Prove : n_ 12 =1 n_ 21 - Vidyadip

Question

Prove : $n_{12}=\frac{1}{n_{21}}$

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Answer

In the figure below, if $n_{21}$ is the refractive index of medium 2 with respect to medium 1, then from Snell's law :
$
n_{21}=\frac{\sin i_1}{\sin r_1} \ldots(i)
$

Fig. : Lateral displacement of a light ray refracted from a slab of parallel faces.
Similarly, if $n_{12}$ is the refractive index of medium 1 relative to medium 2, then from Snell's law :
$
n_{12}=\frac{\sin i_2}{\sin r_2} \ldots(ii)
$
On multiplying equations (i) and (ii) :
$
n_{21} \times n_{12}=\frac{\sin i_1}{\sin r_1} \times \frac{\sin i_2}{\sin r_2}
$
From fig. $\angle r_1=\angle i_2$
And $\quad \angle i_1=\angle r_2$
$
\therefore \quad n_{21} \times n_{12}=\frac{\sin i_1}{\sin i_2} \times \frac{\sin i_2}{\sin i_1}=1
$
or $\quad n_{21} \times n_{12}=1$
or$\quad n_{12}=\frac{1}{n_{21}}$ $\qquad$Hence proved.

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