Question

A beam of protons and $\alpha$-particles are successively accelerated in a cyclotron. The ratio of the normal magnetic field to be applied to the cyclotron so that protons and $\alpha$-particles have the same period of rotation is

• A. $1:4$
• B. $4:1$
• C. $1:2$
• D. $2:1$

Answer 1

Answer: (C)

$1:2$

For cyclotron,

Time period of revolution is given as:

$T=\frac{2\pi m}{Bq}$

Where, m = mass, B = magnetic field intensity and q = charge.

Now, let mass of a proton is 'm' and charge is 'q'.

Then, $T_p=\frac{2\pi m}{B_{p}q}$ ......(i) (For Proton)

Now, for alpha particle, $\left(H{e}^{2+}\right)$

$m_{He}=4\times m_p=4m$

$q_{He}=2q_p=2q$

Now,

$T_{\alpha }=\frac{2\pi \left(4m\right)}{B_{\alpha }\left(2q\right)}$

$\frac{B_p}{B_{\alpha }}=\frac{1}{2}$

Hence, normal magnetioc field for same time period protons and $\alpha -$particle is in ratio $1:2$.