Explanation:

When two coherent sources produce a dark fringe in interference, it means that the waves from the two sources are completely out of phase at that point. In other words, the phase difference between the interfering waves is an odd multiple of half-wavelengths.

Understanding Interference:

Interference occurs when two or more waves overlap and combine to form a resultant wave. In the case of light waves, interference can be observed as a pattern of bright and dark fringes. These fringes are formed due to the constructive and destructive interference between the waves.

Conditions for Dark Fringe:

For a dark fringe to be formed, the condition is that the waves from the two sources should be in complete destructive interference. This means that the crest of one wave should coincide with the trough of the other wave, resulting in cancellation of amplitudes.

Phase Difference and Dark Fringes:

The phase difference between two waves can be calculated using the formula:

Phase Difference (Δϕ) = 2π(Δx/λ)

Where Δx is the path difference between the two waves and λ is the wavelength of the waves.

For a dark fringe to be formed, the phase difference should be an odd multiple of half-wavelengths, i.e., (2n + 1)λ/2, where n is an integer.

Explanation of Option B:

Option B states that the phase difference between the interfering waves is (2n + 1), which satisfies the condition for dark fringe formation.

When n is an integer, (2n + 1) will also be an integer. Multiplying an integer by half-wavelengths (λ/2) will result in an odd multiple of half-wavelengths. Therefore, option B is the correct answer.

Example:

Let's consider an example where the phase difference is (2n + 1)λ/2. For n = 0, the phase difference will be λ/2, which corresponds to a dark fringe. Similarly, for n = 1, the phase difference will be (3λ/2), again resulting in a dark fringe.

In conclusion, option B is the correct answer because it states that the phase difference between the interfering waves is an odd multiple of half-wavelengths, which satisfies the condition for dark fringe formation in interference.