The graph which depicts the results of utherford gold foil experiment with alpha-particles is
theta:

Q: Why do most $\alpha$-particles pass straight through the foil?

Because the atom is mostly empty space! The positive charge and mass are concentrated in a tiny central nucleus, so the chances of an $\alpha$-particle coming close enough to be strongly repelled are very low.

Q: Where does the $\sin^4(\theta/2)$ term come from?

It comes from integrating the Coulomb force over the hyperbolic trajectory of the $\alpha$-particle. By applying conservation of energy and angular momentum, the relationship between the impact parameter and the scattering angle naturally produces this sine term.

Q: Does the graph ever actually touch the axes?

Mathematically, as $\theta \to 0$, $Y \to \infty$, so it's an asymptote (it never touches the y-axis). At $\theta = \pi$, $Y$ reaches a minimum non-zero value, so it doesn't touch the x-axis either, though it gets very close compared to the peak.

Q: Why is the graph in option (a) incorrect if $Y$ decreases as $\theta$ increases?

Option (a) shows a linear decrease. However, the $1/\sin^4(\theta/2)$ relationship is highly non-linear. The number of particles drops off exponentially fast, not at a constant linear rate.

Q: What does a scattering angle of $\theta = \pi$ physically represent?

An angle of $\pi$ radians ($180^\circ$) represents a perfect head-on collision with the nucleus. The $\alpha$-particle stops momentarily and bounces straight back along its original path.

Question

The graph which depicts the results of utherford gold foil experiment with $\alpha$-particles is
$	heta$: Scattering angle
$Y$ : Number of scattered $\alpha$-particles detected
(plots are schematic and not to scale)

Sigma Percentile · Accepted Answer

### The Setup: Bombarding the Gold Foil

Imagine you are standing in a dark room, firing tiny, invisible bullets at a piece of tissue paper. This is essentially what Ernest Rutherford and his team did during their famous gold foil experiment.

They bombarded a remarkably thin sheet of gold with high-energy $\alpha$-particles. To understand the structure of the atom, they needed to see how these particles were deflected, or "scattered," by the gold atoms.

We define two critical variables for this analysis: $	heta$, which is the scattering angle (how far the particle veers off its original straight path), and $Y$, which represents the number of $\alpha$-particles detected at that specific angle.

### The Master Equation: Rutherford's Scattering Formula

Through rigorous mathematical derivation based on Coulomb's law of electrostatic repulsion, Rutherford arrived at a groundbreaking formula.

He found that the number of scattered particles, $Y$, is not just randomly distributed. It follows a very specific, highly sensitive mathematical relationship:

$$Y \propto \frac{1}{\sin^4\left(\frac{	heta}{2}ight)}$$

This equation is the heart of the problem. To find the correct graph, we simply need to analyze how this function behaves at its extremes.

### Analyzing the Extremes: Small and Large Angles

Let's first look at what happens when the scattering angle $	heta$ is very small, approaching $0$.

As $	heta 	o 0$, the term $\sin\left(\frac{	heta}{2}ight)$ also approaches $0$. Because this term is in the denominator and raised to the fourth power, the value of $Y$ explodes towards infinity!

**Physically, this means that the vast majority of $\alpha$-particles pass straight through the gold foil with almost zero deflection.**

Now, what happens at the other extreme? Let's consider large scattering angles, where $	heta$ approaches $\pi$ radians ($180^\circ$).

As $	heta 	o \pi$, the term $\sin\left(\frac{	heta}{2}ight)$ approaches $\sin\left(\frac{\pi}{2}ight)$, which is exactly $1$. Consequently, the denominator becomes $1^4 = 1$, making $Y$ reach its minimum possible value.

**This tells us that very few $\alpha$-particles experience a massive deflection or bounce straight back.**

### The Final Graphical Representation

When we combine these two mathematical realities, the shape of the graph becomes obvious.

The curve must start incredibly high near the vertical axis (when $	heta$ is small) and then drop off precipitously as $	heta$ increases, eventually flattening out near the horizontal axis (when $	heta$ is large).

This describes a rapidly decreasing curve that is asymptotic to both axes. Looking at our options, this perfectly matches the curve shown in option (b).

This simple, elegant curve was the very proof that shattered the plum pudding model and revealed the existence of the dense, positively charged atomic nucleus!

Animated Solution for Physics - Atoms and Nuclei: The graph which depicts the results of utherford gold foil experiment with $α$ -particles is $θ$ : Scattering angle $Y$ : Number of scattered $α$ -particles detected (plots are schematic and not to scale)

Select Answer:

Visualized Solution

$Y vs θ Graph$

$Scattering Formula$

$Small Angles$

$Large Angles$

$Conclusion$

$Physical Significance$

The Sigma Insight: Alpha-Particle Scattering and Rutherford Nuclear Model of Atom

The Setup

The Master Equation

Analyzing the Extremes

The Final Graphical Representation

Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Y vs θ Graph

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Scattering Formula

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Small Angles

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Large Angles

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Physical Significance