Question

With what velocity should an alpha- particle travel toward the nucleus of a copper atom at a distance of ${10}^{-13}m$ from the nucleus of the copper atom ?

Answer 1

Step 1: Given data:

1. We know that mass (m) of α-particle is $4\times massofproton=4\times 1.6\times {10}^{-27}$
2. Also, with velocity u and Z as atomic number, e as electron charge and the required values are,
   $$\begin{gathered} Z\text{(Atomic number of copper)}=29 \\ e=1.6\times {10}^{-19} \\ {\epsilon }_0=8.85\times {10}^{-12} \\ r={10}^{-13} \end{gathered}$$

Step 2: Calculate velocity of Alpha-particle:

The kinetic energy of alpha particle will be equal to potential energy, so
$$\begin{gathered} \frac{1}{2}m{u}^2=\frac{1}{4\pi {ϵ}_{\circ }}\times \frac{2{\mathrm{ze}}^2}{\mathrm{r}} \\ {u}^2=\frac{4z{e}^2}{4\pi {ϵ}_{\circ }.m.r}=\frac{29\times {\left(1.6\times {10}^{-19}\right)}^2}{3.14\times 8.85\times {10}^{-12}\times \left(4\times 1.672\times {10}^{-27}\right)\times {10}^{-13}}\approx 6.3\times {10}^{6}m/sec \end{gathered}$$

Hence, the velocity of an alpha particle is $6.3\times {10}^{6}m/sec$.