An anisotropic material has a coefficient of linear expansion `alpha, 2alpha and 3alpha` along the three co - ordinate axis. Coefficient of cubical expansion of material will be equal to
`2alpha`
`root(3)(6)alpha`
`6alpha`
None of these

Options

A. `2alpha`
B. `root(3)(6)alpha`
C. `6alpha`
D. None of these

Answer

To find the coefficient of cubical expansion (volumetric expansion) of an anisotropic material with given coefficients of linear expansion along three coordinate axes, we can follow these steps:

### Step 1: Understand the Coefficients of Linear Expansion
The coefficients of linear expansion along the three coordinate axes are given as:
- Along the x-axis: \( \alpha \)
- Along the y-axis: \( 2\alpha \)
- Along the z-axis: \( 3\alpha \)

### Step 2: Formula for Coefficient of Cubical Expansion
The coefficient of cubical expansion (volumetric expansion) \( \gamma \) is calculated using the formula:
\[ \gamma = \alpha_x + \alpha_y + \alpha_z \]
where \( \alpha_x \), \( \alpha_y \), and \( \alpha_z \) are the coefficients of linear expansion along the x, y, and z axes respectively.

### Step 3: Substitute the Values
Now, substituting the values of the coefficients of linear expansion into the formula:
\[ \gamma = \alpha + 2\alpha + 3\alpha \]

### Step 4: Simplify the Expression
Combine the terms:
\[ \gamma = (1 + 2 + 3)\alpha = 6\alpha \]

### Step 5: Conclusion
Thus, the coefficient of cubical expansion of the material is:
\[ \gamma = 6\alpha \]

### Final Answer
The correct option is **6α**.

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