Let us consider a compound slab made of two transparent media say water (w) and glass (g) bounded by parallel faces as shown in Fig. Let this compound slab be held in air (a).
A ray AO incident on the slab at O making an angle of incidence i and is refracted at angle r1 along OB. At point B, it is again refracted at angle r2 and finally refracted at C making an angle of refraction i at C because the medium on both sides of the slab is the same. So
At point O
From Snell's law, we have
\(\frac{sin\ i}{\sin\ r_1}\) = aµw ............(1)
At point B
\(\frac{sin\ r_1}{\sin\ r_2}\) = wµg ..........(2)
AT point C
From Snell's law, we have
\(\frac{sin\ r_2}{\sin\ i}\) = gµa .........(3)
Multiplying eqns. (1), (2) and (3), we get
\(\frac{sin\ i}{\sin\ r_1} \times \frac{sin\ r_1}{\sin\ r_2} \times\) \(\frac{sin\ r_2}{\sin\ i}\) = aµw x wµg x gµa
or 1 = aµw x wµg x gµa
or aµw x wµg = \(\frac{1}{^g\mu_a}\)
or aµw x wµg = aµg [∵ aµg = \(\frac{1}{^g\mu_a}\)]
or aµg = aµw x wµg.