Question

The specific charge of proton is \[9.6\times {{10}^{7}}Ck{{g}^{-1}}\], then for an \[\alpha \]-particle it will be:
(a) \[38.4\times {{10}^{7}}Ck{{g}^{-1}}\]
(b) \[19.2\times {{10}^{7}}Ck{{g}^{-1}}\]
(c) \[2.4\times {{10}^{7}}Ck{{g}^{-1}}\]
(d) \[4.8\times {{10}^{7}}Ck{{g}^{-1}}\]

Answer 1

Hint: Proton and alpha particles are both positive species. But they differ in their mass and charge. Alpha particle has more charge and mass when compared to a proton.

Complete step by step solution:
> In the question, it is given that the proton has specific charge of \[9.6\times {{10}^{7}}Ck{{g}^{-1}}\]
> Let us consider Helium nucleus. It is also called an alpha particle and has a charge of +2 on it.
> We know that a specific charge of a proton is the charge on a proton divided by the mass of the proton in the particle. Alpha particle is actually a doubly ionised helium atom with +2 charge. This is twice the charge of the proton and it also has a mass which is almost four times than that of the mass of a proton since there are two protons and two neutrons in the nucleus of Helium.
> Specific charge of a proton can be written as =\[\dfrac{+e}{{{m}_{p}}}\], where \[+e\] is the charge on the proton and \[{{m}_{p}}\] is the mass of the proton.
> Now, for the alpha particle, \[+e\] is double. That is the numerator becomes \[+2e\]. The mass of the alpha particle is 4 x mass of proton, so the denominator of the equation will be \[4{{m}_{p}}\].
Also, Specific charge of alpha particle=\[\dfrac{+2e}{4{{m}_{p}}}\]= \[\dfrac{+e}{{{m}_{p}}}\times \dfrac{1}{2}\]=specific charge of proton \[\times \dfrac{1}{2}\]
Substituting the value of specific charge of proton in the above equation, we get
Specific charge of proton=\[\dfrac{9.6\times {{10}^{7}}}{2}Ck{{g}^{-1}}=4.8\times {{10}^{7}}Ck{{g}^{-1}}\].
Therefore, the correct option to the question is option (d) \[4.8\times {{10}^{7}}Ck{{g}^{-1}}\]

Note: The relation between the alpha particle and proton should be known to solve such problems. Specific charge is the ratio of a particle’s charge and its mass measured in Coulombs per kilogram and charge is measured in coulombs.