The number of alpha particles scattered at `60^(@)` is 100 per minute in an alpha particle scattering experiment. Calculate the number of alpha particles scattered per minute at `90^(@)`.

Explanation

To solve the problem of calculating the number of alpha particles scattered per minute at an angle of \(90^\circ\), given that the number of alpha particles scattered at \(60^\circ\) is 100 per minute, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: The number of alpha particles scattered at an angle \(\theta\) is proportional to \(\frac{1}{\sin^4(\frac{\theta}{2})}\). This means that if we know the number of particles scattered at one angle, we can find it at another angle using this relationship.

2. **Define Variables**:
- Let \(n_1\) be the number of alpha particles scattered at \(60^\circ\).
- Let \(n_2\) be the number of alpha particles scattered at \(90^\circ\).
- Given: \(n_1 = 100\), \(\theta_1 = 60^\circ\), and \(\theta_2 = 90^\circ\).

3. **Apply the Formula**: According to the relationship:
\[ \frac{n_2}{n_1} = \left(\frac{\sin(\frac{\theta_1}{2})}{\sin(\frac{\theta_2}{2})}\right)^4 \]

4. **Calculate \(\sin\) Values**:
- For \(\theta_1 = 60^\circ\):
\[ \sin\left(\frac{60^\circ}{2}\right) = \sin(30^\circ) = \frac{1}{2} \]
- For \(\theta_2 = 90^\circ\):
\[ \sin\left(\frac{90^\circ}{2}\right) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \]

5. **Substitute Values into the Formula**:
\[ \frac{n_2}{100} = \left(\frac{\frac{1}{2}}{\frac{\sqrt{2}}{2}}\right)^4 \]
Simplifying the fraction:
\[ \frac{n_2}{100} = \left(\frac{1}{\sqrt{2}}\right)^4 = \frac{1}{2^2} = \frac{1}{4} \]

6. **Calculate \(n_2\)**:
\[ n_2 = 100 \times \frac{1}{4} = 25 \]

### Final Answer:
The number of alpha particles scattered per minute at \(90^\circ\) is **25**.

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