Prism formula and Related Terms

We must have seen sunlight coming from the sun.

The color of the sunlight looks white but it is made up of several colors.

Prism can split white light into its constituent colors.

Light rays passing through a prism deviates from its original path due to refraction.

Owing to this refraction, the prism disperses white light into a full spectrum of it's constituents.

In order to understand these prisms, let us first understand some terms.

Consider a prism with 2 triangular and 3 rectangular sides

Now a cross-section of the prism, would look something as shown above,

Let's draw two normals and on and respectively.

Now let a ray PQ be incident on at a point such that the angle of incidence is (as shown),

As the ray is travelling from rarer to denser medium; it bends towards the normal.

Then it refracts at and bends away from the normal (denser to rarer) forming an emergent ray.

This emergent ray forms an angle with the normal at known as the angle of emergence.

The angles of refraction are and respectively.

Now if the emergent ray is extended backwards to , then we get the angle of deviation, denoted by .

Let's now derive the prism formula.

Consider quadrilateral .

So, we can say quadrilateral is cyclic


From Eqn 1 and 2 we get,

Consider Let, ,

Now from the diagram, we can write the interior angles as,


Now simplifying the equation we get .

As and

At minimum deviation, the angle of incidence is equal to the angle of emergence

Again from the diagram and the symmetry we can write that the angle refractions are also the same

Now from equation we get,

Again and when

From Snell's law, we can write refractive index ,

The equation gives the relation between the refractive index of the material of the prism, the minimum deviation and the angle of the prism


The relation between and is also can be written as

In case of minimum deviation , . Also can be written as

The prism formula is

The End