A unit scale is to be prepared whose length does not change with temperature and remains 20 cm , using a bimetallic strip made of brass and iron each of different length. The length of both components

Question

A unit scale is to be prepared whose length does not change with temperature and remains $20 cm$ , using a bimetallic strip made of brass and iron each of different length. The length of both components would change in such a way that difference between their lengths remains constant. If length of brass is $40 cm$ and length of iron will be _____ $cm$ .

$(α_{iron} = 1.2 \times 10^{- 5} K^{- 1}$ and $α_{brass} = 1.8 \times 10^{- 5} K^{- 1})$ .

TestHub · Accepted Answer

Linear expansion of solid with temperature is given by $L_{f} = L_{0} (1 + α ∆ T)$
Here, $l_{B} (1 + α_{B} Δ T) - l_{i} (1 + α_{i} Δ T) = l_{B} - l_{i}$
$\Rightarrow α_{B} l_{B} = l_{i} α_{i}$
$\Rightarrow 1.8 \times 10^{- 5} \times 40 = l_{i} \times 1.2 \times 10^{- 5}$
$\Rightarrow l_{i} = \frac{1.8 \times 10^{- 5} \times 40}{1.2 \times 10^{- 5}} = \frac{3 \times 40}{2} = 60$
$\Rightarrow l_{i} = 60 cm$

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