Question

A wedge-shaped thin film is formed using soap-water solution. The refractive index of the film is 1.25. At near normal incidence, when the film is illuminated by a monochromatic light of wavelength 600 nm, 10 interference fringes per cm are observed. The wedge angle (in radians) is ______ \(\times 10^{-5}\). (in integer)

Hint:

For interference in a wedge film, remember that the fringes are of "equal thickness." The fringe width formula \(\beta = \lambda / (2n\theta)\) is fundamental. Note that a smaller wedge angle \(\theta\) leads to wider fringes (larger \(\beta\)).

Answer 1

Correct Answer: 24

Step 1: Understanding the Concept:
In a wedge-shaped thin film, interference fringes are formed due to the path difference between light rays reflected from the top and bottom surfaces of the film. The fringes are parallel to the thin edge of the wedge and are equally spaced. The spacing between the fringes (fringe width) is related to the wavelength of light, the refractive index of the film, and the angle of the wedge.
Step 2: Key Formula or Approach:
For near-normal incidence, the distance between two consecutive bright or dark fringes (the fringe width, \(\beta\)) is given by: \[ \beta = \frac{\lambda}{2n\theta} \] where \(\lambda\) is the wavelength of light in vacuum, \(n\) is the refractive index of the film, and \(\theta\) is the wedge angle in radians.
Step 3: Detailed Explanation:
We are given: Number of fringes = 10 per cm. This means the distance containing 10 fringes is 1 cm. The fringe width \(\beta\) is the distance per fringe. \[ \beta = \frac{1 \text{ cm}}{10} = 0.1 \text{ cm} = 0.001 \text{ m} \] Other given values: Wavelength, \(\lambda = 600 \text{ nm} = 600 \times 10^{-9}\) m. Refractive index, \(n = 1.25\). Now we can rearrange the formula to solve for the wedge angle \(\theta\): \[ \theta = \frac{\lambda}{2n\beta} \] Substitute the values: \[ \theta = \frac{600 \times 10^{-9} \text{ m}}{2 \times 1.25 \times 0.001 \text{ m}} = \frac{600 \times 10^{-9}}{2.5 \times 10^{-3}} \text{ radians} \] \[ \theta = \frac{600}{2.5} \times 10^{-6} = 240 \times 10^{-6} \text{ radians} = 2.4 \times 10^{-4} \text{ radians} \] The question asks for the answer in the form of an integer multiplied by \(10^{-5}\). \[ 2.4 \times 10^{-4} = 24 \times 10^{-5} \] Step 4: Final Answer:
The wedge angle is \(24 \times 10^{-5}\) radians. The integer is 24.