Question

how to derive prism formula​

Answer 1

Derivation of prism formula

μ=sinisinr (by Snell’s law)

δ=i1−r1+i2−r2… (eq.1)

δ=i1+i2−(r1+r2)

∠ALO+∠AMO=2rt∠s (from quadrilateral and ∠ALO=∠AMO=90°)

∠LAM+∠LOM=2rt∠s (sum of four ∠s of a quadrilateral = 4 rt∠s) (eq.2)

∠r1+∠r2+∠LOM=2rt∠s (eq.3)

∠LAM=∠r1+∠r2 (comparing eq.2 and eq.3)

A=∠r1+∠r2

δ=i1+i2−A (substituting A in eq.1)

i1+i2=A+δ

∠i1=∠i2

∠r1=∠r2=∠r

∠ALM=∠LMA=90∘−∠r

Thus, AL = LM and LM ∥ BC

∠A=∠r1+∠r2

A=2r (since, ∠r1=∠r2=∠r)

r=A2

i1+i2=A+δ

i1+i1=A+δm

2i1=A+δm

i1=A+δm2

∴μ=sinA+δm2sinA2

Thus, above is the prism formula.

Answer 2

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