√d₁ and √d₂ are the impact parameters corresponding to scattering angles 60° and 90° respectively, when an α particle is approaching a gold nucleus. For d₁ = x d₂, the value of x will be ____________.

Solution

Physics sir · 4 steps

Impact parameter goes as cot of half the scattering angle, and the d's are its squares, so square the ratio.

1. Rutherford relation. For an α particle scattered by a nucleus, the impact parameter b depends on the scattering angle θ through cot(θ/2). Everything else (charges, kinetic energy) is the same for both cases.

   b = Ze²/(4πε₀ K)cot θ/2 ⇒ b ∝ cot θ/2

2. Take the ratio. Here b₁ = √d₁ at 60° and b₂ = √d₂ at 90°. So √d₁/√d₂ = (cot 30°)/(cot 45°) = √3/1 = √3.
3. Square it. Squaring both sides gives d₁/d₂ = 3, so d₁ = 3d₂.
4. So. x = 3.

The common slip: 1.73. Stopping at the ratio of impact parameters, √d₁/√d₂ = √3, gives 1.73. The question asks for d₁/d₂, so square it to get 3.

The idea to keep. In Rutherford scattering, b ∝ cot(θ/2): a smaller impact parameter means a larger deflection. Always use half the scattering angle inside the cot.

Answer from the NTA final answer key. Solution by PrepWiser.