# Mutually perpendicular rays of light go from the air into the liquid at one angle

**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.