Refraction at Plane Surfaces

TL;DR

When light passes from one transparent medium to another at an angle, it bends; this bending is called refraction. The amount of bending depends on the properties of the two materials and the angle at which the light hits the surface. Snell's Law mathematically describes this phenomenon, relating the angles and refractive indices of the media.

1. The Mental Model

Imagine you're running on a beach and suddenly hit the water at an angle; you'd slow down and change direction. Light behaves similarly: it changes speed when it moves from one transparent material to another (like air to water), causing it to bend or refract.

2. The Core Material

Refraction is the phenomenon where a light ray changes its direction as it passes from one transparent medium to another. This happens because light travels at different speeds in different media.

What causes light to bend?

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When light moves from a medium where it travels fast (like air) to a medium where it travels slower (like water or glass), it bends towards the normal. Conversely, when it moves from a slower medium to a faster one, it bends away from the normal. The "normal" is an imaginary line drawn perpendicular to the surface at the point where the light hits.

Key Terms:

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• Incident Ray: The ray of light falling on the surface separating the two media.
• Refracted Ray: The ray of light that has bent and entered the second medium.
• Normal: An imaginary line perpendicular to the surface at the point of incidence.
• Angle of Incidence (i): The angle between the incident ray and the normal.
• Angle of Refraction (r): The angle between the refracted ray and the normal.
• Refractive Index (n): A measure of how much a medium slows down light. It's the ratio of the speed of light in vacuum (c) to the speed of light in the medium (v): $n = c/v$. A higher refractive index means light travels slower in that medium.

Snell's Law

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Snell's Law quantifies the relationship between the angles of incidence and refraction, and the refractive indices of the two media.

It states: $n_1 \sin i = n_2 \sin r$

Where:
* $n_1$ is the refractive index of the first medium (where the incident ray is).
* $i$ is the angle of incidence.
* $n_2$ is the refractive index of the second medium (where the refracted ray is).
* $r$ is the angle of refraction.

Principle of Reversibility of Light

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This principle states that if a ray of light, after suffering any number of reflections or refractions, has its path reversed, it will retrace its original path.

Apparent Depth and Real Depth

When you look at an object submerged in water, it appears shallower than it actually is. This is due to refraction.
* Real Depth: The actual distance of the object from the surface.
* Apparent Depth: The perceived distance of the object from the surface.

The relationship is: $n = \frac{\text{real depth}}{\text{apparent depth}}$
Here, 'n' is the refractive index of the denser medium (e.g., water) with respect to the rarer medium (e.g., air).

graph TD
    A["Light travels in Medium 1 (n1)"] --> B{"Hits interface at an angle"}
    B --> C{Light changes speed}
    C --> D{Light bends (refracts)}
    D --> E["Enters Medium 2 (n2)"]
    E -- If n1 < n2 --> F["Bends towards Normal"]
    E -- If n1 > n2 --> G["Bends away from Normal"]

3. Worked Example

A ray of light traveling in air (refractive index $n_1 = 1.00$) hits the surface of a glass slab (refractive index $n_2 = 1.50$) at an angle of incidence of $30^\circ$. Calculate the angle of refraction.

Given:
* $n_1 = 1.00$
* $n_2 = 1.50$
* $i = 30^\circ$

Using Snell's Law: $n_1 \sin i = n_2 \sin r$

$1.00 \times \sin(30^\circ) = 1.50 \times \sin r$
$1.00 \times 0.5 = 1.50 \times \sin r$
$0.5 = 1.50 \times \sin r$
$\sin r = \frac{0.5}{1.50} = \frac{1}{3} \approx 0.3333$
$r = \arcsin(0.3333)$
$r \approx 19.47^\circ$

So, the angle of refraction is approximately $19.47^\circ$. Notice how the light bends towards the normal because it's going from a less dense medium (air) to a denser medium (glass).

4. Key Takeaways

• Refraction is the bending of light as it passes from one transparent medium to another due to a change in speed.
• The normal is an imaginary line perpendicular to the surface where the light ray strikes.
• Snell's Law ($n_1 \sin i = n_2 \sin r$) quantitatively describes the relationship between angles and refractive indices.
• When light goes from a rarer to a denser medium, it bends towards the normal.
• When light goes from a denser to a rarer medium, it bends away from the normal.
• The refractive index indicates how much a medium slows down light, with higher values meaning slower speeds.
• Objects submerged in water appear shallower than they are due to the phenomenon of apparent depth.

Common Mistakes to Avoid:
- Don't confuse the angle with the surface with the angle with the normal; always measure angles relative to the normal.
- Make sure to use the correct refractive index for each medium ($n_1$ for the incident medium, $n_2$ for the refracting medium).
- Forgetting that when light speeds up (denser to rarer), it bends away from the normal, and vice-versa.
- Assuming the angle of incidence is always greater than the angle of refraction (this only holds when going from rarer to denser).

5. Now Try It

A fish is 60 cm below the surface of a pond. If you are looking at the fish from directly above the water, how deep does the fish appear to be? (Assume the refractive index of water is 1.33 and air is 1.00). What success looks like is getting an apparent depth that is less than 60 cm, showing the fish appears shallower.