The co-efficient of thermal expansion of a rod is temperature dependent and is given by the formula α=aT, where a is a positive constant and T in 0C. If the length of the rod is l at temperature 00C, then the temperature at which the length will be 2l is

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The co-efficient of thermal expansion of a rod is temperature dependent and is given by the formula $α = aT,$ where a is a positive constant and T in $^{0} C .$ If the length of the rod is l at temperature $0^{0} C,$ then the temperature at which the length will be $2 l$ is

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$\sqrt{\frac{\ln 2}{a}}$

$\sqrt{\frac{\ln 4}{a}}$

$\frac{1}{a}$

$\frac{2}{a}$

answer is B.

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Detailed Solution

dl = l $α$ dT

$\frac{d l}{l} = a T d T \int_{l}^{2 l} \frac{d l}{l} = \int_{0}^{T_{o}} a T . d T {[\ln l]}_{l}^{2 l} = {[\frac{a T^{2}}{2}]}_{0}^{T_{o}} \ln 2 = \frac{a T_{o}^{2}}{2} T_{o} = \sqrt{\frac{\ln 4}{a}}$