Name: Ray Diagram: Lenses
Uploaded: 2019-07-19
Duration: 6 min 35 s
Description: In this video, let us discuss how ray diagrams for lenses can be used to determine the size and location of a refracted image. Get in to know more!

Question

Derive the equation for the angle of deviation produced by a prism and thus obtain the equation for the refractive index of material of the prism

Accepted Answer

Angie of deviation produced by prism:

Let light ray $P Q$ is incident on one of the refracting faces of the prism. The angles of incidence and refraction at the first face AB are $i_{1} and r_{1}$ The path of the light inside the prism is $QR$ The angle of incidence and refraction at the second face AC is $r_{2} and i_{2}$ respectively. RS is the ray emerging from $P$ the second face. Angle $i_{2}$ is called angle of emergence. The angle between the direction of the incident ray $P Q$ and the emergent ray $RS$ is called the angle of deviation

The two normals drawn at the point of incidence Q and emergence R are $QN and RN$ They meet at point $N$ . The incident ray and the emergent ray meet at a point $M$ .

The deviation $d$ at the surface AB is, In the quadrilateral $AQNR$ two of the angles (at the vertices Q and R) are right angles.

$∠ RQM = d_{1} = i_{1} - r_{1}$ and $∠ QRM = d_{2} = i_{2} - r_{2}$

Therefore, the sum of the other angles of the quadrilateral is

$180 ° = ∠ A + ∠ QNR$ ° ................( $1$ )

From the triangle $△ QNR r_{1} + r_{2} + ∠ QNR = 180$ …..........( $2$ )

Comparing these two equations (1) and (2) we get $r_{1} + r_{2} = A$ .............( $3$ )

Substituting this in equation for angle of deviation,

$δ = d_{1} + d_{2} = (i_{1} - r_{1}) + (i_{2} - r_{2}) = (i_{1} + i_{2}) - (r_{1} + r_{2})$

Now , sum of the other angles of the quadrilateral $∠ A + ∠ QNR = 180 or r_{1} + r_{r} + ∠ QNR = 180 + A$

Comparing , we get $r_{1} + r_{2} = A$

Hence, $δ = i_{1} + i_{2} - A$

Thus, the angle of deviation depends on the angle of incidence angle of emergence and the angle for the prism.

Refractive index of the material of the prism:

According to the Snells law $\sin i_{1} = μ \sin r_{1} or i_{1} = {μr}_{1}$

Since these angles are small (in case of thin prism) $i_{2} = {μr}_{2}$

Hence $δ = i_{1} + i_{2} - A = μ_{1} r_{1} + μ_{2} r_{2} - A = μ (r_{1} + r_{2}) - A = Aμ - A$

or $δ = (μ - 1) A$

Derive the equation for the angle of deviation produced by a prism and thus obtain the equation for the refractive index of material of the prism

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