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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 ?

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Solution

Step 1: Given data:We know that mass (m) of α-particle is $4×massofproton=4×1.6×{10}^{-27}$Also, with velocity u and Z as atomic number, e as electron charge and the required values are,$Z\text{(Atomic number of copper)}=29\phantom{\rule{0ex}{0ex}}e=1.6×{10}^{-19}\phantom{\rule{0ex}{0ex}}{\epsilon }_{0}=8.85×{10}^{-12}\phantom{\rule{0ex}{0ex}}r={10}^{-13}\phantom{\rule{0ex}{0ex}}$Step 2: Calculate velocity of Alpha-particle:The kinetic energy of alpha particle will be equal to potential energy, so$\frac{1}{2}m{u}^{2}=\frac{1}{4\pi {ϵ}_{\circ }}×\frac{2{\mathrm{ze}}^{2}}{\mathrm{r}}\phantom{\rule{0ex}{0ex}}{u}^{2}=\frac{4z{e}^{2}}{4\pi {ϵ}_{\circ }.m.r}=\frac{29×{\left(1.6×{10}^{-19}\right)}^{2}}{3.14×8.85×{10}^{-12}×\left(4×1.672×{10}^{-27}\right)×{10}^{-13}}\approx 6.3×{10}^{6}m/sec$Hence, the velocity of an alpha particle is $6.3×{10}^{6}m/sec$.

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