Problem 10 - Arihant AIEEE Physics

Question

If same amount of heat is supplied to two identical spheres (one is hollow and other is solid), then: (a) the expansion in hollow is greater than the solid (b) the expansion in hollow is same as that in solid (c) the expansion in hollow is lesser than the solid (d) the temperature of both must be same to each other.

Step-by-Step Solution

Answer

(b) the expansion in hollow is same as that in solid.

Step 1: Understand Thermal Expansion

Thermal expansion refers to the increase in volume of a material as its temperature increases. For solids, the expansion depends on the material's coefficient of thermal expansion and its initial dimensions. Identical spheres here mean they have the same volume and material.

Step 2: Identify Properties of Hollow and Solid Spheres

Both spheres are made of the same material, implying the same coefficient of thermal expansion. A hollow sphere consists of an outer shell with an empty cavity, while a solid sphere is entirely made of the material.

Step 3: Apply the Concept of Uniform Expansion

For isotropic materials, every part of the material expands uniformly when heated. Therefore, the expansion depends primarily on the initial volume and not whether the sphere is hollow or solid.

Step 4: Analyze the Volume Calculation

Despite differences in shape, the material of both the hollow and solid spheres will expand uniformly throughout their structures. Here, since both start with the same total volume, their expansion will be identical.

Step 5: Conclusion from Analysis

Since both spheres are made of the same material and start with the same initial volume, they will experience the same amount of expansion when the same amount of heat is applied.

Key Concepts

Coefficient of Thermal Expansion

Understanding the coefficient of thermal expansion is crucial for grasping how different materials respond to temperature changes. This coefficient is a measure that indicates how much a material will expand per degree of temperature increase. It's typically expressed in per degree Celsius (°C) or per degree Kelvin (K).
For any material, when temperature increases, its molecules vibrate more vigorously, causing them to take up more space. This leads to an increase in volume. The coefficient of thermal expansion, denoted as \( \alpha \), is mathematically represented by:\[\alpha = \frac{\Delta L}{L_0 \Delta T}\]where:

• \( \Delta L \) is the change in length.
• \( L_0 \) is the original length.
• \( \Delta T \) is the change in temperature.

Different materials have unique values of \( \alpha \), meaning they expand by different amounts when subjected to the same change in temperature. In our exercise, since the spheres are made of identical material, they share the same coefficient of thermal expansion.

Isotropic Materials

Isotropic materials have uniform properties in all directions. This means that when you heat an isotropic material, it expands uniformly in all directions. This uniform expansion is why the shape of the object doesn't matter as much as its overall volume when considering thermal expansion.
For example, imagine blowing up a balloon: it expands uniformly in all directions irrespective of its initial shape. Similarly, when isotropic materials are heated, every part of the material expands at the same rate.
In the context of the exercise, both solid and hollow spheres are assumed to be made of isotropic material. Therefore, they will expand uniformly, making their initial shape less significant. This property simplifies analysis since it allows us to focus on the total volume and not the internal structure when considering thermal expansion.

Solid and Hollow Spheres

Solid and hollow spheres might sound quite different because one is filled entirely with material while the other has a cavity. However, when we analyze thermal expansion for these spheres made of the same material, their expansion behavior is quite similar due to the same total initial volume.
A solid sphere consists purely of the material without any cavities, while a hollow sphere comprises a shell around an empty center. For both spheres, if they have the same external dimensions and are made of the same isotropic material, they should expand identically when exposed to the same increase in temperature.
The reasoning is that the expansion of isotropic materials depends on the volume of material present. Since both hollow and solid spheres contain the same amount of material by volume initially, they expand by the same amount. This demonstrates that for isotropic materials, thermal expansion relies more on volume and less on whether the object is solid or hollow.