When light travels obliquely from one transparent medium into another (e.g., from air into water), its direction of propagation changes. This phenomenon is known as the refraction of light. Refraction occurs because light travels at different speeds in different media.

This relationship is described by Snell’s law:

• Coplanarity: The incident ray, refracted ray, and normal line all lie in the same plane.
• Sine Relationship: The ratio between the sines of the angles of incidence and refraction depends solely on the properties of the two media and is independent of the angle of incidence.
  

Mathematically:: $$n_1 \times \sin(\theta_1) = n_2 \times \sin(\theta_2)$$ where:

• $n_1$ and $n_2$ : Absolute refractive indices of the two media;
• $\theta_1$ : Angle of incidence;
• $\theta_2$: Angle of refraction.

Refractive Index
The refractive index $n$ describes how fast light travels through a medium: Refractive Index

$$n = \frac{c}{v}$$ where:

• $c$: Speed of light in a vacuum
• $v$: Speed of light in the medium

The higher the refractive index, the slower light travels through the medium.

The refractive index of a vacuum is 1. The refractive index of a medium relative to a vacuum is called its absolute refractive index.

• Optically rarer medium: Lower refractive index and higher speed of light
• Optically denser medium: Higher refractive index and lower speed of light
• From an optically rarer medium to an optically denser medium, such as from air to water, the angle of refraction is smaller than the angle of incidence, and the refracted ray bends toward the normal.
• From an optically denser medium to an optically rarer medium, such as from water to air, the angle of refraction is larger than the angle of incidence, and the refracted ray bends away from the normal.

Applications:
Image Formation by a Convex Lens
Light first travels from air, which has a lower refractive index, into the lens, which typically has a higher refractive index. During the first refraction, the light bends toward the normal. As it exits the lens and re-enters the air, it bends away from the normal during the second refraction. These two refractions work together to bend the light rays toward the principal axis, causing them to converge.

Image Formation by a Concave Lens
Because a concave lens is thinner in the center and thicker at the edges, parallel incident rays diverge away from the principal axis after being refracted twice in accordance with the law of refraction. The backward extensions of the diverging rays intersect to form a virtual image.

The “Bending” of a Pen in Water
Phenomenon: Schematic diagram: As shown in the figure, a pencil is partially immersed in water. At the air–water interface, the pencil appears “broken” or displaced. The apparent depth of the water is also shallower than its actual depth.

This phenomenon occurs because light from the submerged portion of the pencil is refracted as it travels from water, with a refractive index of approximately 1.33, into air, with a refractive index of approximately 1. The change in the direction of the light rays causes the eye to perceive the submerged portion at a displaced position, creating the illusion that the pencil is bent.