Question

On the basis of Huygens' wave theory of light prove that velocity of light in a rarer medium is greater than velocity of light in a denser medium.

Answer 1

XY : plane refracting surface

AB : incident plane wavefront

B₁R : refracted wavefront

AA₁ , BB₁ : incident rays

A₁R , B₁R₁ : refracted rays

∠ AA₁M = ∠ BB₁M₁ = ∠ i : angle of incidence

∠ RA₁N = ∠ R₁B₁N₁ = ∠ r : angle of refraction

1.  Let XY be the plane refracting surface separating two media rarer and denser of refractive indices μ₁ and μ₂ respectively.
2. A plane wavefront AB is advancing obliquely towards XY from rarer medium. It is bounded by rays AA₁ and BB₁ which are incident rays.
3. When 'A' reaches at 'A₁' then 'B' will be at 'P' . It still has to cover distance PB₁ to reach XY.
4. According to Huygens' principle , secondary wavelets will originate from A₁ and it will spread over a hemisphere in denser medium.
5. All the rays between AA₁ and BB₁ will reach XY and spread over the hemispheres of increasing radii in denser medium. The surface of tangency of all such hemisphere is RB₁ . This gives rise to refracted wavefront B₁R in denser medium.
6. A₁R and B₁R are refracted rays.
7. Let c₁ and c₂ be the velocities of light in rarer and denser medium respectively.
8. At any instant of time 't' , distance covered by incident wavelength from P to B₁ = PB₁ = c₁t
   Distance covered by secondary wave from A₁  to R = A₁R =c₂t .
9. From above figure ,
   ∠ AA₁M + ∠ MA₁P = 90°      .....(i)  and
   ∠ MA₁P + ∠ PA₁B₁ = 90°    ...........(ii)
   From equation (i) and (ii) , we have,
   ∠ AA₁M = ∠ PA₁B₁ = 90° 
10. Similarly ,
    ∠NA₁R = ∠ N₁B₁R₁ = r
    We have,
    ∠ N₁B₁R₁ + ∠ A₁B₁R =  90°    .........(iii)
    and 
    ∠ N₁B₁R₁ + ∠ A₁B₁R = 90°     .........(iv)
    From equations (iii) and (iv) , we have, 
    ∠ N₁B₁R₁ +  ∠ A₁B₁R = r
11. In Δ A₁PB₁ ,
    sin i = `"PB"_1/("A"_1"B"_1) = ("c"_1"t")/("A"_1"B"_1)`   ........(v)
12. In Δ A₁RB₁ ,
    sin r = `("A"_1"R")/("A"_1"B"_1) = ("c"_2"t")/("A"_1"B"_1)`        ............(vi)
13. Dividing equation (v) bt (vi), we have
    `("sin" "i")/("sin""r") = ((c_1t) / (A_1B_1))/((c_2t)/(A_1B_1))`
    `therefore ("sin" "i")/("sin""r") = c_1/c_2`  .....(vii)
    Also `c_1/c_2 = mu_2/mu_1 = 1 mu_2` .....(viii)
    where 1μ₂ = R.I. of denser medium w.r.t rarer medium.
14. From above figure ,
    ∠ i > ∠ r
    ∴ sin i > sin r
    ∴ `("sin" "i")/("sin" "r") > 1`
    `therefore mu_2/mu_1 > 1`   .........(ix)
    Since , `c_1/c_2 = mu_2/mu_1`    ..........[From ix]
    `therefore c_1/c_2 > 1`
    `therefore c_1 > c_2`
    Hence , velocity of light in rarer medium is greater than veocity in denser medium.