The ratio among coefficient of volume expansion, superficial expansion and linear expansion i.e.,
`gamma` : `beta` : `alpha` is

Options

A. `1:2:3`
B. `3:2:1`
C. `4:3:2`
D. all of these

Answer

To find the ratio among the coefficients of volume expansion (γ), superficial expansion (β), and linear expansion (α), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The coefficient of linear expansion (α) measures how much a material expands per unit length for a unit change in temperature.
- The coefficient of superficial expansion (β) measures how much a material expands per unit area for a unit change in temperature.
- The coefficient of volume expansion (γ) measures how much a material expands per unit volume for a unit change in temperature.

2. **Establish Relationships**:
- It is known that the coefficient of volume expansion (γ) is related to the coefficient of linear expansion (α) by the formula:
\[ γ = 3α \]
- The coefficient of superficial expansion (β) is related to the coefficient of linear expansion (α) by the formula:
\[ β = 2α \]

3. **Express Each Coefficient in Terms of α**:
- From the relationships established:
- γ = 3α
- β = 2α
- α = α

4. **Set Up the Ratio**:
- Now we can express the ratio γ : β : α as:
\[ γ : β : α = 3α : 2α : α \]

5. **Simplify the Ratio**:
- We can simplify this ratio by dividing each term by α:
\[ 3α : 2α : α = 3 : 2 : 1 \]

6. **Conclusion**:
- Therefore, the ratio among the coefficients of volume expansion, superficial expansion, and linear expansion is:
\[ \boxed{3 : 2 : 1} \]