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.