Question

The ratio of time periods of $\alpha$-particle and proton moving on circular path in uniform magnetic field is ___________.

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

Answer 1

The correct option is B $2:1$

We know that
Motion of charged particles in uniform magnetic field
$F=qvB=\frac{m{v}^2}{r}$
$r=\frac{mv}{qB}$ and $K=\frac{1}{2}m{v}^2$
$\Rightarrow v^2=\sqrt{2K/m}$
So, $T=\frac{\sqrt{2mK}}{qB}$
But $T=\frac{2\pi r}{v}=\frac{2\pi m}{qB}$
$\therefore T\propto \frac{m}{q}$
$T_{\alpha }=\frac{m_{\alpha }}{q_{\alpha }},T_p=\frac{m_p}{q_p}$
$\frac{T_{\alpha }}{T_p}=\frac{m_{\alpha }}{q_{\alpha }}\times \frac{q_p}{m_p}$
[Charge on $\alpha$ particle $=3.2\times {10}^{-19}C$
Charge on proton $=1.6\times {10}^{-19}C$
Mass of proton $=1.7\times {10}^{-27}kg$
Mass of $\alpha$-particle $=6.67\times {10}^{-27}kg$]

$=\frac{6.6\times {10}^{-27}\times 1.6\times {10}^{-19}}{2\times 1.6\times {10}^{-19}\times 1.7\times {10}^{-17}}$

$=\frac{66}{34}=\frac{1.94}{1}$

$=\frac{2.00}{1}=2:1$