Young's Double-Slit: Fringe Width Calculation
The fringe width ($\beta$) in Young's double-slit experiment is determined by the wavelength of light ($\lambda$), the distance between the slits and the screen ($D$), and the separation between the slits ($d$). The formula is:
$ \beta = \frac{\lambda D}{d} $
Equal Fringe Width Condition
The problem states that two separate experimental set-ups produce fringes of equal width. Let the parameters for the two set-ups be subscripted with 1 and 2.
Therefore, we have:
$ \beta_1 = \beta_2 $
Substituting the formula for fringe width:
$ \frac{\lambda_1 D_1}{d_1} = \frac{\lambda_2 D_2}{d_2} $
Calculating the Distance Ratio
We need to find the ratio of the distances between the slits and the screens, which is $D_1/D_2$. Rearranging the equation from the previous step:
$ \frac{D_1}{D_2} = \frac{\lambda_2}{\lambda_1} \times \frac{d_1}{d_2} $
Applying Given Ratios
The problem provides the following ratios:
- Ratio of slit separations: $d_1 : d_2 = 2 : 1 \implies \frac{d_1}{d_2} = \frac{2}{1}$
- Ratio of wavelengths: $\lambda_1 : \lambda_2 = 1 : 2 \implies \frac{\lambda_1}{\lambda_2} = \frac{1}{2}$. This implies $\frac{\lambda_2}{\lambda_1} = \frac{2}{1}$.
Substitute these values into the rearranged equation:
$ \frac{D_1}{D_2} = \left(\frac{2}{1}\right) \times \left(\frac{2}{1}\right) $
$ \frac{D_1}{D_2} = \frac{4}{1} $
The corresponding ratio of the distances ($D_1/D_2$) is 4.
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 a Young's double slit experiment set up, the two slits are kept $0.4\text{ mm}$ apart and screen is placed at $1\text{ m}$ from slits. If a thin transparent sheet of thickness $20 \, \mu\text{m}$ is introduced in front of one of the slits then center bright fringe shifts by $20\text{ mm}$ on the screen. The refractive index of transparent sheet is given by $\frac{\alpha}{10}$, where $\alpha$ 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 ________ $\text{cm}$.
- 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 _______.
- An unpolarized light of intensity $I_0$ passes through polarizer and then through a certain optically active solution and finally it goes to analyser. If the angle between analyser and polariser is $0^\circ$ and intensity of light emerged from analyser is $\frac{3}{8} I_0$, the angle of rotation of the light by the solution with respect to analyser is _________ degrees.
A ring is placed at the bottom of a trough containing four immiscible liquids of refractive indices $1.0, 2.0, 3.0$ and $4.0$ poured one above the other of heights $10$ cm, $20$ cm, $30$ cm, and $40$ cm respectively. The apparent depth at which the ring appears, when seen from outside is____________ cm.
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 ?
- A convergent doubled to separated lenses, corrected for spherical aberration, has resultant focal length of 10 cm. The separation between the two lenses is 2 cm. The focal lengths of the component lenses are :
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 a Young's double slit experiment set up, the two slits are kept $0.4\text{ mm}$ apart and screen is placed at $1\text{ m}$ from slits. If a thin transparent sheet of thickness $20 \, \mu\text{m}$ is introduced in front of one of the slits then center bright fringe shifts by $20\text{ mm}$ on the screen. The refractive index of transparent sheet is given by $\frac{\alpha}{10}$, where $\alpha$ 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
