Lens Focal Length Calculation
The problem involves finding the focal length ($f$) of a thin lens using the concept of magnification ($m$) when an object is placed at two different distances, resulting in images of equal size.
Key Principles
- Lens Formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$
- Magnification Formula: $m = \frac{v}{u}$
- Equal Image Size: This implies the magnitude of magnification is the same for both object positions: $|m_1| = |m_2|$.
Derivation for Equal Magnification Magnitudes
The magnification $m$ can also be expressed in terms of $u$ and $f$: $m = \frac{f}{u+f}$.
Given two object distances $u_1 = -8 \text{ cm}$ and $u_2 = -24 \text{ cm}$ (negative by convention as objects are typically placed to the left of the lens).
The condition $|m_1| = |m_2|$ implies $m_1 = \pm m_2$. Since the object distances are different, $m_1$ cannot equal $m_2$. Therefore, $m_1 = -m_2$.
This leads to:
$ \frac{f}{u_1+f} = -\frac{f}{u_2+f} $Dividing both sides by $f$ (assuming $f \neq 0$):
$ \frac{1}{u_1+f} = -\frac{1}{u_2+f} $Cross-multiplying gives:
$ u_2+f = -(u_1+f) $ $ u_2+f = -u_1-f $Rearranging to solve for $f$:
$ 2f = -u_1 - u_2 $ $ f = -\frac{u_1 + u_2}{2} $Calculation of Focal Length
Substitute the given object distances into the derived formula:
$ f = -\frac{(-8 \text{ cm}) + (-24 \text{ cm})}{2} $ $ f = -\frac{-32 \text{ cm}}{2} $ $ f = -(-16 \text{ cm}) $ $ f = 16 \text{ cm} $Verification
Case 1: $u_1 = -8 \text{ cm}$, $f = 16 \text{ cm}$
Using the lens formula $\frac{1}{v_1} = \frac{1}{f} - \frac{1}{u_1}$:
$ \frac{1}{v_1} = \frac{1}{16} - \frac{1}{-8} = \frac{1}{16} + \frac{2}{16} = \frac{3}{16} $Wait, the derivation $m = \frac{f}{u+f}$ is correct. Let's re-verify the calculation based on $f = -\frac{u_1 + u_2}{2}$.
The condition $m_1 = -m_2$ correctly yielded $f = -\frac{u_1 + u_2}{2}$.
Let's recalculate $v$ and $m$ using $f=16$ cm.
Case 1 ($u_1 = -8$ cm):
$ \frac{1}{v_1} = \frac{1}{f} + \frac{1}{u_1} = \frac{1}{16} + \frac{1}{-8} = \frac{1 - 2}{16} = -\frac{1}{16} $ $ v_1 = -16 \text{ cm} $ $ m_1 = \frac{v_1}{u_1} = \frac{-16}{-8} = 2 $Case 2 ($u_2 = -24$ cm):
$ \frac{1}{v_2} = \frac{1}{f} + \frac{1}{u_2} = \frac{1}{16} + \frac{1}{-24} = \frac{3 - 2}{48} = \frac{1}{48} $ $ v_2 = 48 \text{ cm} $ $ m_2 = \frac{v_2}{u_2} = \frac{48}{-24} = -2 $The magnitudes $|m_1| = |2| = 2$ and $|m_2| = |-2| = 2$ are equal. The focal length is indeed $16 \text{ cm}$.
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 ___________.
- 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 __________.
- 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 ___________.
- 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
