Light in a vacuum is incident on a transparent glass slab. The angle of incidence is 35.1°. The slab is then immersed in a pool of liquid. When the angle of incidence for the light striking the slab is 21.1°, the angle of refraction for the light entering the slab is the same as when the slab was in a vacuum. What is the index of refraction of the liquid?

Question

Light in a vacuum is incident on a transparent glass slab. The angle of incidence is 35.1°. The slab is then immersed in a pool of liquid. When the angle of incidence for the light striking the slab is 21.1°, the angle of refraction for the light entering the slab is the same as when the slab was in a vacuum. What is the index of refraction of the liquid?

Answer 1

To find the index of refraction of the liquid, we need to use Snell's law, which relates the angles of incidence and refraction with the refractive indices of the mediums involved.

Snell's law states:

n1 * sin(θ1) = n2 * sin(θ2)

Where:
- n1 is the refractive index of the medium from which light is coming (in this case, the vacuum),
- θ1 is the angle of incidence,
- n2 is the refractive index of the medium in which light is entering (in this case, the transparent glass slab and the liquid), and
- θ2 is the angle of refraction.

In the given scenario, when the light is incident on the glass slab in a vacuum, the angle of incidence is 35.1°. Let's denote the refractive index of the glass slab as ng.

Using Snell's law, we have:

1 * sin(35.1°) = ng * sin(θ2) (eq.1)

Next, the glass slab is immersed in the liquid. When the light strikes the slab at an angle of 21.1°, the angle of refraction is the same as when the slab was in a vacuum. Let's denote the refractive index of the liquid as nl.

Using Snell's law again, we have:

ng * sin(21.1°) = nl * sin(θ2) (eq.2)

Since the angle of refraction (θ2) remains the same in both cases, we can set eq.1 and eq.2 equal:

1 * sin(35.1°) = ng * sin(21.1°)

Now, we need to solve this equation to find the value of ng, the refractive index of the glass slab.

ng = (1 * sin(35.1°)) / sin(21.1°)

After calculating ng, we can use the refractive index of the glass slab to find the refractive index of the liquid. Let's denote the angle of refraction in the liquid as θ2'.

Using Snell's law once more:

ng * sin(21.1°) = nl * sin(θ2')

Since sin(θ2) = sin(θ2') (as given in the problem), we can equate them:

ng * sin(21.1°) = nl * sin(21.1°)

Simplifying, we find that ng = nl.

Therefore, the index of refraction of the liquid is equal to the index of refraction of the glass slab.