathrm{m}_{\mathrm{B}}}}$
$\Rightarrow \quad \frac{\mathrm{m}_{\mathrm{A}}}{\mathrm{m}_{\mathrm{B}}}=\left(\frac{\mathrm{R}_{\mathrm{A}}}{\mathrm{R}_{\mathrm{B}}}\right)^{2}=\left(\frac{2}{3}\right)^{2}=\frac{4}{9}$
