A prism of refractive index $n_1$ and another prism of refractive index $n_2$ are stuck together (as shown in the figure). $n_1$ and $n_2$ depend on $\lambda$, the wavelength of light, according to the relation $n_1 = 1.2 + \dfrac{10.8\times10^{-14}}{\lambda^2}$ and $n_2 = 1.45 + \dfrac{1.8\times10^{-14}}{\lambda^2}$.
The wavelength for which rays incident at any angle on the interface $BC$ pass through without bending at that interface will be ______ nm.
Numerical value type. Enter your answer.
Answer: 600
No bending at $BC$ for every angle of incidence means $n_1 = n_2$:
$$1.2 + \frac{10.8\times10^{-14}}{\lambda^2} = 1.45 + \frac{1.8\times10^{-14}}{\lambda^2} \Rightarrow \frac{9\times10^{-14}}{\lambda^2} = 0.25$$
$\lambda^2 = 3.6\times10^{-13}\ \text{m}^2 \Rightarrow \lambda = 6\times10^{-7}$ m $= 600$ nm.
Solution by Sreeraj P, M.Sc Physics