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.