In an experiment of measuring the refractive index of a glass slab using travelling microscope in physics lab , a student measures real thickness of the glass slab as 5.25 mm and apparent thickness of the glass slab as 5.00 mm . Travelling microscope has 20 divisions in one cm on main scale and 50 divisions on Vernier scale is equal to 49 divisions on main scale .The estimated uncertainty in the measurement of refractive index of the slab is $\frac{x}{10} \times 10^{- 3}$ , where $x$ is _____

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answer is 41.

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Detailed Solution

Refractive index of slab $μ = \frac{Real thicknessof slab}{Apparent thickness of slab}$
$\begin{aligned} μ = \frac{d_{r}}{d_{a}} = \frac{5.25}{5.00} = 1.05 \\ \frac{Δμ}{μ} = \frac{{Δd}_{r}}{d_{r}} + \frac{{Δd}_{a}}{d_{a}} \end{aligned}$
Least count error in measurement of thickness is
L.C=1M.S.D-1V.S.D
$\begin{array}{l} = \frac{1}{20} cm - \frac{49}{50} \times \frac{1}{20} cm \\ = \frac{1}{1000} cm \\ L.C = 0.01 mm \\ \frac{Δμ}{μ} = \frac{0.01}{5.25} + \frac{0.01}{5.00} \\ Δμ = (0.0039) (1.05) = 0.0041 \\ Δμ = \frac{41}{10} \times 10^{- 3} \end{array}$