For scattering by an inverse-square field (such as that produced by a charged nucleus in Rutherford's model) the relation between impact parameter b and the scattering angle `theta` is given by, the scattering angle for `b=0` is

Options

A. `180^(@)`
B. `90^(@)`
C. `45^(@)`
D. `120^(@)`

Explanation

To solve the problem of finding the scattering angle \(\theta\) for an impact parameter \(b = 0\) in the context of Rutherford's model, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Impact Parameter**: The impact parameter \(b\) is defined as the perpendicular distance from the center of the potential field (in this case, the nucleus) to the trajectory of the incoming particle. When \(b = 0\), it implies that the particle is headed directly towards the nucleus.

2. **Use the Relation for Impact Parameter**: The relation given in the problem for the impact parameter \(b\) in terms of the scattering angle \(\theta\) is:
\[ b = \frac{Z e^2 \cot(\theta/2)}{4 \pi \epsilon_0 K} \]
where \(Z\) is the atomic number, \(e\) is the charge of the particle, \(\epsilon_0\) is the permittivity of free space, and \(K\) is the kinetic energy of the incoming particle.

3. **Set Impact Parameter to Zero**: Since we are interested in the case where \(b = 0\), we set the equation to zero:
\[ 0 = \frac{Z e^2 \cot(\theta/2)}{4 \pi \epsilon_0 K} \]

4. **Analyze the Equation**: For the fraction to equal zero, the numerator must be zero (since the denominator cannot be zero as \(Z\), \(e\), \(\epsilon_0\), and \(K\) are all finite values). Therefore, we have:
\[ Z e^2 \cot(\theta/2) = 0 \]
This implies:
\[ \cot(\theta/2) = 0 \]

5. **Solve for Scattering Angle**: The cotangent function is zero when its argument is \(90^\circ\):
\[ \theta/2 = 90^\circ \implies \theta = 180^\circ \]
This indicates a head-on collision where the particle is scattered directly back.

6. **Conclusion**: Thus, the scattering angle \(\theta\) for \(b = 0\) is:
\[ \theta = 180^\circ \]

### Final Answer:
The scattering angle \(\theta\) for \(b = 0\) is \(180^\circ\).

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