Snells Law

Team Physics -
Created by: Team Physics -, Last Updated: July 4, 2024

Snells Law

Introduction of Snells Law

Snell’s Law, also known as the Law of Refraction, is a fundamental principle in Laws of Wave and Optics that describes how light bends, or refracts, as it passes from one medium into another with a different refractive index. The law was named after the Dutch mathematician Willebrord Snellius, who discovered the relationship in 1621. Snell’s Law provides the mathematical relationship between the angles of incidence and refraction when considering the refraction of light at the boundary between two different isotropic media. This law is crucial for understanding and designing optical devices such as lenses, prisms, and fiber optic cables, and it plays a significant role in fields ranging from physics and engineering to vision sciences and photography.

snell’s law formula

The formula for Snell’s Law, which describes the relationship between the angles of incidence and refraction when light passes between two media with different refractive indices, is given by:

n₁ sin(θ₁ ) = n₂ sin(θ₂ )


n₁ and 𝑛₂ are the refractive indices of the first and second media, respectively.
𝜃₁ is the angle of incidence — the angle between the incident light ray and the normal (perpendicular) to the surface at the point of incidence. 𝜃₂ is the angle of refraction — the angle between the refracted light ray and the normal to the surface at the point of refraction.

Derivation of snell’s law formula

Derivation of Snell’s Law using Fermat’s Principle:
Step1: Setting the Scenario:
Consider a light ray passing from one medium with refractive index
𝑛1 into another medium with refractive index 𝑛2. The ray strikes the boundary at an angle 𝜃1to the normal and refracts at an angle 𝜃2 in the second medium.

Step 2: Fermat’s Principle:
Fermat’s Principle of Least Time states that the path taken by the light ray between any two points (in this case, from a point in medium 1 to a point in medium 2) minimizes the travel time.

Step 3: Mathematical Formulation:
Let’s assume the light ray travels from point 𝐴 in medium 1 to point 𝐵 in medium 2, crossing the interface at point 𝐶. The distances 𝐴𝐶 and 𝐶𝐵 depend on where the ray crosses the boundary.

Step 4: Travel Time Calculation:
The time taken to travel from A to 𝐶 is 𝑡1=𝐴𝐶/𝑣1 and from 𝐶 to 𝐵 is 𝑡2=𝐶𝐵/𝑣2 where 𝑣1=𝑐/𝑛1 and 𝑣2=𝑐𝑛2/v2​ are the speeds of light in medium 1 and 2, respectively (with 𝑐 being the speed of light in vacuum).

Step 5: Expressing Distances Using Trigonometry:
Using the definitions of sine, 𝐴 𝑑1sin(𝜃1)AC=d1sin(θ1) and CB=d2​sin(θ2​), where 𝑑1and 𝑑2 are the perpendicular distances from points 𝐴 and𝐵 to the boundary.

Step 6: Minimizing the Total Time:
The total travel time 𝑇 = d1sin(θ1)/ 𝑣1 + d2​sin(θ2​)/𝑣2 According to Fermat’s Principle, 𝑇 should be a minimum. Using calculus, particularly the method of Lagrange multipliers or simply setting the derivative of 𝑇 with respect to the path variable (like the position of 𝐶 along the boundary) to zero, leads to the condition: n₁ sin(θ₁ ) = n₂ sin(θ₂ )

This final condition is the mathematical expression of Snell’s Law. It shows that the path of light changes at the boundary in such a way that the time to travel between two points is minimized, accounting for changes in the speed of light due to differing media.

Applications of snell’s law

1.Design of Lenses
Snell’s Law is crucial in designing optical lenses used in cameras, microscopes, and eyeglasses. By understanding how light refracts through different materials, manufacturers can create lenses that precisely focus light to improve image quality or correct vision.

2.Fiber Optic Communication
In fiber optics, Snell’s Law helps engineers design the core and cladding of optical fibers to achieve total internal reflection. This principle ensures that light signals travel long distances with minimal loss, which is essential for high-speed internet and telecommunications.

3.Optical Instruments
Snell’s Law is used in the development of various optical instruments like telescopes, binoculars, and periscopes. It helps in calculating the necessary angles and materials for prisms and mirrors to correctly direct light and produce clear images.

4.Underwater Imaging
Snell’s Law explains how light bends when entering water from air, which is critical for designing underwater cameras and instruments. Understanding refraction helps correct distortions in underwater photography and videography.

In gemology, Snell’s Law assists in cutting precious stones at angles that enhance their inherent brilliance. Proper cutting techniques that consider light refraction increase the sparkle and value of gemstones.

Snell’s Law helps meteorologists understand phenomena like mirages, where variations in the refractive index of air due to temperature gradients cause the bending of light rays from distant objects, creating optical illusions.

7.Medical Imaging
Some medical imaging techniques, such as optical coherence tomography, rely on understanding how light refracts through different tissues. Snell’s Law assists in interpreting the data to provide accurate images.


What is a real life example of Snell’s law?

A real-life example of Snell’s Law is the bending of light when it passes from air into water, visibly altering the appearance of objects submerged in a pool.

Where is Snell’s law not applicable?

Snell’s Law is not applicable when light encounters non-linear, anisotropic, or non-homogeneous media where the refractive index varies with direction, polarization, or position.

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