A Ray of light falls normally on a flat surface of a glass hemisphere of radius 4 cm. The distance from the point of incidence of the beam to the axis of the hemisphere a = 2 cm , index of refraction

Question

A Ray of light falls normally on a flat surface of a glass hemisphere of radius 4 cm. The distance from the point of incidence of the beam to the axis of the hemisphere a = 2 cm , index of refraction n = 2 ​ . At what distance x (in cm) from the curved surface of the hemisphere does the beam cross the axis ?

TestHub · Accepted Answer

Inside the hemisphere the ray travels at $sin^{- 1} \frac{a}{R} = 3 0^{\circ}$ to the axis-normal; refraction out: $2 sin 3 0^{\circ} = 1 \cdot sin ϕ \Rightarrow ϕ = 4 5^{\circ}$ .

Sine rule in the ray triangle: $\frac{R + x}{s i n 13 5 ^{\circ}} = \frac{R}{s i n 1 5 ^{\circ}}$ .

$R + x = \frac{2 R}{3 - 1}$ , so $x = R [\frac{2}{3 - 1} - 1] = R \frac{3 - 3}{3 - 1} = 6.92$ - $6.93$ cm

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