If `alpha, beta` and `gamma` coefficient of linear, superficial and volume expansion respectively, tehn

Options

A. `(beta)/(alpha) = 1/2`
B. `(beta)/(gamma) = 2/3`
C. `(gamma)/(alpha) = 3/2`
D. `(beta)/(alpha) = (gamma)/(beta)`

Explanation

To solve the problem regarding the relationship between the coefficients of linear, superficial, and volume expansion (denoted as α, β, and γ respectively), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Coefficients**:
- The coefficient of linear expansion (α) describes how a material's length changes with temperature.
- The coefficient of superficial (or area) expansion (β) describes how a material's area changes with temperature.
- The coefficient of volume expansion (γ) describes how a material's volume changes with temperature.

2. **Establishing Relationships**:
- The relationship between these coefficients can be expressed as:
\[ \beta = 2\alpha \]
This means that the superficial expansion coefficient is twice the linear expansion coefficient.

- Similarly, the relationship between the volume expansion coefficient and the linear expansion coefficient is:
\[ \gamma = 3\alpha \]
This indicates that the volume expansion coefficient is three times the linear expansion coefficient.

3. **Combining the Relationships**:
- From the above relationships, we can express β and γ in terms of α:
\[ \beta = 2\alpha \quad \text{and} \quad \gamma = 3\alpha \]
- We can also express α in terms of β and γ:
\[ \alpha = \frac{\beta}{2} \quad \text{and} \quad \alpha = \frac{\gamma}{3} \]

4. **Finding a Common Relation**:
- By substituting the expressions for β and γ in terms of α, we can derive a common relationship:
\[ 6\alpha = 3\beta = 2\gamma \]

5. **Conclusion**:
- Thus, the relationships can be summarized as:
\[ 2\beta = \alpha \quad \text{(incorrect)} \]
\[ 3\beta = 2\gamma \quad \text{(correct)} \]
\[ 2\gamma = 3\alpha \quad \text{(incorrect)} \]
\[ \beta^2 = \alpha\gamma \quad \text{(incorrect)} \]
- Therefore, the correct option based on the relationships derived is **3β = 2γ**.