Consider three equally spaced and equal-intensity coherent sources of light (such as adding a third slit to the two slits of Fig. 34–12). Determine the positions of minima.

Verified step by step guidance

1. Understand the problem: We are dealing with three equally spaced and equal-intensity coherent sources of light. The goal is to determine the positions of minima in the interference pattern. Minima occur where destructive interference happens, meaning the waves from the sources cancel each other out.
2. Write the condition for destructive interference for three sources. For three equally spaced sources, the condition for minima is that the path difference between adjacent sources must result in a phase difference of \( \frac{2\pi}{3} \) or \( \frac{4\pi}{3} \), leading to complete cancellation.
3. Express the path difference in terms of the angle \( \theta \) and the wavelength \( \lambda \). The path difference between adjacent sources is given by \( d \sin \theta \), where \( d \) is the spacing between the sources. For minima, the condition becomes \( d \sin \theta = m \frac{\lambda}{3} \), where \( m \) is an integer (but not a multiple of 3).
4. Solve for \( \sin \theta \): Rearrange the equation to find \( \sin \theta = \frac{m \lambda}{3d} \). Here, \( m \) takes on values such as \( \pm 1, \pm 2, \pm 4, \pm 5, \dots \) (avoiding multiples of 3).
5. Interpret the result: The angles \( \theta \) corresponding to these values of \( m \) will give the positions of the minima in the interference pattern. To find the exact angles, substitute the known values of \( \lambda \) and \( d \) into the equation.

Key Concepts

Interference of Light

Interference occurs when two or more coherent light waves overlap, resulting in a new wave pattern. This can lead to regions of constructive interference, where waves add together, and destructive interference, where they cancel each other out. The pattern of bright and dark spots, or fringes, is a direct result of this phenomenon.

Path Difference

Path difference refers to the difference in distance traveled by two waves arriving at a point. For minima to occur in an interference pattern, the path difference must equal an odd multiple of half the wavelength (λ/2). This condition leads to destructive interference, resulting in dark spots on the screen.

Conditions for Minima

The conditions for minima in a multi-slit interference pattern can be derived from the formula d sin(θ) = (m + 1/2)λ, where d is the distance between slits, θ is the angle of the minima, m is an integer (0, ±1, ±2,...), and λ is the wavelength of the light. This equation helps determine the specific angles at which dark fringes appear in the interference pattern.