Understanding the Problem
We are given a convex lens focusing sunlight onto paper. The fastest burning occurs when the lens is placed at a distance equal to its focal length from the paper. This distance is given as $30\text{ cm}$. We are also provided with the radius of curvature of the lens ($60\text{ cm}$) and need to find the refractive index ($n$) of the lens material, which is expressed as $\frac{\alpha}{10}$.
Applying the Lens Maker's Formula
The Lens Maker's Formula relates the focal length ($f$) of a lens to its refractive index ($n$) and the radii of curvature of its surfaces ($R_1$ and $R_2$):
$ \frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) $- Focal Length ($f$): Since the burning is fastest at $30\text{ cm}$, this is the focal length. For a convex lens, $f = +30\text{ cm}$.
- Radii of Curvature ($R_1, R_2$): For a biconvex lens with equal curvatures, the first radius $R_1 = +60\text{ cm}$ (surface facing incident light) and the second radius $R_2 = -60\text{ cm}$ (surface facing away from incident light, according to sign convention).
Calculation Steps
- Substitute the known values into the Lens Maker's Formula: $ \frac{1}{30\text{ cm}} = (n-1) \left( \frac{1}{60\text{ cm}} - \frac{1}{-60\text{ cm}} \right) $
- Simplify the expression within the parenthesis: $ \frac{1}{30} = (n-1) \left( \frac{1}{60} + \frac{1}{60} \right) $ $ \frac{1}{30} = (n-1) \left( \frac{2}{60} \right) $ $ \frac{1}{30} = (n-1) \left( \frac{1}{30} \right) $
- Solve for $(n-1)$ by dividing both sides by $\frac{1}{30}$: $ 1 = n-1 $
- Calculate the refractive index ($n$): $ n = 1 + 1 = 2 $
- Relate the calculated refractive index to the given form $n = \frac{\alpha}{10}$: $ 2 = \frac{\alpha}{10} $
- Solve for $\alpha$: $ \alpha = 2 \times 10 = 20 $
The value of $\alpha$ is 20.
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 ________ $\text{cm}$.
- 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 __________.
- 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 ________ $\text{cm}$.
