Calculate the Poisson's ratio for steel. Given that Young's modulus is `2 xx 10^(11) Nm^(-2)` and rigidity modulus is `8 xx 10^(10) Nm^(-2)`.

Explanation

To calculate the Poisson's ratio for steel, we will use the relationship between Young's modulus (E), rigidity modulus (G), and Poisson's ratio (σ). The formula we will use is:

\[ E = 3G(1 - 2\sigma) \]

Where:
- \( E \) is Young's modulus,
- \( G \) is the rigidity modulus,
- \( \sigma \) is Poisson's ratio.

### Step-by-Step Solution:

1. **Identify the given values**:
- Young's modulus \( E = 2 \times 10^{11} \, \text{N/m}^2 \)
- Rigidity modulus \( G = 8 \times 10^{10} \, \text{N/m}^2 \)

2. **Write down the relationship**:
\[ E = 3G(1 - 2\sigma) \]

3. **Substitute the known values into the equation**:
\[ 2 \times 10^{11} = 3 \times (8 \times 10^{10}) \times (1 - 2\sigma) \]

4. **Calculate \( 3G \)**:
\[ 3G = 3 \times (8 \times 10^{10}) = 24 \times 10^{10} \]

5. **Rearranging the equation**:
\[ 2 \times 10^{11} = 24 \times 10^{10} \times (1 - 2\sigma) \]

6. **Divide both sides by \( 24 \times 10^{10} \)**:
\[ \frac{2 \times 10^{11}}{24 \times 10^{10}} = 1 - 2\sigma \]

7. **Simplify the left side**:
\[ \frac{2}{24} = \frac{1}{12} \]
So,
\[ \frac{1}{12} = 1 - 2\sigma \]

8. **Rearranging to find \( 2\sigma \)**:
\[ 2\sigma = 1 - \frac{1}{12} \]

9. **Finding a common denominator**:
\[ 1 = \frac{12}{12} \]
Thus,
\[ 2\sigma = \frac{12}{12} - \frac{1}{12} = \frac{11}{12} \]

10. **Finally, solve for \( \sigma \)**:
\[ \sigma = \frac{11}{24} \]

11. **Calculating the numerical value**:
\[ \sigma \approx 0.4583 \]

### Final Answer:
The Poisson's ratio for steel is approximately \( \sigma \approx 0.4583 \).