As shown in the figure, in Young’s double slit experiment, a thin plate of thickness \(t = 10 \, \mu\text{m}\) and refractive index \(\mu = 1.2\) is inserted in front of slit \(S_1\). The experiment is conducted in air (\(\mu = 1\)) and uses a monochromatic light of wavelength \(\lambda = 500 \, \text{nm}\). Due to the insertion of the plate, central maxima is shifted by a distance of \(x \beta_0\). \(\beta_0\) is the fringe-width before the insertion of the plate. The value of \(x\) is _________.
Show Hint
Remember the formula for fringe shift in Young’s double slit experiment when a thin plate is introduced. Ensure consistent units while substituting values.
Correct Answer: 4
Approach Solution - 1
Step 1: Recall the Formula for Fringe Shift
When a thin plate of thickness \(t\) and refractive index \(\mu\) is introduced in the path of one of the slits in Young’s double slit experiment, the fringe pattern shifts. The fringe shift (\(\Delta x\)) is given by:
\[ \Delta x = \frac{t (\mu - 1)}{\lambda} \beta_0 \]
where \(\lambda\) is the wavelength of light and \(\beta_0\) is the fringe width.
Step 2: Convert Units and Substitute Values
Given \(t = 10 \, \mu \text{m} = 10 \times 10^{-6} \, \text{m}\), \(\mu = 1.2\), and \(\lambda = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} = 5 \times 10^{-7} \, \text{m}\):
\[ \Delta x = \frac{10 \times 10^{-6} (1.2 - 1)}{5 \times 10^{-7}} \beta_0 \]
\[ \Delta x = \frac{10 \times 10^{-6} \times 0.2}{5 \times 10^{-7}} \beta_0 \]
\[ \Delta x = \frac{2 \times 10^{-6}}{5 \times 10^{-7}} \beta_0 = \frac{20 \times 10^{-7}}{5 \times 10^{-7}} \beta_0 = 4 \beta_0 \]
Step 3: Find the Value of \(x\)
The central maxima is shifted by a distance of \(x \beta_0\). We found that \(\Delta x = 4 \beta_0\). Therefore, \(x = 4\).
Conclusion: The value of \(x\) is 4.
Approach Solution -2
Fringe shift x
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Concepts Used:
Wave Optics
- Wave optics are also known as Physical optics which deal with the study of various phenomena such as polarization, interference, diffraction, and other occurrences where ray approximation of geometric optics cannot be done. Thus, the section of optics that deals with the behavior of light and its wave characteristics is known to be wave optics.
- In wave optics, the approximation is carried out by utilizing ray optics for the estimation of the field on a surface. Further, it includes integrating a ray-estimated field over a mirror, lens, or aperture for the calculation of the transmitted or scattered field.
- Wave optics stands as a witness to a famous standoff between two great scientific communities who devoted their lives to understanding the nature of light. Overall, one supports the particle nature of light; the other supports the wave nature.
- Sir Isaac Newton stood as a pre-eminent figure that supported the voice of particle nature of light, he proposed a corpuscular theory which states that “light consists of extremely light and tiny particles, called corpuscles which travel with very high speeds from the source of light to create a sensation of vision by reflecting on the retina of the eye”.