Lens Power Calculation for Vision Correction
This problem requires calculating the power of a lens needed to correct vision so that a person can read a book placed at 25 cm, given their minimum distance of distinct vision is 50 cm.
Understanding the Vision Problem
- The person's near point (minimum distance of distinct vision) is 50 cm. This means they can see objects clearly only if they are at or beyond 50 cm.
- The goal is to read a book at 25 cm.
- A corrective lens is needed to make the book at 25 cm appear as if it is at 50 cm, allowing the person to see it clearly. This means the lens must form a virtual image of the object (book) at the person's near point.
Applying the Lens Formula
We use the lens formula to find the focal length ($f$). The formula is:
$ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} $- Object Distance ($u$): The distance of the book from the lens. Since the book is placed in front of the lens, $u = -25 \text{ cm}$.
- Image Distance ($v$): The distance where the virtual image should be formed for clear vision. This is the person's near point. Since it's a virtual image on the same side as the object, $v = -50 \text{ cm}$.
Calculating Focal Length
Substitute the values into the lens formula:
$ \frac{1}{f} = \frac{1}{(-50 \text{ cm})} - \frac{1}{(-25 \text{ cm})} $
$ \frac{1}{f} = -\frac{1}{50 \text{ cm}} + \frac{1}{25 \text{ cm}} $
Find a common denominator (50):
$ \frac{1}{f} = -\frac{1}{50 \text{ cm}} + \frac{2}{50 \text{ cm}} $
$ \frac{1}{f} = \frac{-1 + 2}{50 \text{ cm}} = \frac{1}{50 \text{ cm}} $
Thus, the focal length $f = 50 \text{ cm}$.
Calculating Lens Power
The power ($P$) of a lens is defined as the reciprocal of its focal length ($f$) in meters.
$ P = \frac{1}{f} $
First, convert the focal length to meters:
$ f = 50 \text{ cm} = 0.50 \text{ m} $
Now, calculate the power:
$ P = \frac{1}{0.50 \text{ m}} = 2 \text{ D} $
The required power of the lens is 2 Diopters.
Similar Questions
- 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
- Beyond what distance, the ray optics is sufficiently valid when the aperture is $6\text{ mm}$ wide and the wavelength is $6000\text{ \AA}$?
- A plano-convex lens fits exactly into a plano-concave lens. Their plane surfaces are parallel to each other. If lenses are made of different materials of refractive indices $\mu_1$ and $\mu_2$ and $R$ is the radius of curvature of the curved surface of the lenses, then the focal length of the combination is
- An electromagnetic wave, whose wave normal makes an angle of $45^{\circ}$ with the vertical, is travelling in air and strikes a horizontal liquid surface. While travelling through the liquid, it gets deviated by $15^{\circ}$. If the speed of electromagnetic wave in air is $3 \times 10^{8}\text{ m/s}$, then the speed of electromagnetic wave in the liquid will be
Two points of monochromatic and coherent sources of light of wavelength $\lambda$ each, are placed as shown in figure. The initial phase difference between the sources is zero, ($D \gg d$). Mark the correct statement(s).
X-rays of wavelength λ gets reflected from parallel planes of atoms in a crystal with spacing d between two planes as shown in the figure. If the two reflected beams interfere constructively, then the condition for maxima will be, (n is the order of interference fringe)
A ray of monochromatic light is incident on the plane surface of separation between two media X and Y with angle of incidence ‘i’ in medium X and angle of refraction ‘r’ in medium Y. The given graph shows the relation between sin i and sin r. If VX and VY are the velocities of the ray in media X and Y respectively, then which of the following is true ?
Three identical convex lenses each of focal length f are placed in a straight line separated by a distance f from each other. An object is located at f/2 in front of the leftmost lens. Then,
An interference pattern is obtained with two coherent sources of intensity ratio n:1. The ratio \(\dfrac{I_{Max} - I_{Min}}{I_{Max} + I_{Min}}\) will be maximum if
Figure shows the graph of angle of deviation δ versus angle of incidence i for a light ray striking a prism. The prism angle is
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 ________ $\text{cm}$.
