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 $5.25\ \text{mm}$ and apparent thickness of the glass slab at $5.00\ \text{mm}$. 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 $\dfrac{x}{10}\times10^{-3}$, where $x$ is ______.
Numerical value type. Enter your answer.
Answer: 41
Least count: $1\ \text{MSD}=\dfrac{1\ \text{cm}}{20}=0.5\ \text{mm}$ and $1\ \text{VSD}=\dfrac{49}{50}\ \text{MSD}$, so
$$\text{LC}=1\ \text{MSD}-1\ \text{VSD}=\frac{0.5}{50}=0.01\ \text{mm}$$
$\mu=\dfrac{\text{real}}{\text{apparent}}=\dfrac{5.25}{5.00}=1.05$, and
$$\frac{\Delta\mu}{\mu}=\frac{0.01}{5.25}+\frac{0.01}{5.00}=0.0019+0.0020=0.0039$$
$$\Delta\mu=1.05\times0.0039\approx4.1\times10^{-3}=\frac{41}{10}\times10^{-3}\ \Rightarrow\ x=41$$
Solution by Sreeraj P, M.Sc Physics