Question
An $$\alpha $$ particle passes rapidly through the exact centre of a hydrogen molecule, moving on a line perpendicular to the inter-nuclear axis. The distance between the nuclei is $$b.$$ Where on its path does the $$\alpha $$ particle experience the greatest force? (Assume that the nuclei do not move much during the passage of the $$\alpha $$ particle. Also neglect the electric field of the electrons in the molecule.)
An $$\alpha $$ particle passes rapidly through the exact centre of a hydrogen molecule, moving on a line perpendicular to the inter-nuclear axis. The distance between the nuclei is $$b.$$ Where on its path does the $$\alpha $$ particle experience the greatest force? (Assume that the nuclei do not move much during the passage of the $$\alpha $$ particle. Also neglect the electric field of the electrons in the molecule.)
A.
$$\frac{b}{2}$$
B.
$$\frac{b}{{2\sqrt 2 }}$$
C.
$$\frac{b}{{\sqrt 2 }}$$
D.
None of these
Answer :
$$\frac{b}{{2\sqrt 2 }}$$
Solution :
$${E_p} = \frac{{2{K_{{\text{ex}}}}}}{{{{\left( {{x^2} + \frac{{{b^2}}}{4}} \right)}^{\frac{3}{2}}}}}$$
For maximum $${E_P}$$
$$\eqalign{ & \frac{{d{E_P}}}{{dx}} = 0 \cr & \Rightarrow x = \frac{b}{{2\sqrt 2 }} \cr} $$
$${E_p} = \frac{{2{K_{{\text{ex}}}}}}{{{{\left( {{x^2} + \frac{{{b^2}}}{4}} \right)}^{\frac{3}{2}}}}}$$
For maximum $${E_P}$$
$$\eqalign{ & \frac{{d{E_P}}}{{dx}} = 0 \cr & \Rightarrow x = \frac{b}{{2\sqrt 2 }} \cr} $$