The relation between `beta and gamma` of a solid is

Options

A. `3 beta=2gamma`
B. `beta=2gamma`
C. `gamma=2beta`
D. `gamma=3beta`

Answer

To find the relation between the aerial expansion coefficient (β) and the volumetric expansion coefficient (γ) of a solid, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Coefficients**:
- The linear expansion coefficient is denoted by α.
- The aerial expansion coefficient (β) is related to the linear expansion coefficient as it pertains to area changes.
- The volumetric expansion coefficient (γ) relates to the linear expansion coefficient in terms of volume changes.

2. **Relate Aerial Expansion Coefficient to Linear Expansion Coefficient**:
- Since the area of a solid is affected by its linear dimensions, we can express the aerial expansion coefficient as:
\[ \beta = 2\alpha \]
- This is because when an object expands in two dimensions, each dimension contributes a factor of α, hence the factor of 2.

3. **Relate Volumetric Expansion Coefficient to Linear Expansion Coefficient**:
- The volumetric expansion coefficient relates to the three dimensions of a solid, so we can express it as:
\[ \gamma = 3\alpha \]
- This is because when an object expands in three dimensions, each dimension contributes a factor of α, hence the factor of 3.

4. **Set Up the Relation**:
- Now we can set up the relation between β and γ using the expressions derived:
\[ \beta = 2\alpha \quad \text{and} \quad \gamma = 3\alpha \]

5. **Express β in terms of γ**:
- To find the relation between β and γ, we can express α in terms of β and γ:
\[ \alpha = \frac{\beta}{2} \quad \text{and} \quad \alpha = \frac{\gamma}{3} \]
- Setting these two expressions for α equal gives:
\[ \frac{\beta}{2} = \frac{\gamma}{3} \]

6. **Cross-Multiply to Find the Relation**:
- Cross-multiplying gives:
\[ 3\beta = 2\gamma \]
- Rearranging this gives the final relation:
\[ \frac{\beta}{\gamma} = \frac{2}{3} \]

### Final Relation:
Thus, the relation between the aerial expansion coefficient (β) and the volumetric expansion coefficient (γ) of a solid is:
\[ 3\beta = 2\gamma \]