In Rutherford's experiment, number of particles scattered at `90^(@)` angel are x per second. Number particles scattered per second at angle `60^(@)` is

Options

A. x
B. 4x
C. 8x
D. 16x

Answer

To solve the problem of finding the number of particles scattered per second at an angle of 60 degrees in Rutherford's experiment, we can use the relationship derived from the experiment regarding the scattering of alpha particles.

### Step-by-Step Solution:

1. **Understand the Relationship**:
In Rutherford's experiment, the number of particles scattered at an angle θ is given by the formula:
\[ n(\theta) \propto \csc^4\left(\frac{\theta}{2}\right) \]
This means that the number of particles scattered at angle θ is directly proportional to the cosecant of half the angle raised to the fourth power.

2. **Set Up the Known Values**:
We know that the number of particles scattered at 90 degrees is \( n(90^\circ) = x \) particles per second. We need to find \( n(60^\circ) \).

3. **Calculate \( n(90^\circ) \)**:
Using the formula:
\[ n(90^\circ) \propto \csc^4\left(\frac{90^\circ}{2}\right) = \csc^4(45^\circ) \]
Since \( \csc(45^\circ) = \sqrt{2} \):
\[ n(90^\circ) \propto (\sqrt{2})^4 = 4 \]
Therefore, we can write:
\[ n(90^\circ) = k \cdot 4 \]
where \( k \) is a proportionality constant.

4. **Calculate \( n(60^\circ) \)**:
Now, calculate \( n(60^\circ) \):
\[ n(60^\circ) \propto \csc^4\left(\frac{60^\circ}{2}\right) = \csc^4(30^\circ) \]
Since \( \csc(30^\circ) = \frac{2}{\sqrt{3}} \):
\[ n(60^\circ) \propto \left(\frac{2}{\sqrt{3}}\right)^4 = \frac{16}{9} \]

5. **Set Up the Ratio**:
Now we can set up the ratio of the number of particles scattered at 60 degrees to those scattered at 90 degrees:
\[ \frac{n(60^\circ)}{n(90^\circ)} = \frac{\frac{16}{9}}{4} = \frac{16}{36} = \frac{4}{9} \]

6. **Find \( n(60^\circ) \)**:
Since \( n(90^\circ) = x \):
\[ n(60^\circ) = x \cdot \frac{4}{9} \]

7. **Final Calculation**:
To find the number of particles scattered at 60 degrees:
\[ n(60^\circ) = \frac{4}{9}x \]

### Conclusion:
The number of particles scattered per second at an angle of 60 degrees is \( \frac{4}{9}x \).