Question

If the ratio of the intensity of two coherent sources is 4 then the visibility \(\left[\left(I_{\max }-I_{\min }\right) /\left(I_{\max }+I_{\min }\right)\right]\) of the fringes is - (1) 4 (2) \(4 / 5\) (3) \(3 / 5\) (4) 9

Short answer

The visibility of the fringes is \( \frac{4}{5} \). Option (2) is correct.

Step-by-step solution

- Identify Given Data

The ratio of the intensities of the two coherent sources is given as 4. Let one intensity be denoted as \(I_1\) and the other as \(I_2\). Therefore, \( I_1/I_2 = 4 \). Thus, \( I_1 = 4I_2 \).

- Determine Maximum and Minimum Intensities

For coherent sources, \( I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \) and \( I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2 \). Substituting \( I_1 = 4I_2 \) into these equations, we get: \[ I_{\text{max}} = (\sqrt{4I_2} + \sqrt{I_2})^2 = (2\sqrt{I_2} + \sqrt{I_2})^2 = (3\sqrt{I_2})^2 = 9I_2 \] \[ I_{\text{min}} = (\sqrt{4I_2} - \sqrt{I_2})^2 = (2\sqrt{I_2} - \sqrt{I_2})^2 = (\sqrt{I_2})^2 = I_2 \]

- Calculate Intensity Visibility

The visibility of the fringes is given by \[ V = \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}} \] Substituting \( I_{\text{max}} = 9I_2 \) and \( I_{\text{min}} = I_2 \) into the expression, we get: \[ V = \frac{9I_2 - I_2}{9I_2 + I_2} = \frac{8I_2}{10I_2} = \frac{8}{10} = \frac{4}{5} \]

Key concepts

coherent sources

In the context of interference in physics, coherent sources are essential. Coherent sources are light sources that have a constant phase difference and the same frequency. They ensure a stable and clear interference pattern.
For instance, if you shine light from two separate bulbs, the light waves produced are generally not coherent. However, in experiments with lasers or carefully controlled setups, the light can be coherent.
Coherence is crucial for interference patterns because it ensures consistent constructive and destructive interference.
To sum up, without coherent sources, it would be impossible to observe clear and stable interference fringes.

intensity ratio

Understanding the intensity ratio is essential for solving problems involving interference. The intensity ratio of two sources describes how much stronger one source is compared to the other.
In the given exercise, the intensity ratio is 4. This tells us that one source's intensity is four times that of the other. Mathematically, if we let one intensity be about 4 times the other, i.e., if one intensity is denoted as \(I_1\) and the other as \(I_2\), then \(I_1/I_2 = 4\) or \(I_1 = 4I_2\).
Knowing this ratio helps us determine the maximum and minimum intensities observed in the interference pattern.

fringe visibility

Fringe visibility in an interference pattern measures how clearly the maximum and minimum points (fringes) are distinguished.
It is quantified by the formula: \( V = \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}} \). Here, \(V\) denotes visibility.
In the exercise, we determined \(I_{\text{max}}\) and \(I_{\text{min}}\) using the given intensity ratio. The resulting visibility can thus indicate how pronounced the interference fringes are.
Fringe visibility close to 1 (or 100%) means very clear fringes, whereas lower values indicate weaker fringes. In this case, with a visibility of 4/5 or 0.8, the interference fringes are quite noticeable, making it easy to study interference effects.