Total initial length at 30°C = 120 cm.
Since rods have same lengths:
\[ L_{Al} = L_{Steel} = 60 \text{ cm} \]
Temperature rise:
\[ \Delta T = 100 - 30 = 70^\circ C \]
Linear expansion formula:
\[ \Delta L = L \alpha \Delta T \]
Expansion of aluminium:
\[ \Delta L_{Al} = 60 \times 24 \times 10^{-6} \times 70 \]
\[ = 0.1008 \text{ cm} \]
Expansion of steel:
\[ \Delta L_{Steel} = 60 \times 1.2 \times 10^{-5} \times 70 \]
\[ = 0.0504 \text{ cm} \]
Total expansion:
\[ \Delta L = 0.1008 + 0.0504 = 0.1512 \text{ cm} \]
Final length:
\[ L = 120 + 0.1512 \approx 120.15 \text{ cm} \]
Therefore correct answer is (C).
Thermal expansion is one of the most fundamental topics in thermal physics and plays a very important role in JEE Main and JEE Advanced examinations. When a substance is heated, its temperature increases, leading to an increase in the average kinetic energy of its molecules. As a result, intermolecular distances increase slightly, causing expansion in dimensions.
There are three types of thermal expansion: linear expansion, area expansion and volume expansion. For rods and wires, linear expansion is most relevant.
The formula for linear expansion is:
\[ \Delta L = L_0 \alpha \Delta T \]
Where L₀ is initial length, α is coefficient of linear expansion and ΔT is change in temperature.
The coefficient of linear expansion depends on material. Aluminium has larger α than steel, which means aluminium expands more for the same temperature rise.
For composite rods joined end to end, each part expands independently. Total expansion equals sum of individual expansions. This is because they are in series mechanically.
Important conceptual understanding: Expansion is proportional to original length. If two rods have same length but different α, the one with larger α expands more.
In JEE problems, composite rods are common. Sometimes rods are constrained, leading to thermal stress. In such cases, expansion is restricted and internal forces develop.
Thermal stress formula (when prevented from expansion):
\[ Stress = Y \alpha \Delta T \]
Where Y is Young’s modulus.
In unrestricted expansion like this question, no stress develops because rod is free.
Exam relevance: Linear expansion questions are direct but require careful calculation and unit consistency.
Common mistakes:
• Forgetting to divide total length equally
• Using wrong coefficient powers
• Arithmetic mistakes
• Confusing 10⁻6 and 10⁻5 values
Applications in real life:
• Railway tracks have gaps to allow expansion
• Bridges have expansion joints
• Bimetallic strips work on differential expansion principle
• Thermostats use expansion difference
In bimetallic strip, two metals with different α are joined. When heated, unequal expansion causes bending. This principle is widely used in electrical switches.
In JEE Advanced, composite rod problems may involve stress when ends are fixed. Students must apply condition that total change in length is zero and solve for stress distribution.
Volume expansion relation:
\[ \gamma = 3\alpha \]
For isotropic solids, coefficient of volume expansion is three times linear expansion.
Area expansion coefficient is 2α.
Understanding microscopic reason: As temperature increases, potential energy curve of interatomic forces becomes asymmetric, leading to greater average separation.
This topic links thermodynamics, solid state physics and mechanics concepts.
Practice numerical variations including unequal lengths, different temperature changes and constraint cases to master this topic.
With conceptual clarity and calculation accuracy, thermal expansion questions become easy scoring areas in JEE Main.
This educational content is prepared for learning and practice for competitive examinations such as JEE Main, JEE Advanced and NEET. Students should verify calculations independently during preparation.