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, we will follow these steps:

### Step 1: Understand the relationship between scattering angles and the number of particles.
In Rutherford's experiment, the number of particles scattered at different angles is related to the sine of half the scattering angle. Specifically, the number of particles scattered per second (N) is inversely proportional to the sine of half the angle raised to the fourth power. This can be expressed as:
\[ N \propto \frac{1}{\sin^4(\phi/2)} \]

### Step 2: Define the variables.
Let:
- \( N_1 \) be the number of particles scattered at \( 90^\circ \) (which is given as \( x \)).
- \( N_2 \) be the number of particles scattered at \( 60^\circ \).
- \( \phi_1 = 90^\circ \)
- \( \phi_2 = 60^\circ \)

### Step 3: Set up the proportionality equation.
From the relationship established, we can write:
\[ \frac{N_1}{N_2} = \frac{\sin^4(\phi_2/2)}{\sin^4(\phi_1/2)} \]

### Step 4: Calculate the sine values for the angles.
- For \( \phi_1 = 90^\circ \):
\[ \sin(90^\circ/2) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \]

- For \( \phi_2 = 60^\circ \):
\[ \sin(60^\circ/2) = \sin(30^\circ) = \frac{1}{2} \]

### Step 5: Substitute the sine values into the equation.
Now substituting these values into the equation gives:
\[ \frac{x}{N_2} = \frac{\left(\frac{1}{2}\right)^4}{\left(\frac{\sqrt{2}}{2}\right)^4} \]

### Step 6: Simplify the equation.
Calculating the right-hand side:
- \( \left(\frac{1}{2}\right)^4 = \frac{1}{16} \)
- \( \left(\frac{\sqrt{2}}{2}\right)^4 = \frac{2^2}{16} = \frac{4}{16} = \frac{1}{4} \)

So the equation becomes:
\[ \frac{x}{N_2} = \frac{\frac{1}{16}}{\frac{1}{4}} = \frac{1}{16} \times 4 = \frac{1}{4} \]

### Step 7: Solve for \( N_2 \).
Cross-multiplying gives:
\[ x \cdot 4 = N_2 \]
Thus:
\[ N_2 = 4x \]

### Final Answer:
The number of particles scattered per second at an angle of \( 60^\circ \) is \( 4x \).

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