Snell’s Law – Learn

Refraction is the bending of light as it enters a different medium. This is a direct result of the change in speed of the light ray in the new medium. The degree to which the light ray refracts is dependent on how much its speed changes.

Refractive Index:

The refractive index of medium x, { n }_{ x }, is defined as the ratio of the speed of light in a vacuum, c, to the speed of light in the medium { v }_{ x }. Mathematically, the refractive index of medium x is given by:

{ n }_{ x }=\cfrac { c }{ { v }_{ x } }

Where:

{ n }_{ x } = the refractive index of medium x

c = the speed of light in a vacuum

{ v }_{ x } = the speed of light in the medium

As this applies to any material and c is a constant:

{ n }_{ 1 }{ v }_{ 1 }={ n }_{ 2 }{ v }_{ 2 }

Snell’s Law:

Snell’s Law is a mathematical relationship for incident and refracted rays passing across a boundary between two medium:

{ n }_{ 1 }\sin\theta _{ 1 }={ n }_{ 2 }\sin\theta _{ 2 }

Where:

{ n }_{ 1 } = the refractive index of the incident medium

{ n }_{ 2 } = the refractive index of the refracted medium

sin\theta _{ 1 } = the angle of incidence

sin\theta _{ 1 } = the angle of refraction

Snell’s Law is illustrated below:

Example 1: Changes to speed.

A ray of light travels from water (n=1.33) with a speed of 2.25\times 10^{ 8 }{ m }/{ s } into glass (n=1.52). Calculate the speed of the light in glass:

Using: { n }_{ 1 }{ v }_{ 1 }={ n }_{ 2 }{ v }_{ 2 }

1.33\times 2.25\times 10^{ 8 }=1.52\times { v }_{ 2 }

{ v }_{ 2 }=\cfrac { 1.33\times 2.25\times 10^{ 8 } }{ 1.52 }

{ v }_{ 2 }={ 1.97×10^{ 8 } }{ m }/{ s }

Example 2: Snell’s Law.

A ray of light in air strikes water (n=1.33) at an incident angle of 24º to the normal. Calculate the angle of refraction:

Using: { n }_{ 1 }\sin\theta _{ 1 }={ n }_{ 2 }\sin\theta _{ 2 }

1.00\times \sin { 24 } =1.33\times \sin { \theta } _{ 2 }

\sin { \theta _{ 2 } } =\cfrac { 1.00\times \sin { 24 } }{ 1.33 }

\sin { \theta _{ 2 } } =0.3058

{ \theta }_{ 2 }=\sin ^{ -1 }{ 0.3058 }

{ \theta }_{ 2 }=17.8^{ \circ }