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
Let's solve the problem step-by-step to find the fringe spacing of the interference pattern produced by the two coherent plane electromagnetic waves.
Given:
- Wavelength of the waves, \(\lambda = 0.5 \ \mu\text{m}\).
- Angle between each wave vector and the normal to the plane, \(\theta = 30^\circ\).
The fringe spacing \(d\) in the interference pattern can be calculated using the formula:
\(d = \frac{\lambda}{2 \sin\theta}\)
Substituting the given values:
\(d = \frac{0.5 \ \mu\text{m}}{2 \sin 30^\circ}\)
Since \(\sin 30^\circ = 0.5\), we have:
\(d = \frac{0.5 \ \mu\text{m}}{2 \times 0.5} = \frac{0.5 \ \mu\text{m}}{1} = 0.5 \ \mu\text{m}\)
Therefore, the fringe spacing is \(0.5 \ \mu\text{m}\).
Thus, the correct answer is:
- \(0.5 \ \mu\text{m}\)
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
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?
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
