© All Rights reserved @ LearnWithDash
Step-by-Step Explanation
Step 1: Recall the Rutherford Gold Foil Experiment
In Rutherford’s famous experiment, a beam of energetic $\alpha$-particles (helium nuclei) was directed at a thin gold foil. Most $\alpha$-particles passed through with minimal deflection, but a small fraction were deflected through large angles. This observation led to the conclusion that atoms have a tiny, positively charged nucleus.
Step 2: State the Rutherford Scattering Formula
Rutherford derived a quantitative relation describing the number of $\alpha$-particles scattered at an angle $\theta$. The formula is:
$Y_{\theta} = \frac{K}{\sin^4 \bigl(\frac{\theta}{2}\bigr)}$
where $K$ is a constant (dependent on factors like the foil thickness and the incident $\alpha$-particle flux), and $\theta$ is the scattering angle.
Step 3: Understand How the Formula Affects the Graph
According to $Y_{\theta} = \frac{K}{\sin^4 \bigl(\frac{\theta}{2}\bigr)}$, the number of scattered particles $Y$ decreases sharply for larger angles $\theta$. This means that:
For small $\theta$, $\sin(\theta/2)$ is small, so $Y_{\theta}$ is relatively large.
For larger $\theta$, $\sin(\theta/2)$ increases, hence $\sin^4(\theta/2)$ increases significantly, causing $Y_{\theta}$ to fall rapidly.
Graphically, this gives a steep downward curve as $\theta$ increases.
Step 4: Identify the Correct Graph
The correct plot must show a high count of $\alpha$-particles at smaller scattering angles and a rapid drop in the number of $\alpha$-particles as the angle increases. Comparing the provided options, the graph in
matches this characteristic shape most accurately.
Step 5: Conclusion
Therefore, the correct representation of Rutherford’s scattering data is the third option, consistent with the inverse $\sin^4(\theta/2)$ dependence.
