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Optics
Short-Sightedness (Myopia)
Short-sightedness, or myopia, is a refractive defect of the eye in which parallel rays of light from a distant object are focused in front of the retina rather than directly on it. Consequently, distant objects appear blurred while near objects are seen clearly.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition, anatomical cause (elongation of eyeball or excessive curvature of cornea), identification of corrective lens (concave lens), and basic numericals involving far point and lens power.
PGT extension
Detailed optical ray tracing, calculation of focal length and power for a given far point, accommodation range alterations, and combination of lenses for complete correction.
Prerequisites
Refraction through spherical lenses
Lens Maker's Formula and thin lens equation
Anatomy and working of the human eye
Anatomical and Optical Causes of Myopia
The eyeball gets elongated along its optical axis, increasing the distance between the eye lens and the retina.
The curvature of the cornea or the eye lens becomes too great, yielding excessive converging power.
Because of the high convergence, parallel rays from infinity form a real image in the vitreous humor before reaching the retina.
Far Point of a Myopic Eye
For a normal human eye, the far point is at infinity ($u = -\infty)$.
For a myopic eye, the far point (x) is at a finite distance in front of the eye.
Rays coming from any point beyond this finite far point cannot be focused on the retina without optical correction.
Correction Using a Concave Lens
A diverging (concave) lens is used to correct myopia.
The concave lens intercepts parallel rays from infinity and diverges them such that they appear to come from the patient's actual far point.
The virtual image of a distant object formed by the corrective lens lies at the far point of the myopic eye, enabling the eye lens to focus it sharply on the retina.
Optical Power and Focal Length Sign Convention
The focal length of the correcting concave lens must be negative.
The magnitude of the focal length of the corrective lens is precisely equal to the distance of the myopic far point from the eye.
The power of the corrective lens is negative, expressed in diopters (D).
Derivations
Derivation of Focal Length of Corrective Concave Lens
Conditions: Object at infinity ($u = -\infty) must form a virtual image at the eye's far point (v = -x)$.
State the thin lens formula: \frac{1}{f} = \frac{1}{v} - \frac{1}{u}
Substitute the object distance for a distant object requiring correction: $u = -\infty$
Substitute the required image position at the patient's far point: $v = -x$
n = refractive index, R_1, $R_2 = radii of curvature$ meters^{-1}
Symmetrical or asymmetrical thin lens
Graphs and experiments
Myopic Eye Ray Diagram
Draw: Parallel rays entering an elongated eyeball and converging to a focal point in front of the retina.
Interpret: Illustrates the anatomical defect where distant objects form blurred images before reaching the retina.
Exam use: Frequently asked in TGT/PGT subjective and diagram-based identification questions.
Correction of Myopia Using Concave Lens
Draw: Parallel rays diverging through a concave lens before entering the myopic eye, forming a virtual image at the far point and final sharp image on the retina.
Interpret: Demonstrates how a diverging lens adjusts the incoming wavefront vergence to match the eye's reduced accommodation range.
Exam use: Tested in multi-part numerical and conceptual optical design questions.
Power vs. Far Point Distance Graph
Draw: A hyperbola plot of Lens Power P (in D) on the y-axis versus Far Point x (in meters) on the x-axis.
Interpret: Shows that as the far point moves closer to the eye (severe myopia), the required corrective power increases non-linearly in magnitude.
Exam use: Interpreting graphical trends in advanced PGT screening questions.
Solved numericals
TGT core
A myopic person cannot see objects distinctly beyond 80 cm from the eye. What is the nature and power of the lens required to enable him to see distant objects clearly?
Identify the far point: x = 80 cm = 0.8 m.
Determine the nature of the lens: Since it is myopia, a concave (diverging) lens is required.
A myopic physicist has a far point of 50 cm. She wishes to read a book placed at 25 cm without glasses, but her near point has also shifted. However, focusing strictly on her myopia correction for viewing an object at 150 cm, what power lens is required?
Far point x = 50 cm = 0.5 m. Desired object distance $u = -150 cm = -1.5 m$.
Apply lens formula: 1 / $f = (1 / v) - (1 / u)$.
Set image distance v to the far point: $v = -50 cm = -0.5 m$.
In a myopic eye, parallel rays from infinity form a real image in front of the retina due to excessive converging power or elongation of the eyeball.
The far point ($x$) of a myopic eye is at a finite distance in front of the eye, unlike the normal eye where it is at infinity.
Myopia is corrected using a diverging lens (concave lens) whose focal length magnitude equals the distance of the far point ($f = -x$).
The optical power of the corrective lens for myopia is always negative, expressed in diopters ($P = -1/x$).
A myopic person can see nearby objects clearly, but distant objects appear blurred because their far point is limited.
Glossary
Myopia (Short-Sightedness)A defect of vision where distant objects are not seen clearly because images are formed in front of the retina.
Far Point ($x$)The maximum distance up to which a myopic eye can clearly see objects without optical correction.
Concave LensA diverging lens used as a corrective optical element for short-sightedness to diverge incoming rays.
Optical Power ($P$)The degree of convergence or divergence of a lens, measured in diopters ($D$), given by $P = 1/f$.
Vitreous HumorThe transparent gelatinous tissue filling the eyeball behind the lens, where the premature image forms in myopia.
Myopia vs. Normal Vision
Feature
Normal Eye
Myopic Eye
Far Point Position
At infinity ($u = -\infty$)
At a finite distance ($x$) in front of the eye
Image Formation for Infinite Object
Exactly on the retina
In front of the retina (in vitreous humor)
Corrective Lens Required
None
Concave (diverging) lens
Power of Corrective Lens
Zero
Negative ($P = -1/x$)
Practice Quiz
Choose an option to reveal the explanation.
1. Which type of lens is used to correct short-sightedness (myopia)?
Explanation: A concave (diverging) lens is used to diverge incoming parallel rays so that they appear to originate from the patient's far point, enabling the eye lens to focus them on the retina.
2. What is the position of the far point for a normal human eye?
Explanation: For a normal human eye, the far point is located at infinity ($u = -\infty$). For a myopic eye, it is restricted to a finite distance.
3. The far point of a myopic person is at $80\text{ cm}$ in front of the eye. What should be the focal length of the corrective lens?
Explanation: The focal length of the corrective concave lens is given by $f = -x$. Since the far point $x = 80\text{ cm} = 0.8\text{ m}$, the focal length must be $f = -80\text{ cm}$.
4. What is the optical power of a concave lens required to correct myopia for a person whose far point is $2\text{ m}$?
Explanation: The power of the corrective lens is $P = -\frac{1}{x}$. Given $x = 2\text{ m}$, $P = -\frac{1}{2} = -0.5\text{ D}$.
5. Which of the following anatomical changes causes myopia?
Explanation: Myopia is caused by the elongation of the eyeball along its optical axis or excessive curvature of the cornea, increasing the distance between the lens and the retina.
6. Where does the image of a distant object form in an uncorrected myopic eye?
Explanation: Due to excessive converging power, parallel rays from infinity form a real image in the vitreous humor in front of the retina before reaching it.
7. A student states that a myopic person cannot see near objects clearly. Is this statement correct?
Explanation: This is a common misconception. A myopic person can see near objects clearly; their far point is restricted, making distant objects blurry.
8. What is the sign of the focal length of a corrective lens for short-sightedness?
Explanation: A concave lens is diverging, and by convention, its focal length is always negative ($f = -x$). mols
9. If the far point of a myopic patient is $50\text{ cm}$, what is the power of the lens needed to read distant signboards?
Explanation: Using $P = -\frac{1}{x}$, where $x = 0.5\text{ m}$, we get $P = -\frac{1}{0.5} = -2.0\text{ D}$.
10. When a corrective concave lens is placed in front of a myopic eye, where does it form a virtual image of a distant object?
Explanation: The concave lens forms a virtual image of a distant object at the patient's actual far point, allowing the eye lens to focus it sharply on the retina.
11. Which of the following relations correctly expresses the thin lens equation?
Explanation: The standard thin lens equation is $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$.
12. What happens to the optical power of the eye-lens system in a myopic person compared to a normal eye?
Explanation: Myopia is characterized by excessive converging power, meaning the optical power of the eye system is higher than normal.
13. If a myopic person uses a lens of power $-1.25\text{ D}$, what is their far point?
14. Which formula relates the combined power when the eye and corrective lens are in contact?
Explanation: For thin lenses in contact, the total optical power is the algebraic sum of the individual powers: $P_{total} = P_{eye} + P_{lens}$.
15. What is the primary optical condition required for a clear image in a corrected myopic eye?
Explanation: The corrective lens must form a virtual image of the distant object right at the patient's far point so the eye can accommodate it onto the retina.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A person with myopia can see objects clearly up to a distance of $1.5\text{ m}$. What power of lens is required to enable him to see distant objects clearly?
Explanation: Given far point $x = 1.5\text{ m}$. The power of the corrective lens is $P = -\frac{1}{x} = -\frac{1}{1.5} \approx -0.67\text{ D}$ (concave lens).
2. The far point of a myopic eye is $100\text{ cm}$. In order to view distant objects clearly, the focal length of the lens to be used should be:
Explanation: For a myopic eye with far point $x$, the required focal length of the concave lens is $f = -x$. Thus, $f = -100\text{ cm}$.
3. A myopic person uses a lens of power $-2.0\text{ D}$. The maximum distance of objects that he can see clearly without glasses is:
Explanation: The far point $x = \frac{1}{|P|} = \frac{1}{2.0} = 0.5\text{ m}$. Hence, the maximum distance is $0.5\text{ m}$ or $50\text{ cm}$.
4. In myopia, parallel rays from infinity form an image:
Explanation: Excessive convergence causes parallel rays to converge and form a real image in front of the retina within the vitreous humor.
5. Which of the following lens parameters is fundamentally negative for correcting short-sightedness?
Explanation: A concave lens is required for correction, which inherently possesses a negative focal length ($f = -x$).
6. If the eyeball is elongated along its optical axis, the resulting optical defect is:
Explanation: Elongation of the eyeball increases the lens-retina distance, resulting in myopia (short-sightedness).
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. A myopic eye has its far point at $40\text{ cm}$. What is the power of the lens needed to view an object at infinity?
2. Why is a concave lens capable of correcting myopia?
Explanation: A concave lens diverges incoming parallel rays from infinity so they seem to originate from the patient's far point, allowing the eye to focus them.
3. If a person requires a lens of power $-0.5\text{ D}$ for correcting his distant vision, what is his far point?
Explanation: Far point $x = \frac{1}{|P|} = \frac{1}{0.5} = 2.0\text{ m}$.
4. An anatomical cause of myopia other than eyeball elongation is:
Explanation: Excessive curvature of the cornea or eye lens yields excessive converging power, contributing to myopia.
5. What is the nature of the image formed by a corrective concave lens for a distant object in myopic vision correction?
Explanation: A concave lens forms a virtual and erect image of a distant object at the patient's far point.
6. A student calculates the focal length of a corrective lens as $-50\text{ cm}$. What is the far point of the patient?
Explanation: The magnitude of the focal length of the corrective lens equals the distance of the myopic far point: $x = |f| = 50\text{ cm}$.
7. In the lens formula applied to myopia, when viewing an object at infinity ($u = -\infty$), the image distance $v$ corresponds to:
Explanation: The corrective lens forms the virtual image at the far point ($v = -x$) when the object is at infinity.
8. Which of the following statements about myopia is incorrect?
Explanation: Statement 3 is incorrect because a myopic person has clear near vision and impaired distant vision.
9. Calculate the combined power if an eye with power $P_{eye}$ is fitted with a corrective lens of power $-3\text{ D}$ in contact.
Explanation: Total power $P_{total} = P_{eye} + P_{lens} = P_{eye} + (-3\text{ D}) = P_{eye} - 3\text{ D}$.
10. What is the primary unit of optical power of a corrective lens in standard SI representation?
Explanation: Optical power is expressed in diopters ($D$), which is equivalent to inverse meters ($m^{-1}$).