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Write the formula to get the refractive index of the prism and write the terms in it.

Answer
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Hint Draw a ray diagram showing the refraction inside and outside a prism. Use exterior angle property, angle sum property of a triangle and a quadrilateral and hence find a relation between incident angle, angle of deviation, and prism angle. Consider the ray of light to undergo a minimum angle of deviation and approximate the incident angle and emergent angle. Finally, substitute the derived value of incident and refracted angle into Snell’s law.

Complete step by step answer:
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Let us consider a prism ABC with a refractive index μ. Let QN and RN be normal lines to the faces AB and AC respectively. Let the incident angle on the face AB be i1 and the refracted angle on this face be r1. Let the incident angle on the face AC be r2 and the refracted ray after passing through AC be i2. We will extend the ray lines- PQ and SR to the point O. Let the angle of deviation be d.
In the quadrilateral AQNR, the sum of all four angles must be 360 .
A+N+AQN+ARN=360
But here, AQN=ARN=90 as they are normal rays.
A+N=360180
Rearranging the above equation gives us
A=180N
A is an exterior angle in the triangle, ΔQRN .
Therefore, using the exterior angle property for the triangle ΔQRN gives us r1+r2=A .
From the figure, let i1r1=d1 and i2r2=d2
By using the exterior angle property for the triangle ΔQOR ,
d1+d2=d
By substituting d1 as i1r1 and d2 as i2r2, we get
i1r1+i2r2=d
But we have already proved that r1+r2=A,
d=i1+i2A
In the case of minimum deviation, i1i2i and d=dm .
Substituting these values in the above equation, we get
i=A+dm2
Now let us write Snell’s law for the prism,
μ=sinisinr
From the already calculated values of i and r, we get
μ=sin(A+dm2)sin(A2)
This is the equation of the refractive index for a prism with the prism angle - A and angle of minimum deviation -dm .

Note
A minimum angle of deviation in a prism means that the incident and the emergent angles will be equal, the refracted angles inside will be equal and the refracted ray will run parallel to the base of the prism. We use Snell’s law as the medium is getting changed.