Question

Obtain the equation for resultant intensity due to interference of light.

Answer 1

1. The phenomenon of addition or superposition of two light waves which produces increase in intensity at some points and a decrease in intensity at some other points is called interference of light.
2. Let us consider two light waves from the two sources S₁ and S₂ meeting at a point P as shown
3. The wave from S₁ at an instant t and P is, y₁ = a₁ sin ωt
   The wave from S₂ an instant t at P is
   y₂ = a₂ sin(ωt + Φ)
   
   Superposition principle
4. The two waves have different amplitudes a₁ and a₂, same angular frequency ω’ and a phase difference of Φ between them. The resultant displacement will be given by.
   y = y₁ + y₂ = a₁ sin ωt + a₁ sin₂ (ωt + Φ) y = A sin (ωt + Φ)
   Where, A = `sqrt("a"_1^2 + "a"_2^2 + 2"a"_1"a"_2 cos phi)`  .....(1)
   `theta = tan^-1  ("a"_2 sin phi)/("a"_1 + "a"_2 cos phi)`   ......(2)
5. The resultant amplitude is maximum.
   A_max = `sqrt(("a"_1 + "a"_2)^2)`;
   when Φ = 0, ±2π, ± 4π …….(3)
6. The resultant amplitude is minimum.
   A_min = `sqrt(("a"_1 - "a"_2)^2)`;
   when Φ = 0, ±π, ± 3π ± 5π …..(4)
7. The intensity of light is proportional to the square of amplitude.
8. I α A² ……(5)
   Now equation (1) becomes
   I α I₁ + I₂ + 2`sqrt("I"_1"I"_2)` cos Φ .(6)
9. 9. If the phase difference, Φ = 0, ± 2π, ± 4π., it corresponds to the condition for maximum intensity of light called as constructive interference.
10. The resultant maximum intensity is,
    I_max α (a₁ + a₂)² …….(7)
11. If the phase difference, Φ = + π, ± 3π, ± 5π …., it corresponds to the condition for the minimum intensity of light called destructive interference.
12. The resultant minimum intensity is I_min α
    (a₁ – a₂)² α I₁ + I₂ – 2`sqrt("I"_1"I"_2)`   ......(8)
    As a special case, if a₁ = a₂ = a, then equation (1) becomes,
    A = `sqrt(2"a"^2 + 2"a"^2 cos phi)`
    = `sqrt(2"a"^2 (1 + cos phi))`
    = `sqrt(2"a"^2 2cos^2 (phi//2))`

13. A = 2 a cos(Φ/2) ….(9)

    I α 4a² cos² (Φ/2) [∴ I α A²] ……(10)
    I α 4 I₀ cos² (Φ/ 2) [ΦI₀ α a²] …….(11)
    I_Max = 4I₀ when, Φ = o, ± 2π, 4π  …..(12)
    I_min = 0 when, Φ = ± π, ± 3π, ± 5π …..(13)
    Conclusion:
    The phase difference between the two waves decides the intensity of light meet at a point.