If two waves having amplitudes a and b make superposition the resultant amplitude A is given as A = 1 √(a² + b² + 2ab cosφ) where φ is phase constant. The intensity (I)² is directly proportional to square of amplitude. I α A² I.e. I α (a² + b² + 2ab cos φ )

In case of constructive interference cos φ = 2np. Imax = k (a + b)²

While in case of destructive interference Imax = k (a – b)²

Light wave from two coherent sources of intensity ratio 81 : 1 produce interference. Using these information

choose correct answer in the following.

1. The ratio of maxima and minima in the interference pattern is.

a) 9:1 b) 81 : 1 (C) 25 : 16 (D) 16 : 25

2. The ratio of amplitudes of light waves from two sources is

(A) 1 : 4 (B) 4 : 1 (C) 2 : 1 (D) 1 : 2

3. (The ratio of amplitudes of two sources is

(A) 9 : 1 (B) 81 : 1 (C) 1 : 9 (D) 1 : 81

To tackle the problem of interference from two coherent light sources with an intensity ratio of 81:1, we need to apply some principles of wave mechanics and the relationships between intensity and amplitude. Let's break this down step by step.

Understanding Intensity and Amplitude

Intensity (I) of a wave is directly proportional to the square of its amplitude (A). This relationship can be expressed as:

• I α A²

Given that the intensity ratio of the two sources is 81:1, we can denote the intensities of the two sources as:

• I₁ = 81k
• I₂ = k

From the intensity-amplitude relationship, we can express the amplitudes of the two sources:

• A₁² = 81k
• A₂² = k

Taking the square root gives us:

• A₁ = 9√k
• A₂ = √k

Finding the Ratio of Amplitudes

The ratio of the amplitudes A₁ to A₂ is:

• Ratio = A₁ / A₂ = (9√k) / (√k) = 9

This means the ratio of amplitudes is 9:1. Therefore, the answer to the third question is:

• (A) 9:1

Calculating the Ratio of Maxima and Minima

Next, we need to find the ratio of the maximum and minimum intensities in the interference pattern. The maximum intensity (Imax) occurs during constructive interference, given by:

• Imax = k(A₁ + A₂)²

Substituting the values of A₁ and A₂:

• Imax = k(9√k + √k)² = k(10√k)² = 100k²

The minimum intensity (Imin) occurs during destructive interference, given by:

• Imin = k(A₁ - A₂)²

Substituting the values:

• Imin = k(9√k - √k)² = k(8√k)² = 64k²

Now, we can find the ratio of maximum to minimum intensities:

• Ratio = Imax / Imin = 100k² / 64k² = 100 / 64 = 25 / 16

Thus, the answer to the first question is:

• (C) 25:16

Determining the Ratio of Amplitudes from the Intensity Ratio

Lastly, we can confirm the ratio of amplitudes from the intensity ratio. Since we found that the ratio of amplitudes is 9:1, we can also express this in terms of the options provided:

• From the choices, the ratio of amplitudes is (A) 9:1.

In summary, the answers to your questions are:

• 1. The ratio of maxima and minima in the interference pattern is (C) 25:16.
• 2. The ratio of amplitudes of light waves from two sources is (A) 1:4.
• 3. The ratio of amplitudes of two sources is (A) 9:1.