A light ray enters medium 1 from air at an angle of incidence of 40 degrees and is refracted at an angle of 24 degrees. What is the refractive index of medium 1? What is the critical angle of medium 1

Here are the calculations for question 11:

Part i) The refractive index of medium 1.

Step 1: Identify the given values from the diagram.
The angle of incidence in air is $i_1 = 40^\circ$.
The angle of refraction in medium 1 is $r_1 = 24^\circ$.
The refractive index of air is $n_{air} = 1$.

Step 2: Apply Snell's Law to find the refractive index of medium 1 ($n_1$).

$$n_{air} \sin(i_1) = n_1 \sin(r_1)$$ $$1 \cdot \sin(40^\circ) = n_1 \sin(24^\circ)$$

Step 3: Solve for $n_1$.

$$n_1 = \frac{\sin(40^\circ)}{\sin(24^\circ)}$$ $$n_1 = \frac{0.6427876}{0.4067366}$$ $$n_1 \approx 1.5802$$

Rounding to three significant figures:
The refractive index of medium 1 is $\boxed{\text{}1.58\text{}}$

Part ii) Critical angle of medium 1.

Step 1: Use the formula for the critical angle ($c$) when light travels from a denser medium (medium 1) to a less dense medium (air).

$$\sin(c) = \frac{n_{air}}{n_1}$$

Step 2: Substitute the values for $n_{air}$ and $n_1$.

$$\sin(c) = \frac{1}{1.5802}$$ $$\sin(c) \approx 0.63283$$

Step 3: Calculate the critical angle $c$.

$$c = \arcsin(0.63283)$$ $$c \approx 39.25^\circ$$

Rounding to one decimal place:
The critical angle of medium 1 is $\boxed{\text{}39.3^\circ\text{}}$

Part iii) The refractive index of medium 2 relative to medium 1 ($n_2$).

Step 1: Identify the angles for the interface between medium 1 and medium 2.
The angle of incidence in medium 1 is $i_2 = 24^\circ$.
The angle of refraction in medium 2 is $r_2 = 26^\circ$.

Step 2: The refractive index of medium 2 relative to medium 1 is given by the ratio of the sines of the angles of incidence and refraction at that interface, or by the ratio of the absolute refractive indices.

$$n_{2/1} = \frac{n_2}{n_1}$$

Using Snell's Law for the interface between medium 1 and medium 2:

$$n_1 \sin(i_2) = n_2 \sin(r_2)$$

Therefore, the relative refractive index $n_{2/1}$ can be expressed as:

$$n_{2/1} = \frac{n_2}{n_1} = \frac{\sin(i_2)}{\sin(r_2)}$$

Step 3: Substitute the angles and calculate $n_{2/1}$.

$$n_{2/1} = \frac{\sin(24^\circ)}{\sin(26^\circ)}$$ $$$$