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.
Apparent Depth of Ring Calculation
This problem involves calculating the apparent depth of an object (a ring) at the bottom of a container filled with multiple immiscible liquids. The apparent depth differs from the real depth due to the refraction of light as it travels from the liquid to the observer's eye in the air.
Apparent Depth Physics Formula
When observing from air (refractive index $n_{air} \approx 1$) into a single liquid layer of depth $h$ and refractive index $n$, the apparent depth $d_{app}$ is calculated as: $ d_{app} = \frac{h}{n} $ For multiple liquid layers stacked one above the other, the total apparent depth ($D_{app}$) is the sum of the apparent depths of each individual layer. The formula becomes: $ D_{app} = \sum_{i=1}^{N} \frac{h_i}{n_i} $ where $h_i$ is the real depth of the $i$-th layer and $n_i$ is its refractive index.
Trough Liquid Layers Application
The given information is:
| Layer | Real Depth ($h_i$) | Refractive Index ($n_i$) |
|---|---|---|
| 1 | 10 cm | 1.0 |
| 2 | 20 cm | 2.0 |
| 3 | 30 cm | 3.0 |
| 4 | 40 cm | 4.0 |
Calculating Ring's Apparent Depth
Apply the formula for total apparent depth using the values provided:
$ D_{app} = \frac{h_1}{n_1} + \frac{h_2}{n_2} + \frac{h_3}{n_3} + \frac{h_4}{n_4} $
Substitute the depths and refractive indices:
$ D_{app} = \frac{10 \text{ cm}}{1.0} + \frac{20 \text{ cm}}{2.0} + \frac{30 \text{ cm}}{3.0} + \frac{40 \text{ cm}}{4.0} $
Perform the division for each term:
$ D_{app} = 10 \text{ cm} + 10 \text{ cm} + 10 \text{ cm} + 10 \text{ cm} $
Sum the results to find the total apparent depth:
$ D_{app} = 40 \text{ cm} $
The ring appears at an apparent depth of 40 cm from the surface of the top liquid layer.
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 __________.
- 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 ________ $\text{cm}$.
- 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 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 __________.
- 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 ________ $\text{cm}$.
