sqrt{d_{1}} and sqrt{d_{2}} are the impact pa | vedclass

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(A) The impact parameter $b$ is related to the scattering angle $	heta$ by the formula: $b = \frac{1}{4\pi\epsilon_0} \frac{Ze^2 \cot(	heta/2)}{K}$,

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$3$

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(A) The impact parameter $b$ is related to the scattering angle $	heta$ by the formula: $b = \frac{1}{4\pi\epsilon_0} \frac{Ze^2 \cot(	heta/2)}{K}$,where $K$ is the kinetic energy.This implies $b \propto \cot(	heta/2)$.Given $\sqrt{d_1}$ and $\sqrt{d_2}$ are the impact parameters for angles $	heta_1 = 60^{\circ}$ and $	heta_2 = 90^{\circ}$ respectively,we have:$\sqrt{d_1} \propto \cot(60^{\circ}/2) = \cot(30^{\circ}) = \sqrt{3}$.$\sqrt{d_2} \propto \cot(90^{\circ}/2) = \cot(45^{\circ}) = 1$.Squaring both sides,we get $d_1 \propto 3$ and $d_2 \propto 1$.Thus,$d_1 = 3 d_2$.Comparing this with $d_1 = x d_2$,we find $x = 3$.

$\sqrt{d_{1}}$ and $\sqrt{d_{2}}$ are the impact parameters corresponding to scattering angles $60^{\circ}$ and $90^{\circ}$ respectively,when an $\alpha$-particle is approaching a gold nucleus. For $d_{1} = x d_{2}$,the value of $x$ will be . . . . . .

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