# A ray of light AO in vacuum is incident on a glass slab at angle $60^{\circ}$ andrefracted at angle $30^{\circ}$ along OB as shown in the figure. The optical path length of light ray from A to B is :  Option 1) $\frac{2\sqrt3}{a}+2b$ Option 2) $2a+\frac{2b}{3}$ Option 3) $2a+\frac{2b}{\sqrt3}$ Option 4) $2a+2b$

Optical Path -

$x{}'=\mu \cdot x$

- wherein

$x{}'=$ Distance travelled in vacuum

$x=$ Distance travelled in a medium of refractive index $\mu$

Optical path = AO + $\mu$ ( OB )

Apply Snells law ,

$1\cdot sin60^{\circ}=\mu \cdot sin30^{\circ}$

$\mu=\frac{\sqrt3}{2}/\frac{1}{2}=\sqrt3$

$sin 30^{\circ}=\frac{a}{AO}$

So, $AO = 2a$

Similarly,

$cos30^{\circ}=\frac{b}{OB}$

=> $OB=\frac{b}{cos30^{\circ}}$

=> $OB=\frac{2}{\sqrt3}b$

So, optical path = AO + $\mu$ ( OB )

= $2a+(\sqrt3\times \frac{2}{\sqrt3}b)$

= 2a+2b

Option 1)

$\frac{2\sqrt3}{a}+2b$

Option 2)

$2a+\frac{2b}{3}$

Option 3)

$2a+\frac{2b}{\sqrt3}$

Option 4)

$2a+2b$

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