Light takes the same time to pass normally through a glass plate of thickness 6 mm as through a layer of water of thickness x. If the refractive indices of glass and water are 1.5 and 4/3 respectively, the value of x is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 6.75 mm
In a medium of refractive index \( n \) light travels with the speed \( v=c/n \). So light crosses a layer of thickness \( d \) in the time \( t=nd/c \). Two layers take equal times when the product \( nd \) is the same for both.
The time taken in the glass plate is:
\( \displaystyle t_g=\frac{n_g\,d_g}{c} \)
The time taken in the water layer is:
\( \displaystyle t_w=\frac{n_w\,x}{c} \)
Equating the two times, \( n_w\,x=n_g\,d_g \), so the thickness of water is:
\( \begin{aligned} x &= \frac{n_g\,d_g}{n_w} \\ &= \frac{1.5\times6\,\mathrm{mm}}{4/3} \\ &= 6.75\,\mathrm{mm} \end{aligned} \)
Light moves faster in water than in glass, so it needs a thicker layer of water to spend the same time.