Question

A proton and $\alpha -$ particle enter a uniform magnetic field perpendicularly with the same speed. If proton takes $25\mu$ second to make $5$ revolutions, then time period for the $\alpha -$ particle would be
A. $50\mu \,\sec$
B. $25\mu \,\sec$
C. $10\mu \,\sec$
D. $5\mu \,\sec$

Answer 1

A proton is a subatomic particle in a nucleus of an atom which is having a positive charge. An $\alpha -$ particle is a particle that is emitted in radioactive decay and it consists of two protons and two neutrons. Here, we will calculate the time period for one revolution of $\alpha -$ particle by using the time period of a charged particle.

FORMULA USED: Now, the formula for time period of one revolution of a charged particle is given by
$T = \dfrac{{2\pi m}}{{Bq}}$
Here, $T$ is the time period, $m$ is the mass of the particle, $B$ is the magnetic field and $q$ is the charge on the particle.

COMPLETE STEP BY STEP ANSWER:
Consider a proton and $\alpha -$ particle are entering in a uniform magnetic field perpendicularly. The speed of both the particles is the same.

Now, the time taken by proton to make $5$ revolutions $= 25\mu \,\sec$
Therefore, the time taken by the proton to make one revolution $= \,\dfrac{{25}}{5} = 5\mu \,\sec$
The formula used for calculating the time period for one revolution of the charged particle is given by
$T = \dfrac{{2\pi m}}{{Bq}}$

Therefore, the time period for one revolution of photon is, ${T_p} = \dfrac{{2\pi {m_p}}}{{B{q_p}}}$
Also, the time period for one revolution of photon is, ${T_\alpha } = \dfrac{{2\pi {m_\alpha }}}{{B{q_\alpha }}}$

Now, dividing time period of $\alpha -$ particle by time period of proton, we get
$\dfrac{{{T_\alpha }}}{{{T_p}}} = \dfrac{{\dfrac{{2\pi {m_\alpha }}}{{B{q_\alpha }}}}}{{\dfrac{{2\pi {m_p}}}{{B{q_p}}}}}$
$\Rightarrow \,\dfrac{{{T_\alpha }}}{{{T_p}}} = \dfrac{{{m_\alpha }}}{{{q_\alpha }}} \times \dfrac{{{q_p}}}{{{m_p}}}$
Now, as we know that the mass of $\alpha -$ particle is four times the mass of a proton. Therefore,
if we consider
Mass of a proton, ${m_p} = m$
Then, mass of $\alpha -$ particle, ${m_\alpha } = 4m$
Also, we know that the charge on $\alpha -$ particle is double the charge on proton. Therefore, if we consider
Charge of proton, ${q_p} = q$
Then, the charge on $\alpha -$ particle, ${q_\alpha } = 2q$
Therefore, putting these values in the above equation, we get

$\dfrac{{{T_\alpha }}}{{{T_p}}} = \dfrac{{4m}}{{2q}} \times \dfrac{q}{m}$
$\Rightarrow \,\dfrac{{{T_\alpha }}}{{{T_p}}} = 2$
$\Rightarrow \,{T_\alpha } = 2{T_p}$
$\Rightarrow \,{T_\alpha } = 2 \times 5\mu$
$\Rightarrow \,{T_\alpha } = 10\mu \,\sec$
Therefore, the time period for one revolution is $10\mu \,\sec$ .

Hence, option (C) is the correct option.

NOTE: An alpha particle is four times the mass of protons because it consists of two protons and two neutrons. As the mass of the proton is nearly the same as the mass of the neutron. Therefore, we can say that an alpha particle consists of four particles.