A monochromatic beam of light is incident at $$60$$° on one face of an equilateral prism of refractive index n and emerges from the opposite face making an angle θn with the normal (see$$ the $$figure). For $$n=3$$ the value of θ is $$60$$° and dθ$$dn=m$$. The value of m is

Question

A monochromatic beam of light is incident at  $$60$$° on one face of an equilateral prism of refractive index n  and emerges from the opposite face making an angle  θn with the normal (see$$ the $$figure).  For $$n=3$$  the value of  θ is  $$60$$° and  dθ$$dn=m$$.  The value of  m is

Infinity Learn · Accepted Answer

We know, $$r1+r2=A=60$$∘  (equilateral$$ $$prism) Snell's Law at P, (1) $$sin$$⁡$$60$$∘$$=(n)$$$$sin$$⁡$$r1$$⇒$$32=(n)$$$$sin$$⁡$$60$$∘−$$r2$$⇒$$32=(n)$$$$sin$$⁡$$60$$∘$$cos$$⁡$$r2$$−$$cos$$⁡$$60$$∘$$sin$$⁡$$r2$$⇒$$32=(n)321$$−$$sin2$$⁡$$r2$$−$$12sin$$⁡$$r2$$−−−−−(1) Snell's Law at Q , nsin⁡$$r2=(1)$$$$sin$$⁡θ⇒$$sin$$⁡$$r2=$$$$sin$$⁡θn−−−−−−(2)Put$$⁡$$(2)in$$⁡$$(1)⇒$$32=(n)321$$−$$sin2$$⁡θ$$n2$$−$$12sin$$⁡θnAfter squaring both sides,⇒$$4sin2$$⁡θ$$+23sin$$⁡θ−$$3n2+3=0$$ Differentiating wrt n , ⇒$$4(2sin$$⁡θ$$)$$$$cos$$⁡θdθ$$dn+23cos$$⁡θdθdn−$$6n=0$$ Put θ$$=60$$∘ and $$n=3$$⇒dθ$$dn=2$$ So ,$$m=2$$

Prism

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A monochromatic beam of light is incident at $60 °$ on one face of an equilateral prism of refractive index n and emerges from the opposite face making an angle $θ (n)$ with the normal (see the figure). For $n = \sqrt{3}$ the value of $θ$ is $60 °$ and $\frac{d θ}{d n} = m$ . The value of m is

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Prism

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A monochromatic beam of light is incident at 60° on one face of an equilateral prism of refractive index n and emerges from the opposite face making an angle θn with the normal (see the figure). For n=3 the value of θ is 60° and dθdn=m. The value of m is

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free study material

A monochromatic beam of light is incident at $60 °$ on one face of an equilateral prism of refractive index n and emerges from the opposite face making an angle $θ (n)$ with the normal (see the figure). For $n = \sqrt{3}$ the value of $θ$ is $60 °$ and $\frac{d θ}{d n} = m$ . The value of m is