A screen has two slits, each of width $w$, with their centres at a distance $2w$ apart. It is illuminated by a monochromatic plane wave travelling along the $x$-axis.
The intensity of the interference pattern, measured on a distant screen, at an angle $\theta = n\lambda/w$ to the $x$-axis is
The given problem involves the interference pattern produced by two closely spaced slits of equal width \(w\), with their centers separated by a distance \(2w\). This configuration is illuminated by a monochromatic plane wave along the \(x\)-axis. We are to find the condition under which the intensity of the interference pattern at an angle \(\theta = n\lambda/w\) is zero.
The intensity of the interference pattern is determined by the principle of superposition of the light waves from the two slits, resulting in constructive and destructive interference.
- The path difference (\(\Delta x\)) between the light from the two slits at angle \(\theta\) is given by: \(\Delta x = 2w \sin \theta\).
- The condition for destructive interference (zero intensity) is when the path difference is an odd multiple of half-wavelengths: \(\Delta x = (m + \frac{1}{2}) \lambda\), where \(m\) is an integer.
- Substituting the path difference formula: \(2w \sin \theta = (m + \frac{1}{2}) \lambda\).
- Given that \(\theta = n\lambda/w\), we substitute: \(\sin \theta \approx \theta = n\lambda/w\).
- Recalculate: \(2w \cdot \frac{n\lambda}{w} = (m + \frac{1}{2})\lambda\) \(2n\lambda = (m + \frac{1}{2})\lambda\) \(2n = m + \frac{1}{2}\).
- From this equation, for integer \(n\), \((m + \frac{1}{2})\) is never an integer unless \(n\) is a half-integer like \(1/2, 3/2\), etc., i.e., odd integers cancel with half.
- Thus, zero intensity occurs for integer multiples of \(n\): \(n = 1, 2, 3, \ldots\).
Therefore, the correct answer is that the intensity is zero for \(n = 1, 2, 3 \ldots\).
Correct Option: Zero for \(n = 1, 2, 3 \ldots\)
Important Questions from Optics and Diffraction
- A highly collimated laser beam with a diameter of 1 cm and wavelength 500 nm is directed from the earth's surface towards the moon (~384,000 km away from the earth). Assuming ideal diffraction limited propagation in vacuum, which of the following best estimates the diameter of the beam upon returning to the earth after reflection from an ideal reflector installed on the moon.
Three identical pinholes separated by distance $a$ along the x-axis are illuminated by a collimated monochromatic coherent beam of light (wavelength $\lambda$) as shown in the figure below.
The intensity (in arbitrary units) pattern of fringes obtained on a screen kept at distance $D$ ($D>>a$) along the z- axis is best represented by
Two coherent plane electromagnetic waves of wavelength $0.5 \ \mu\text{m}$ (both have the same amplitude and are linearly polarized along the $z$-direction) fall on the $y = 0$ plane. Their wave vectors $\mathbf{k}_1$ and $\mathbf{k}_2$ are as shown in the figure.
If the angle $\theta$ is $30^\circ$, the fringe spacing of the interference pattern produced on the plane is
The figure below describes the arrangement of slits and screens in a Young's double slit experiment. The width of the slit in $\text{S}_1$ is $a$ and the slits in $\text{S}_2$ are of negligible width.
If the wavelength of the light is $\lambda$, the value of $d$ for which the screen would be dark is- The separation between the energy levels of a two-level atom is 2 eV. Suppose that $4 \times 10^{20}$ atoms are in the ground state and $7 \times 10^{20}$ atoms are pumped into the excited state just before lasing starts. How much energy will be released in a single laser pulse?
