A ray of light AO in vacuum is incident on a glass slab at angle 60° and refracted at angle 30° along OB as shown in the figure. The optical path length of light ray from A to B is:

 Concept: Optical Path Length

 Optical path length (OPL) = μ×actual path length

• In vacuum: μ=1
• In medium: μ>1

 Step 1: Apply Snell’s Law

1⋅sin⁡60∘=μ⋅sin⁡30∘1 

$$μ=\ \frac{\sin60∘\ }{\sin30∘​}=\sqrt{\ 3}​$$

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 Step 2: Write optical path

OPL=AO+μ⋅OB

 Step 3: Express in terms of given lengths

$$AO=\ \frac{\ a}{\cos60∘}​,\ OB=\ \frac{\ b}{\cos30∘}​$$

$$OPL=\ \frac{\ a}{\cos60∘}​+\sqrt{\ 3}​⋅\ \frac{\ b}{\cos30∘}​$$

 Step 4: Simplify

$$\cos60∘=\ \frac{\ 1}{2}​,\cos30∘=\ \frac{\ \sqrt{\ 3}}{2}​​$$

$$OPL=2a+2b$$