Young's Double Slit Fringe Shift Analysis
The problem involves calculating the refractive index of a transparent sheet placed in a Young's double-slit experiment, given the fringe shift.
Given Data
- Slit separation, $d = 0.4\text{ mm} = 0.4 \times 10^{-3}\text{ m}$
- Screen distance, $D = 1\text{ m}$
- Sheet thickness, $t = 20 \, \mu\text{m} = 20 \times 10^{-6}\text{ m}$
- Fringe shift, $\Delta y = 20\text{ mm} = 20 \times 10^{-3}\text{ m}$
- Refractive index, $\mu = \frac{\alpha}{10}$
Fringe Shift Formula
When a thin transparent sheet of thickness $t$ and refractive index $\mu$ is introduced in front of one slit, the central bright fringe shifts. The formula for this shift $\Delta y$ is:
$ \Delta y = \frac{D}{d}(\mu - 1)t $Calculation of Refractive Index
- Substitute the given values into the fringe shift formula: $ 20 \times 10^{-3}\text{ m} = \frac{1\text{ m}}{0.4 \times 10^{-3}\text{ m}} (\mu - 1) (20 \times 10^{-6}\text{ m}) $
- Simplify the equation: $ 20 \times 10^{-3} = (2.5 \times 10^3) (\mu - 1) (20 \times 10^{-6}) $ $ 20 \times 10^{-3} = (50 \times 10^{-3}) (\mu - 1) $
- Solve for $(\mu - 1)$: $ \mu - 1 = \frac{20 \times 10^{-3}}{50 \times 10^{-3}} = \frac{20}{50} = 0.4 $
- Solve for the refractive index $\mu$: $ \mu = 1 + 0.4 = 1.4 $
Determining Alpha
The refractive index is given as $\mu = \frac{\alpha}{10}$. Equating this with the calculated value:
$ 1.4 = \frac{\alpha}{10} $Solve for $\alpha$:
$ \alpha = 1.4 \times 10 = 14 $Therefore, the value of $\alpha$ is 14.
Similar Questions
- In a microscope the objective is having focal length $f_o = 2 \text{ cm}$ and eye-piece is having focal length $f_e = 4 \text{ cm}$. The tube length is $32 \text{ cm}$. The magnification produced by this microscope for normal adjustment is ____________.
- A collimated beam of light of diameter $2 \text{ mm}$ is propagating along x-axis. The beam is required to be expanded in a collimated beam of diameter $14 \text{ mm}$ using a system of two convex lenses. If first lens has focal length $40 \text{ mm}$, then the focal length of second lens is ____________ $\text{mm}$.
- In two separate Young's double-slit experimental set-ups and two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slits separations and that of the wavelengths of light used are $2 : 1$ and $1 : 2$ respectively. The corresponding ratio of the distances between the slits and the respective screens ($D_1/D_2$) is ___________.
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- If sunlight is focused on a paper using convex lens, it starts burning the paper in shortest time when the lens is kept at $30\text{ cm}$ above the paper. If the radius of curvature of the lens is $60\text{ cm}$ then the refractive index of the lens material is $\frac{\alpha}{10}$. The value of $\alpha$ is __________.
- In a double slit experiment, when one of the slits is covered by a transparent mica sheet of refractive index 1.56, the central fringe shifts to the position of $7^{\text{th}}$ bright fringe, obtained with both slits uncovered. If the light source wavelength is 450 nm, the thickness of mica sheet is $\alpha \times 10^{-9} \text{ m}$. The value of $\alpha$ is _______.
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A plane polarized light is incident on a polariser with its pass axis making angle $\theta$ with x-axis, as shown in the figure. At four different values of $\theta$, $\theta = 8^\circ, 38^\circ, 188^\circ$ and $218^\circ$, the observed intensities are same. What is the angle between the direction of polarization and x-axis ?
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Important Questions from Optics
- In a microscope the objective is having focal length $f_o = 2 \text{ cm}$ and eye-piece is having focal length $f_e = 4 \text{ cm}$. The tube length is $32 \text{ cm}$. The magnification produced by this microscope for normal adjustment is ____________.
- A collimated beam of light of diameter $2 \text{ mm}$ is propagating along x-axis. The beam is required to be expanded in a collimated beam of diameter $14 \text{ mm}$ using a system of two convex lenses. If first lens has focal length $40 \text{ mm}$, then the focal length of second lens is ____________ $\text{mm}$.
- In two separate Young's double-slit experimental set-ups and two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slits separations and that of the wavelengths of light used are $2 : 1$ and $1 : 2$ respectively. The corresponding ratio of the distances between the slits and the respective screens ($D_1/D_2$) is ___________.
- The size of the images of an object, formed by a thin lens are equal when the object is placed at two different positions $8 \text{ cm}$ and $24 \text{ cm}$ from the lens. The focal length of the lens is ________ $\text{cm}$.
- A ray of light travelling in air is incident on one face of a parallel glass slab of thickness $t$ and refractive index $\mu$ at an angle of incidence $i$. Total time spent by the ray inside the slab is
