Question

Two rods of length ${{l}_{1}}$ and ${{l}_{2}}$ are made of material whose coefficients of linear expansion are ${{\alpha }_{1}}$ and ${{\alpha }_{2}}$ . If the difference between their length is independent of temperature then.
(a) $\dfrac{\alpha _{1}^{2}}{{{l}_{1}}}=\dfrac{\alpha _{2}^{2}}{{{l}_{2}}}$
(b) $\dfrac{{{l}_{1}}}{{{l}_{2}}}=\dfrac{{{\alpha }_{1}}}{{{\alpha }_{2}}}$
(c) $\dfrac{{{l}_{1}}}{{{l}_{2}}}=\dfrac{{{\alpha }_{2}}}{{{\alpha }_{1}}}$
(d) $1l_{2}^{2}{{\alpha }_{1}}=l_{1}^{2}{{\alpha }_{2}}$

Answer 1

Hint : First find the change in length of both rods and then find ${{l}_{1}}$ apply this concept to determine values of this question ${{l}_{2}}$
$\Delta l=l\alpha \Delta \theta$
Where,
$\Delta \theta =$ change in temperature
$\Delta l=$ change in length.

Complete Step By Step Answer:
As per data given in the question we have,
Length of rods are ${{l}_{1}}$ and ${{l}_{2}}$
Coefficient of linear are ${{\alpha }_{1}}\And {{\alpha }_{2}}$
By increasing the temperature , the length of both rods will increase too.
When temperature is increased and changes in lengths the difference between their length is independent of temperature.
So, increase in the length of first rod will be
$\Delta {{l}_{1}}={{l}_{1}}{{\alpha }_{1}}\Delta \theta$
And increase in the length of second rod will be
$\Delta {{l}_{2}}={{l}_{2}}{{\alpha }_{2}}\Delta \theta$
$\left( \Delta {{l}_{1}} \right)=\left( \Delta {{l}_{2}} \right)$
${{l}_{1}}{{\alpha }_{1}}\Delta \theta ={{l}_{2}}{{\alpha }_{2}}\Delta \theta$
${{l}_{1}}{{\alpha }_{1}}={{l}_{2}}{{\alpha }_{2}}$
$\dfrac{{{l}_{1}}}{{{l}_{2}}}=\dfrac{{{\alpha }_{2}}}{{{\alpha }_{1}}}$
Hence option C is the correct answer.

Note :
Change in length or increase in length is expansion. When change in length is along one dimension over the volume then it is called linear expansion.
So definition of coefficient of linear expansion is,
Ratio of change of unit length per unit to change in temperature is called coefficient of linear expansion.
Coefficient of linear expansion is different for different materials.
For aluminium coefficient of linear expansion at $20{}^\circ C$ is $23.1\times {{10}^{-6}}{{K}^{-1}}$
For copper coefficient of linear expansion at $20{}^\circ C$ is $17\times {{10}^{-6}}{{K}^{-1}}$
Best example of a coefficient of linear expansion is rail tracks. When temperature is more than rail tracks expands its shape.
Read the question properly, we have to prove that the difference between lengths of both rods are independent of temperature.