Question

Prove that ratio of the intensities at maxima and minima is Imax/Imin =(r+1/r-1)^2 ; where r= a^1/a^2 is the ratio of amplitudes
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Answer 1

We have to prove that the ratio of the intensities at maximum to minimum is I_max/I_min = (r + 1)²/(r - 1)² , where r = A₁/A₂ is the ratio of amplitudes.

proof : we know, intensity is given by, I = I₁ + I₂ +2√I₁I₂ cosΦ

also intensity is directly proportional to square of its amplitude.

so, I = kA²

so, I = k[A₁² + A₂ + 2A₁A₂cosΦ]

for maximum intensity, Φ = 0°

then, I_max = k[A₁ + A₂]²

for minimum intensity, Φ = 180°

then, I_min = K[A₁ - A₂]²

now, ratio of maximum to minimum intensity = I_max/I_min = [A₁ + A₂]²/[A₁ - A₂]2

= [A₁/A₂ + A₂/A₂]²/[A₁/A₂ - A₂/A₂]²

= (r + 1)²/(r - 1)² hence proved