Atoms & Nuclei

JEE Main level, numerical answer, medium

14 of 239

Question 14 of 239

d1\sqrt{{d}_{1}} and d2\sqrt{{d}_{2}} are the impact parameters corresponding to scattering angles 60∘60^{\circ} and 90∘90^{\circ} respectively, when an α\alpha particle is approaching a gold nucleus. For d1=xd2{d}_{1}=x{d}_{2}, the value of xx will be _____ .

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Answer: 3

The relation between impact parameter and scattering angle is b=kze2Ecot⁡(θ2)b=\dfrac{kz{e}^{2}}{E}\cot \left(\dfrac{\theta }{2}\right)
⇒b∝cot⁡(θ2)\Rightarrow b\propto \cot \left(\dfrac{\theta }{2}\right)
⇒d1∝cot⁡(60∘2)\Rightarrow \sqrt{{d}_{1}}\propto \cot \left(\dfrac{60^{\circ}}{2}\right)
⇒d2∝cot⁡(90∘2)\Rightarrow \sqrt{{d}_{2}}\propto \cot \left(\dfrac{90^{\circ}}{2}\right)
d1d2=cot⁡(60∘2)cot⁡(90∘2)=3⇒d1=3d2⇒x=3\dfrac{\sqrt{{d}_{1}}}{\sqrt{{d}_{2}}}=\dfrac{\cot \left(\dfrac{60^{\circ}}{2}\right)}{\cot \left(\dfrac{90^{\circ}}{2}\right)}=\sqrt{3}\Rightarrow {d}_{1}=3{d}_{2}\Rightarrow x=3

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