Refraction of Light and Refractive Index

Refraction Of Light

Refraction of light is the phenomenon of bending of light as it passes from one transparent medium to another. This bending occurs because light travels at different speeds in different media. When a ray of light enters a medium where its speed changes, its path bends (unless it is incident normally to the boundary).

Refraction is responsible for many common optical phenomena, such as the apparent bending of a spoon in a glass of water, the shimmering effect of heat rising from a road, the formation of rainbows, and the working of lenses in cameras, spectacles, and telescopes.

---

Why Refraction Occurs

When light travels from one medium to another, its frequency remains unchanged (determined by the source), but its speed and wavelength change. The change in speed causes the light ray to bend.

• If light travels from an optically rarer medium (where speed of light is higher, e.g., air) to an optically denser medium (where speed of light is lower, e.g., water, glass), it bends towards the normal.
• If light travels from an optically denser medium to an optically rarer medium, it bends away from the normal.

The normal is a line perpendicular to the boundary between the two media at the point of incidence.

---

Laws of Refraction

Refraction of light obeys two laws, known as the laws of refraction (or Snell's Laws):

1. The incident ray, the refracted ray, and the normal to the interface of two transparent media at the point of incidence, all lie in the same plane. (Similar to the first law of reflection).
2. Snell's Law: The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media and a given colour of light.

   $ \frac{\sin i}{\sin r} = \text{constant} $

   where $i$ is the angle of incidence (angle between incident ray and the normal) and $r$ is the angle of refraction (angle between refracted ray and the normal). This constant is called the refractive index of the second medium with respect to the first medium.

The angle of incidence is always in the first medium, and the angle of refraction is always in the second medium.

(Image Placeholder: A diagram showing a boundary between Medium 1 and Medium 2. An incident ray strikes the boundary. A normal is drawn at the point of incidence. A refracted ray passes into Medium 2, bending at the boundary. Label the incident ray, refracted ray, normal, angle of incidence (i), angle of refraction (r), Medium 1 (e.g., Air, n1), Medium 2 (e.g., Glass, n2).)

---

Refraction Through A Rectangular Glass Slab

When a ray of light passes through a rectangular glass slab with parallel faces, it undergoes refraction twice: first at the interface between air and glass, and second at the interface between glass and air.

(Image Placeholder: A rectangular glass slab. An incident ray enters from air into glass at the first surface, bending towards the normal. This refracted ray travels through the glass and strikes the second parallel surface, exiting back into air and bending away from the normal. Show the original direction of the incident ray (as a dotted line). The emergent ray is parallel to the original incident ray but laterally shifted.)

Let a ray of light be incident on the first surface (Air-Glass interface) at an angle of incidence $i_1$. It refracts into the glass at an angle of refraction $r_1$. The refracted ray travels through the glass and is incident on the second surface (Glass-Air interface). Since the two faces of the slab are parallel, the normal at the second surface is parallel to the normal at the first surface. The angle of incidence at the second surface, $i_2$, is equal to the angle of refraction at the first surface, $r_1$ (alternate interior angles).

The ray then refracts out into the air at an angle of refraction $r_2$. Applying Snell's Law at both interfaces:

At Air-Glass interface: $\frac{\sin i_1}{\sin r_1} = n_{glass, air}$ (refractive index of glass with respect to air)

At Glass-Air interface: $\frac{\sin i_2}{\sin r_2} = n_{air, glass}$ (refractive index of air with respect to glass)

We know that $i_2 = r_1$ and $n_{glass, air} = 1/n_{air, glass}$. So, $\frac{\sin r_1}{\sin r_2} = \frac{1}{n_{glass, air}}$, which means $n_{glass, air} \sin r_1 = \sin r_2$.

Since $\sin i_1 = n_{glass, air} \sin r_1$, and $n_{glass, air} \sin r_1 = \sin r_2$, we have $\sin i_1 = \sin r_2$. This implies $i_1 = r_2$ (assuming both angles are less than 90°). The angle of emergence ($r_2$) is equal to the angle of incidence ($i_1$).

Thus, the emergent ray is parallel to the incident ray. However, the emergent ray is laterally shifted from the original path of the incident ray. The amount of lateral shift depends on the angle of incidence, the thickness of the slab, and the refractive index of the glass.

---

The Refractive Index ($ n = c/v $)

The refractive index ($n$) of a transparent medium is a fundamental optical property that describes how much light slows down when it passes through that medium compared to its speed in a vacuum.

---

Absolute Refractive Index

The absolute refractive index of a medium is defined as the ratio of the speed of light in a vacuum ($c$) to the speed of light in that medium ($v$).

$ n = \frac{\text{Speed of light in vacuum}}{\text{Speed of light in medium}} = \frac{c}{v} $

Since $c$ is the maximum speed of light, the absolute refractive index of any medium is always greater than or equal to 1. The refractive index of vacuum is $n_{vacuum} = c/c = 1$. The refractive index of air is very close to 1 (approx. 1.0003) and is often approximated as 1 for practical purposes.

A medium with a higher refractive index is called optically denser, and light travels slower in it. A medium with a lower refractive index is called optically rarer, and light travels faster in it.

---

Relative Refractive Index

The refractive index of medium 2 with respect to medium 1 ($n_{21}$ or $\mu_{21}$) is the ratio of the speed of light in medium 1 ($v_1$) to the speed of light in medium 2 ($v_2$).

$ n_{21} = \frac{\text{Speed of light in medium 1}}{\text{Speed of light in medium 2}} = \frac{v_1}{v_2} $

We can express the speeds in terms of absolute refractive indices: $v_1 = c/n_1$ and $v_2 = c/n_2$.

$ n_{21} = \frac{c/n_1}{c/n_2} = \frac{n_2}{n_1} $

So, the refractive index of medium 2 with respect to medium 1 is the ratio of the absolute refractive index of medium 2 to the absolute refractive index of medium 1. Note that $n_{12} = 1/n_{21}$.

Snell's Law in terms of Refractive Indices: Using $n_{21} = \frac{\sin i}{\sin r}$ and $n_{21} = \frac{n_2}{n_1}$, we get:

$ \frac{\sin i}{\sin r} = \frac{n_2}{n_1} $

$ n_1 \sin i = n_2 \sin r $

This is the most common form of Snell's Law, relating the angles of incidence and refraction to the absolute refractive indices of the two media. Here, $i$ is the angle in medium 1 (with refractive index $n_1$) and $r$ is the angle in medium 2 (with refractive index $n_2$).

The refractive index of a medium depends slightly on the wavelength (colour) of light, a phenomenon called dispersion. This is why white light splits into its constituent colours when passing through a prism.

Refraction (Snell's Law $ n_1\sin\theta_1 = n_2\sin\theta_2 $)

As discussed, refraction is the bending of light as it passes from one transparent medium to another. The degree of bending is governed by the laws of refraction, with the second law, Snell's Law, being the quantitative relationship.

---

Snell's Law Statement

Snell's Law states that for a ray of light passing from medium 1 to medium 2, the ratio of the sine of the angle of incidence ($\theta_1$) in medium 1 to the sine of the angle of refraction ($\theta_2$) in medium 2 is equal to the ratio of the refractive index of medium 2 ($n_2$) to the refractive index of medium 1 ($n_1$).

$ \frac{\sin\theta_1}{\sin\theta_2} = \frac{n_2}{n_1} $

Rearranging this gives the more commonly remembered form:

$ n_1 \sin\theta_1 = n_2 \sin\theta_2 $

where:

• $\theta_1$ is the angle of incidence in medium 1.
• $\theta_2$ is the angle of refraction in medium 2.
• $n_1$ is the absolute refractive index of medium 1.
• $n_2$ is the absolute refractive index of medium 2.

The angles $\theta_1$ and $\theta_2$ are always measured with respect to the normal to the interface.

---

Direction of Bending from Snell's Law

• If light goes from a rarer medium to a denser medium ($n_1 < n_2$), then $\sin\theta_1 / \sin\theta_2 = n_2/n_1 > 1$. This implies $\sin\theta_1 > \sin\theta_2$. Since sine is increasing in the first quadrant, $\theta_1 > \theta_2$. The refracted ray bends towards the normal.
• If light goes from a denser medium to a rarer medium ($n_1 > n_2$), then $\sin\theta_1 / \sin\theta_2 = n_2/n_1 < 1$. This implies $\sin\theta_1 < \sin\theta_2$. So, $\theta_1 < \theta_2$. The refracted ray bends away from the normal.
• If the light is incident normally ($\theta_1 = 0^\circ$), $\sin\theta_1 = 0$. By Snell's law, $n_1 \sin(0^\circ) = n_2 \sin\theta_2 \implies 0 = n_2 \sin\theta_2$. Since $n_2 \ne 0$, $\sin\theta_2 = 0$, so $\theta_2 = 0^\circ$. A ray incident normally passes straight through the boundary without bending.

---

Relation to Speed and Wavelength

Since $n = c/v$, Snell's law can also be written in terms of the speeds of light in the two media: $ \frac{c}{v_1} \sin\theta_1 = \frac{c}{v_2} \sin\theta_2 $. So, $ \frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} $. This confirms that the ratio of sines of angles is equal to the ratio of speeds. The speed of light is different in different media, which is the fundamental cause of refraction.

Also, since the frequency $\nu$ of light remains constant when it passes from one medium to another, and $v = \nu \lambda$, we have $\lambda_1 = v_1/\nu$ and $\lambda_2 = v_2/\nu$.

$ \frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{\nu\lambda_1}{\nu\lambda_2} = \frac{\lambda_1}{\lambda_2} $

So, Snell's law also relates the angles to the wavelengths in the two media. When light enters an optically denser medium, its speed and wavelength decrease, and it bends towards the normal. When it enters a rarer medium, its speed and wavelength increase, and it bends away from the normal.

Answer: