Derive i+e=A+`delta` for a triangular glass prism.

Explanation

In figure the cross-section perpendicular to the rectangular surface of a prism made up of a transparent medium is shown.

A ray of monochromatic light is incident at point Q on the AB surface of the prism.
According to Snell.s Law, it is refracted and travels along path QR. It experiences deviation `delta_1`, at point Q.
Ray QR is incident on surface AC at point R and suffering deviation `delta_2`, due to refraction emerges in the direction RS.
If the incident ray PQ is extended it advances in QE direction. When the emergent ray RS is extended backwards it meets PE in D.
Incidence angle : Angle made by incident ray at a point on surface with normal is called incidence angle.
`anglePQK = i`
Refracted angle : Angle made by refracted ray at a point on surface with normal is called refracted angle. `angleRQL = r_1`, and `angleQRL = r_2`
Emergence angle : Angle made by emergent ray with normal of surface is called emergence angle.
`angle`TRS = e
Angle of Deviation : “Angle between incident ray and emergent ray is called angle of deviation `(delta)`”.
`angle epsilon DS =delta`
In the figure in `squareAQLR,`
`angleAQL = 90^@` and `angleARL= 90^@` are right angle.
`therefore angleA+angleQLR=180^@` ... (1)
Now, in `triangleQLR`
`r_1+r_2+angleQLR=180^@` ...(2)
From equation (1) and (2),
`r_1+r_2+angleQLR=angleA+angleQLR=180^@`
`therefore r_1+r_2=angleA` ... (3)
From the geometry of the figure angle of deviation `delta` is the exterior angle for `angleDQR i.e. angleEDS = angleDOR`
`therefore angleEDS=angleDQR+angleDRQ`
`therefore delta=delta_1+delta_2`
but, `angleDQR = angleDQL – angleRQL `
`delta_1=i-r_1` [`because angle PQK=angleDQL=i]`
and,`angleDRQ=angleDRL-angleQRL`
`delta_2=e-r_2` So, equation (4)
`delta=i-r_1+e-r_2`
`delta=i+e-(r_1+r_2)`
`thereforedelta=i+e-A` [`because` equation (3)]
`thereforei+e=A+delta`
This equation gives the relation between angle of deviation, angle of incident, angle of emergence and angle of prism.
From above equation we say that angle of deviation depends on the angle of incidence.