Problem 29 - Arihant AIEEE Physics

Question

In an anisotropic medium, the coefficients of linear expansion of a solid are \(\alpha_{1}, \alpha_{2}\) and \(\alpha_{3}\) in three mutually perpendicular directions. The coefficient of volume expansion for the solid is (a) \(\alpha_{1}-\alpha_{2}+\alpha_{3}\) (b) \(\frac{\alpha_{1}+\alpha_{2}+\alpha_{3}}{3}\) (c) \(\alpha_{1}+\alpha_{2}+\alpha_{3}\) (d) none of these

Step-by-Step Solution

Answer

The correct option is (c) \( \alpha_1 + \alpha_2 + \alpha_3 \).

Step 1: Understand the Problem

The question gives us three coefficients of linear expansion for a solid: \( \alpha_1, \alpha_2, \alpha_3 \) in three perpendicular directions. We need to find the coefficient of volume expansion.

Step 2: Recall Formula for Volume Expansion

The coefficient of volume expansion \( \beta \) in an anisotropic medium is the sum of the linear expansion coefficients in the three mutually perpendicular directions. Therefore, we use the formula: \( \beta = \alpha_1 + \alpha_2 + \alpha_3 \).

Step 3: Compare with Given Options

Now that we know \( \beta = \alpha_1 + \alpha_2 + \alpha_3 \), we compare this with the options provided. The correct option should match this result, which is option (c).

Step 4: Validate the Conclusion

Verify that the formula for volume expansion is accurately applied for anisotropic media as given: \( \beta = \alpha_1 + \alpha_2 + \alpha_3 \). This confirms the correctness.

Key Concepts

Linear Expansion Coefficients

In the context of thermal physics, when we talk about linear expansion coefficients, we are referring to how much a material expands in one dimensional direction as the temperature changes. For anisotropic materials, these coefficients can differ based on the direction in which they are measured. This is because an anisotropic material has different physical properties along different axes.

Each linear expansion coefficient

• -denoted as \(\alpha_1, \alpha_2, \alpha_3\) for instance- indicate the fractional change in length per degree temperature change for the respective directional axis.
  
  
• They are significant because these coefficients determine how the material's structure changes with varying temperatures.
• The coefficients are specific to the material and depend on the material's molecular structure and bonding.
  
  

Understanding linear expansion coefficients is crucial, especially in engineering and construction, because different types of materials will expand differently, potentially leading to stress and structural failure if not properly accounted for.

Volume Expansion

Volume expansion refers to the increase in volume of a substance when it is heated. This concept is central when considering materials in three dimensions, as opposed to linear expansion which only considers one dimension.

For an anisotropic medium, each direction has its own linear expansion coefficient: \( \alpha_1, \alpha_2, \alpha_3 \).

• The coefficient of volume expansion, usually represented as \( \beta \), is essentially the sum of these linear expansion coefficients: \( \beta = \alpha_1 + \alpha_2 + \alpha_3 \).
  
  
• This formula means the overall volume increase is related to each linear direction’s expansion behavior.
• Volume expansion is relevant in various real-life situations, such as when designing objects that need to fit together tightly, like parts in machinery or construction materials in a building.

The concept and calculations are essential in predicting how much space a material will occupy after a temperature change.

Thermal Expansion

Thermal expansion is a general term that covers the increase in size—be it in one, two, or three dimensions—of a material when subjected to temperature changes. This physical property is vital for understanding the behavior of materials in varying thermal environments.

Thermal expansion involves both linear and volume expansion:

• Linear expansion pertains to changes in one dimension, which is significant for structures like metal rods or pipes.
• Volume expansion, discussed earlier, considers three-dimensional changes, crucial for applications involving solids that occupy a given space, like containers and structural components.

Materials expand due to increased kinetic energy at higher temperatures, which causes their molecules to move further apart. Tracking thermal expansion is important in:

• Construction, where materials must fit properly regardless of temperature.
• Manufacturing, especially where precision components are made.
• Everyday real-world scenarios, such as when a lid on a jar becomes loose when heated.

Understanding both linear and volume expansion within thermal expansion allows us to predict and manage the effects of temperature on different materials.