Mutually perpendicular rays of light go from the air into the liquid at one angle of refraction 30 ° at the other 45 ° find the refractive index of the liquid.
Let the rays fall at angles a and b.
The angles of refraction of the rays are 45 ° and 30 °, respectively.
Snell’s law for each of the rays:
sin a / sin 45 ° = sin a / (√2 / 2) = n,
where n is the refractive index of the liquid.
sin b / sin 30 ° = n
After replacement b = 90 ° – a:
sin (90 ° – a) / sin 30 ° = cos a / 0.5 = n;
From the expressions for n, select sin a and cos a:
sin a = (√2 / 2) n;
cos a = 0.5n.
We square both sides of the equalities:
sin2 a = 0.5n ^ 2;
cos2 a = 0.25n ^ 2.
Let’s summarize the last two equalities:
sin ^ 2 a + cos ^ 2 a = 0.5n ^ 2 + 0.25n ^ 2;
1 = 0.75n ^ 2;
n = √ (1 / 0.75) = 1.15.
Answer: n = 1.15.
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