Question

The x-z plane separates two media A and B with refractive indices $n_{1}and{n}_2$ respectively. A ray of light travels from A to B. Its directions in the two media are given by the unit vectors, $\overline{r_A}=a\hat{i}+b\hat{j}\&\overline{r_B}=\alpha \hat{i}+\beta \hat{j}$ respectively where $\hat{i}and\hat{j}$ are unit vectors in the x and y directions. Then

• A. $n_{1}a=n_2\alpha$
• B. $n_1\alpha =n_{2}a$
• C. $n_{1}b=n_2\beta$
• D. $n_1\beta =n_{2}b$

Answer 1

The correct option is A $n_{1}a=n_2\alpha$

The normal is along Y axis,
$tani=\frac{a}{b}\Rightarrow sini=\frac{a}{\sqrt{a^2+b^2}}=\frac{a}{1},sincea\hat{i}+b\hat{j}$ is an unit vector.
$tanr=\frac{\alpha }{\beta }\Rightarrow sinr=\frac{\alpha }{{\alpha }^2+{\beta }^2}=\alpha since\alpha \hat{i}+\beta \hat{j}$ is an unit vector.
Snell’s law gives,
$n_{1}sini=n_{2}sinr$
$\Rightarrow n_{1}a=n_2\propto$