Question

Two cyclotrons of different radii $R$ and $2R$ are used to accelerate $\alpha$- particles from rest and the same magnetic field is used in both cyclotrons. Find the ratio of de Broglie wavelength of $\alpha$ particles ejected from cyclotrons.

• A. 2 : 1
• B. 4 : 1
• C. 8 : 1
• D. 16 : 1

Answer 1

The correct option is A 2 : 1

We know K.E. of particle ejected from cyclotron $=\frac{q^2{B}^2{r}^2}{2m}$

$K.E\propto r^2$

$\frac{K{E}_1}{K{E}_2}=\frac{r_1^2}{r_2^2}=\frac{1}{4}$

Now, $\frac{{\lambda }_1}{{\lambda }_2}=\frac{\frac{h}{\sqrt{2mK{E}_1}}}{\frac{h}{\sqrt{2mK{E}_2}}}=\sqrt{\frac{K{E}_2}{K{E}_1}}=\sqrt{\frac{4}{1}}=\frac{2}{1}$