Which of the following may be representing graph between number of scattered particles detected (N) and scattering angle` (theta)` in Rutherford's experiment?

Options

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Answer

To determine the graph representing the relationship between the number of scattered particles detected (N) and the scattering angle (θ) in Rutherford's experiment, we can follow these steps:

### Step 1: Understand the Relationship
In Rutherford's gold foil experiment, the number of particles scattered at an angle θ is inversely related to the sine of the angle raised to the fourth power. Mathematically, this can be expressed as:
\[ N \propto \frac{1}{\sin^4(\theta)} \]

### Step 2: Analyze the Behavior of the Function
As θ approaches 0, sin(θ) approaches 0, which means that sin^4(θ) also approaches 0. Consequently, \( \frac{1}{\sin^4(\theta)} \) approaches infinity. Therefore, at θ = 0, the number of detected particles (N) becomes infinitely large.

### Step 3: Examine the Range of θ
The angle θ varies from 0 to π (0 to 180 degrees). As θ increases from 0 to π, sin(θ) increases from 0 to 1 and then decreases back to 0. This means that \( \sin^4(\theta) \) will initially increase, leading to a decrease in N until it reaches a maximum point, after which N will decrease again as θ approaches π.

### Step 4: Sketch the Graph
1. At θ = 0, N approaches infinity.
2. As θ increases from 0 to π/2, N decreases.
3. At θ = π/2, N reaches a minimum value.
4. As θ continues to increase towards π, N increases again but remains finite.

The graph will start from infinity at θ = 0, decrease to a minimum at θ = π/2, and then increase again as θ approaches π, but will not reach infinity again.

### Step 5: Conclusion
The graph representing the relationship between the number of scattered particles detected (N) and the scattering angle (θ) will show a trend starting from infinity, decreasing to a minimum, and then increasing again, resembling a curve that approaches the axes but never touches them.