If scattering particles are `56` for `90^(@)` angle than this will be at `60^(@)` angle

Options

A. 224
B. 256
C. 98
D. 108

Explanation

To solve the problem of finding the number of scattered particles at a \(60^\circ\) angle given that there are \(56\) particles scattered at a \(90^\circ\) angle, we can use the relationship between the number of scattered particles and the sine function.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The number of scattered particles \(n\) is inversely proportional to \(\sin^4(\theta/2)\). This can be expressed as:
\[ n \propto \frac{K}{\sin^4(\theta/2)} \]
where \(K\) is a constant.

2. **Set Up the Ratio**:
For two angles, \( \theta_1 = 90^\circ \) and \( \theta_2 = 60^\circ \), we can write:
\[ \frac{n_1}{n_2} = \frac{\sin^4(\theta_2/2)}{\sin^4(\theta_1/2)} \]
Here, \(n_1 = 56\) (for \(90^\circ\)) and \(n_2\) is what we want to find (for \(60^\circ\)).

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

- For \(\theta_2 = 60^\circ\):
\[ \sin\left(\frac{60^\circ}{2}\right) = \sin(30^\circ) = \frac{1}{2} \]
Therefore,
\[ \sin^4(30^\circ) = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

4. **Substitute Values into the Ratio**:
Now substitute the values into the ratio:
\[ \frac{56}{n_2} = \frac{\frac{1}{16}}{\frac{1}{4}} = \frac{1}{16} \times \frac{4}{1} = \frac{4}{16} = \frac{1}{4} \]

5. **Solve for \(n_2\)**:
Rearranging gives:
\[ n_2 = 56 \times 4 = 224 \]

6. **Conclusion**:
Therefore, the number of scattered particles at a \(60^\circ\) angle is \(224\).

### Final Answer:
The number of scattered particles at \(60^\circ\) is \(224\).