Question

The focal length of a spherical mirror made of steel is $150 \,cm$. If the temperature of the mirror increases by $200\, K$, its focal length become (coefficient of linear expansion of steel $\alpha=12 \times 10^{-6 \circ} C ^{-1} .$ )

• A. 186.3 cm
• B. 156.3 cm
• C. 150.036 cm
• D. 150.36 cm

Answer 1

The Correct Option is D

Given,
focal length of spherical mirror, $f=150\,cm$
coefficient of linear expansion of steel,
$\alpha=12 \times 10^{-60} C ^{-1}$
As we know that,
coefficient of linear expansion, $\alpha=\frac{\Delta R}{R \times \Delta T}$
$\frac{\Delta R}{R} =\alpha \Delta T $
$\frac{\Delta f}{f} =\alpha \Delta T$
$\left(\because f=\frac{R}{2}\right) $
$\Delta\, f =f \,\alpha \,\Delta \,T $
$f'-f =f \,\alpha\, \Delta \,T$
where, $f=$ initial focal length of the mirror and $f^{\prime}=$ final focal length of mirror
$f'=f+f \,\alpha \,\Delta \,T$
$f'=f(1+\alpha \,\Delta\, T)$
$f'=150\left(1+12 \times 10^{-6} \times 200\right)$
$(\because \Delta \,T=200 \,K )$
$f'=150.36 \,cm$