Question

Light from two coherent Sources of the same amplitude \(\mathrm{A}\) and wavelength \(\lambda\), illuminates the Screen. The intensity of the central maximum is Io. If the sources were incoherent, the intensity at the same point will be (A) \(\left(\mathrm{I}_{0} / 2\right)\) (B) \(\left(\mathrm{I}_{0} / 4\right)\) (C) \(4 \mathrm{I}_{0}\) (D) \(2 \mathrm{I}_{0}\)

Short answer

The intensity at the central maximum when the sources are incoherent is half of that when they are coherent. Therefore, the correct answer is (A) \(\left(\mathrm{I}_{0} / 2\right)\).

Step-by-step solution

Recall the intensity formula for coherent sources

When two coherent light sources of the same amplitude A and wavelength λ interfere, the resulting intensity at any point on the screen is given by the formula: \[ I = 4A^2\mathrm{cos}^2\left(\frac{\delta}{2}\right) \] where I is the intensity and δ is the phase difference between the two waves. Since the sources are coherent and the interference pattern occurs at the central maximum, the phase difference δ is 0, and cos(δ/2) equals 1. Now we can find the intensity for the coherent light sources.

Calculate the intensity for coherent light sources

Using the formula and substituting with δ=0, we get: \[ I = 4A^2\mathrm{cos}^2\left(\frac{0}{2}\right) = 4A^2 \] This value is the intensity of the central maximum, which is denoted as Io.

Determine the intensity for incoherent sources

Incoherent light sources don't have a stable phase relationship, so they don't produce any interference pattern on the screen. Instead, the intensity at any point is simply the sum of the intensities from each source. Since both sources have the same amplitude A, their individual intensities are: \[ I_1 = A^2 \text{ and } I_2 = A^2 \] The total intensity at the same point for incoherent sources is the sum of their individual intensities: \(I_{total} = I_1 + I_2\).

Calculate the total intensity for incoherent sources

Substitute the values we found for I1 and I2, and calculate the total intensity: \[ I_{total} = A^2 + A^2 = 2A^2 \]

Compare the coherent and incoherent intensity values

Now we have the intensity Io for the coherent light sources (4A²) and the total intensity I_total for the incoherent light sources (2A²). To find the relation between these values, we can write: \[ I_{total} = \frac{1}{2} Io \] So, the intensity at the central maximum when the sources are incoherent is half of that when they are coherent. Therefore, the correct answer is (A) \(\left(\mathrm{I}_{0} / 2\right)\).