Coefficient of Linear Expansion Derivation – Rankers Physics
Topic: Thermal Physics
Subtopic: Thermal Expansion

Coefficient of Linear Expansion Derivation

Assertion (A): The temperature of a metallic rod is raised by a temperature \(\Delta t\) so that its length becomes double. The value of \(\alpha\) (coefficient of linear expansion) is given by \(\frac{\log_e (2)}{\Delta t}\).
Reason (R): Coefficient of linear expansion is defined as \(\frac{1}{l} \frac{dl}{dt}\).
 
(1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
(2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
(3) (A) is true but (R) is false
(4) Both (A) and (R) are false

Solution:

Reason (R) is the correct definition for the instantaneous coefficient of linear expansion (assuming \(t\) is temperature), so it is true. Integrating \(dl/l = \alpha dT\) with constant \(\alpha\) gives \(ln(l/l_0) = \alpha \Delta T\), so \(l = l_0 e^{\alpha \Delta T}\). If \(l = 2l_0\), then \(2 = e^{\alpha \Delta T}\), leading to \(\alpha = \frac{ln 2}{\Delta T}\).

So Assertion (A) is true. (R) provides the foundational definition from which (A) is derived, thus it's the correct explanation.

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