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) \(\left(\mathrm{I}_{0} / 2\right)\).

Key concepts

Coherent Sources

In the realm of optics, coherent sources play a vital role in phenomena like interference and diffraction. Coherent sources are essentially light sources that have a constant phase difference and maintain the same frequency. This constant phase relationship is crucial because it allows the light waves from these sources to effectively interact in a stable manner.

When two coherent sources of light overlap, they can interfere constructively or destructively based on their phase difference. Constructive interference leads to an increase in light intensity, while destructive interference results in a decrease. This interaction leads to patterns of light and dark bands, commonly observed in phenomena such as double-slit experiments. Easily expressed through formulas, the constructive case at the central maximum, where the phase difference is zero, results in an intensity formula of \( I = 4A^2 \), showcasing the amplification effect due to perfect coherence.

Understanding coherent sources is fundamental when considering light as waves, rather than mere rays. For instance, lasers are designed as coherent sources, allowing for practical applications in various fields.

Incoherent Sources

In contrast to coherent sources, incoherent sources lack a stable phase relationship. This means there is no constant phase difference between the waves emanating from these sources. Often, everyday light sources like bulbs or the sun are considered incoherent because their photons are emitted randomly.

Without a stable phase relationship, the waves from incoherent sources do not create a stable interference pattern. Instead, they simply add up, intensity-wise, at any given point. Unlike coherent light, where the interference pattern can lead to pronounced bands of light and dark, incoherent light tends to spread the energy more evenly across space. The result is a uniform illumination rather than a pattern.

The total intensity from incoherent sources is simply the sum of intensities from each individual source. If two incoherent sources have amplitudes \(A\), their combined intensity becomes \(I = 2A^2\). This distinct lack of interference pattern makes incoherent sources reliable for general illumination purposes but unsuitable for precise optical interference measurements.

Intensity Formula

The intensity formula for light interference is a mathematical representation of how wave-like properties of light result in observable intensity variations. For coherent sources, the formula given is \( I = 4A^2 \mathrm{cos}^2\left(\frac{\delta}{2}\right) \), where \( I \) is the resulting intensity, \( A \) is the amplitude of the waves, and \( \delta \) is the phase difference between them. At the central maximum, where the phase difference \( \delta \) is zero, \( \mathrm{cos}^2(0) \) equals 1, simplifying the intensity to \( I = 4A^2 \).

When dealing with incoherent sources, the intensity formula adjusts since there's no particular phase relationship. Each source's intensity remains independent. Thus, their combined intensity at a point is the sum of their individual intensities, expressed as \( I_{total} = I_1 + I_2 = 2A^2 \), assuming each source has the same amplitude \( A \).

This difference in intensity calculation highlights the raw power of coherence in light to create dramatically different outcomes, explaining why similar starting conditions (same amplitude and wavelength) produce different light intensities when comparing coherent and incoherent sources.