Home › Physics › Optics › Geometrical Optics: Lenses

Optics

Geometrical Optics: Lenses

A lens is a transparent optical medium bounded by at least one curved spherical surface that refracts light to form real or virtual images. Lenses operate on the principle of Snell's Law of refraction at curved boundaries.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Refraction at spherical surfaces, lens maker's formula, thin lens formula, linear magnification, power of a lens, combination of thin lenses in contact, and simple optical instruments like magnifying glasses.

PGT extension

Lens displacement method (displacement method for focal length measurement), equivalent focal length of two lenses separated by a distance, thick lenses, chromatic and spherical aberrations in lenses, and cardinal points of optical systems.

Prerequisites

Laws of reflection and refraction (Snell's Law)
Real and virtual images sign conventions (Cartesian sign convention)
Geometrical properties of triangles and spherical geometry

Refraction at a Single Spherical Surface

Relates object distance (u), image distance (v), radius of curvature (R), refractive index of the medium containing the object (n₁), and refractive index of the medium containing the image (n₂).
Assumes paraxial ray approximation where angles of incidence and refraction are small.
Sign convention dictates distances measured against the direction of incident light are negative.

Lens Maker's Formula and Thin Lenses

Relates the focal length (f) of a thin lens to the refractive index of the lens material (n) relative to the surrounding medium (n_m) and the radii of curvature (R₁, R₂).
A double convex lens with n > n_m converges light, whereas it diverges light if immersed in a medium with a higher refractive index than the lens.
The thin lens approximation assumes the thickness of the lens is negligible compared to object and image distances.

Combination of Thin Lenses

For two thin lenses of focal lengths f₁ and f₂ in contact, the equivalent focal length F is given by 1/$F = 1/f_{1} + 1/f_{2}$.
The net power of lenses in contact is the algebraic sum of individual powers: $P = P_{1} + P_{2}$.
When lenses are separated by a distance d, the equivalent focal length is modified to account for the spatial separation.

Optical Aberrations in Lenses

Chromatic aberration arises due to the dependence of refractive index on wavelength (dispersion), causing different colors to focus at different points.
Spherical aberration occurs because marginal rays focus closer to the lens than paraxial rays for large apertures.
Can be minimized by using stopped-down apertures, crossed lenses, or a combination of suitable lenses.

Derivations

Lens Maker's Formula

Conditions: Thin lens of refractive index n₂ placed in a medium of refractive index n₁ with spherical surfaces of radii R₁ and R₂.

Apply refraction formula for the first spherical surface: (n₂/v₁) - (n₁/u) = (n₂ - n₁)/R₁, where v₁ is the intermediate image distance.
Treat the intermediate image as a virtual object for the second refracting surface: (n₁/v) - (n₂/v₁) = (n₁ - n₂)/R₂.
Add the two equations to eliminate v₁: (n₁/v) - (n₁/u) = (n₂ - n₁)(1/R₁ - 1/R₂).
Divide by n₁ and set $u = -infinity$ and v = f to obtain the final expression.

Result: (1/f) = ((n₂/n₁) - 1)(1/R₁ - 1/R₂)

Equivalent Focal Length of Two Thin Lenses Separated by Distance d

Conditions: Two thin lenses of focal lengths f₁ and f₂ separated by distance d in air.

Trace a paraxial ray parallel to the principal axis through the first lens to get deviation.
Determine the point of emergence from the second lens.
Apply the lens formula successively for each lens accounting for the coordinate shift d.
Equate the overall system to a single equivalent lens of focal length F.

Result: (1/F) = (1/f₁) + (1/f₂) - (d / (f₁ f₂))

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Refraction at Spherical Surface	$$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$$	n₁, n₂: refractive indices; u: object distance; v: image distance; R: radius of curvature metres (m)	Paraxial rays, spherical surface
Lens Maker's Formula	$$\frac{1}{f} = \left(n_{rel} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$	f: focal length; n_rel = n_lens/n_medium; R₁, R₂: radii of curvature metres (m)	Thin lens, paraxial approximation
Thin Lens Formula	$$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$$	f: focal length; v: image distance; u: object distance metres (m)	Thin lens in air or uniform medium
Linear Magnification	$$m = \frac{h_i}{h_o} = \frac{v}{u}$$	h_i: image height; h_o: object height; v: image distance; u: object distance Dimensionless	Real and virtual images formed by thin lenses
Power of a Lens	$$P = \frac{1}{f}$$	P: power; f: focal length in metres Dioptre (D, m^-1)	Lens in air
Lenses in Contact	$$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} + \dots$$	F: equivalent focal length; f₁, f₂: individual focal lengths metres (m)	Thin lenses in direct contact
Lenses Separated by Distance	$$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$$	F: equivalent focal length; d: separation distance metres (m)	Coaxial thin lenses separated by distance d
Displacement Method (Upto-v method)	$$f = \frac{a^2 - d^2}{4a}$$	a: fixed distance between object and screen; d: separation between two lens positions metres (m)	a > 4f for real images to exist

Graphs and experiments

u-v Graph for a Convex Lens

Draw: A rectangular hyperbola plotted in the second quadrant for negative u and positive v, asymptotic to lines u = f and v = f.

Interpret: Shows the inverse nonlinear relation between object and image distances for a converging lens.

Exam use: Used to find focal length graphically by finding the intercept where u = v = 2f.

1/u vs 1/v Graph

Draw: A straight line with a negative slope crossing the axes at 1/f.

Interpret: Linearizes the thin lens equation (1/v - 1/$u = 1/f)$ to yield intercept equal to 1/f.

Exam use: Extensively used in laboratory optical bench experiments to determine focal length accurately.

Optical Bench Experiment for Convex Lens Focal Length

Draw: Experimental setup showing illuminated pin, lens holder, and movable screen on an optical bench scale.

Interpret: Direct measurement of conjugate foci and displacement method parameters.

Exam use: Core practical component evaluated in viva voce and numerical problem formulation.

Solved numericals

TGT core

A convex lens of glass (n = 1.5) has both surfaces of same radius of curvature R = 20 cm. Find its focal length in air.

Identify given parameters: n = 1.5, R₁ = +20 cm, R₂ = -20 cm.
Substitute values into the lens maker's formula: 1/$f = (1.5 - 1) * (1/20 - (-1/20))$.
Calculate: 1/$f = 0.5 * (2/20) = 0.5 * (1/10) = 1/20$.

Answer: 20 cm

PGT extension

A convex lens of focal length 10 cm and a concave lens of focal length 20 cm are placed in contact. Find the power and nature of the combination.

Identify focal lengths with signs: f₁ = +10 c$m = +0.1 m$, f₂ = -20 c$m = -0.2 m$.
Calculate individual powers: P₁ = 1 / (+0.1) = +10 D, P₂ = 1 / (-0.2) = -5 D.
Find net power: $P = P_{1} + P_{2} = +10 D - 5 D = +5 D$.
Determine net focal length: $F = 1 / P = +1/5 m = +20 cm$.

Answer: +5 Dioptre (Converging/Convex nature)

PGT extension

In a displacement method experiment, the distance between an object and a screen is 100 cm. A lens produces real images on the screen at two positions separated by 20 cm. Calculate the focal length of the lens.

Use the displacement method formula: $f = (a^{2} - d^{2}) / (4a)$.
Substitute given values: a = 100 cm, d = 20 cm.
Calculate numerator: 100² - 20² = 10000 - 400 = 9600.
Calculate denominator: 4 * 100 = 400.
Divide: $f = 9600 / 400 = 24 cm$.

Answer: 24 cm

Assumptions and limitations

Paraxial approximation (sin theta approx theta) holds true, limiting validity to small aperture lenses.
Thin lens approximation assumes thickness t approaches zero, ignoring refraction losses within the bulk material.
Monochromatic light is assumed unless chromatic dispersion is explicitly accounted for.

Common misconceptions

Common error: A convex lens always forms a real image.

Correct: A convex lens forms a virtual image when the object is placed between the optical centre and the principal focus.

Ray divergence exceeds convergence capacity when the object is within the focal length.

Common error: The focal length of a glass lens remains unchanged if it is submerged in water.

Correct: The focal length increases (power decreases) when a lens is immersed in water because the relative refractive index decreases.

Lens maker's formula depends on the ratio of refractive index of the lens to that of the surrounding medium.

Quick revision

A convex lens converges light in air and diverges it if placed in a medium of higher refractive index than itself.
Refraction at Spherical Surface: \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}
Lens Maker's Formula: \frac{1}{f} = \left(n_{rel} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)

Must remember

The refraction formula at a single spherical surface is $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$.
Lens Maker's Formula is given by $\frac{1}{f} = (n_{rel} - 1)(\frac{1}{R_1} - \frac{1}{R_2})$.
A convex lens behaves as a diverging lens if immersed in a medium with a refractive index greater than that of the lens.
The equivalent focal length of two thin lenses separated by a distance $d$ is $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$.
Chromatic aberration occurs due to the dependence of refractive index on the wavelength of light.

Glossary

Paraxial RayRays that make very small angles with the principal axis and are close to it.

Optical AberrationDeparture of image from the ideal geometric approximation, leading to blurring or distortion.

DioptreThe SI unit of optical power of a lens, equivalent to $m^{-1}$.

Refraction at Spherical Surface\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} (Paraxial rays, spherical surface)

Chromatic vs Spherical Aberration

Feature	Chromatic Aberration	Spherical Aberration
Cause	Dispersion (variation of refractive index with wavelength)	Large aperture causing marginal rays to focus closer than paraxial rays
Nature of image defect	Coloured fringes around images	Blurring and loss of sharpness of image
Elimination method	Achromatic doublet of lenses	Stopped-down aperture, crossed lenses, or aspheric surfaces

Practice Quiz

Choose an option to reveal the explanation.

1. What is the formula for refraction at a single spherical surface separating two media of refractive indices $n_1$ and $n_2$?

Explanation: The standard relation for refraction at a single spherical surface is $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$ under paraxial approximation.

2. When a double convex glass lens ($n = 1.5$) is immersed in water ($n = 1.33$), its focal length:

Explanation: The relative refractive index $n_{rel} = n_{lens}/n_{medium}$ decreases when placed in water, which decreases the refracting power and increases the focal length.

3. What is the equivalent focal length $F$ of two thin lenses of focal lengths $f_1$ and $f_2$ placed in direct contact?

Explanation: For thin lenses in contact, inverse focal lengths add up: $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$.

4. What is the SI unit of power of a lens?

Explanation: Power is defined as $P = 1/f$ (where $f$ is in metres), and its unit is the Dioptre ($D$ or $m^{-1}$).

5. Which optical aberration is caused by the variation of refractive index with the wavelength of light?

Explanation: Chromatic aberration arises because different wavelengths (colors) bend by different amounts due to dispersion, focusing at separate points.

6. A convex lens forms a virtual image when the object is placed:

Explanation: When an object is within the focal length of a convex lens ($u < f$), the rays diverge and appear to come from a virtual, magnified image on the same side.

7. What is the expression for linear magnification ($m$) of a thin lens in terms of object distance ($u$) and image distance ($v$)?

Explanation: Linear magnification for thin lenses is given by $m = \frac{h_i}{h_o} = \frac{v}{u}$ with proper sign conventions.

8. What happens to a converging convex lens if its refractive index is less than that of the surrounding medium?

Explanation: According to the Lens Maker's Formula, if $n_{lens} < n_{medium}$, $n_{rel} - 1$ becomes negative, reversing the sign of power and making a convex lens diverge light.

9. For two coaxial thin lenses separated by a distance $d$, the equivalent focal length $F$ is given by:

Explanation: The formula for two lenses separated by distance $d$ is $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$.

10. Spherical aberration in lenses can be minimized by:

Explanation: Restricting marginal rays using stopped-down apertures or employing crossed lenses helps minimize spherical aberration.

11. In the displacement method for finding the focal length of a convex lens, if the distance between object and screen is $a$ and separation between lens positions is $d$, the focal length is:

Explanation: The displacement method formula is $f = \frac{a^2 - d^2}{4a}$, valid when $a > 4f$.

12. What is the thin lens approximation assumption?

Explanation: The thin lens approximation assumes the thickness along the principal axis is very small compared to $u$ and $v$.

13. If two lenses of power $+2\text{ D}$ and $-3\text{ D}$ are kept in contact, the net power of the combination is:

Explanation: Net power for lenses in contact is $P = P_1 + P_2 = 2 + (-3) = -1\text{ D}$.

14. Why do marginal rays focus closer to the lens than paraxial rays in spherical aberration?

Explanation: Greater refraction occurs for marginal rays because the surface slope is steeper further from the axis, causing them to cross the axis closer to the lens.

15. What is the condition for real images to exist in the displacement method where $a$ is the distance between object and screen?

Explanation: Real images are formed by a convex lens on a screen only when the distance $a$ between the object and screen is greater than $4f$.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A convex lens of glass ($n = 1.5$) has a focal length of $10\text{ cm}$ in air. What is its focal length when immersed in water ($n = 1.33$)?

Explanation: Using Lens Maker's Formula, $\frac{1}{f_a} = (1.5 - 1)(\frac{1}{R_1} - \frac{1}{R_2})$ and $\frac{1}{f_w} = (\frac{1.5}{1.33} - 1)(\frac{1}{R_1} - \frac{1}{R_2})$. Taking the ratio gives $f_w \approx 40\text{ cm}$.

2. Two thin lenses of power $+4\text{ D}$ and $-2\text{ D}$ are placed in contact. What is the focal length of the combination?

Explanation: Net power $P = 4 + (-2) = +2\text{ D}$. Equivalent focal length $F = \frac{1}{P} = \frac{1}{2} = +0.5\text{ m}$.

3. In a displacement method experiment, the distance between the object and the screen is $100\text{ cm}$. For two positions of the lens, real images are formed on the screen separated by $20\text{ cm}$. What is the focal length of the lens?

Explanation: Using $f = \frac{a^2 - d^2}{4a}$, with $a = 100$ and $d = 20$: $f = \frac{100^2 - 20^2}{4(100)} = \frac{10000 - 400}{400} = \frac{9600}{400} = 24\text{ cm}$ (Wait, calculation: $9600/400 = 24$. Let's check options. Option B is 24 cm). Correct index is 1.

4. If a symmetrical biconvex lens has radii of curvature $R$ and refractive index $n$, its focal length is given by:

Explanation: For a symmetrical biconvex lens, $R_1 = R$ and $R_2 = -R$. Substituting into Lens Maker's Formula: $\frac{1}{f} = (n - 1)(\frac{1}{R} - \frac{1}{-R}) = \frac{2(n - 1)}{R}$, so $f = \frac{R}{2(n - 1)}$.

5. What is the primary cause of chromatic aberration in single lenses?

Explanation: Chromatic aberration is exclusively caused by dispersion—the refractive index of lens material varies with wavelength ($\mu_v > \mu_r$).

6. An object is placed at a distance of $30\text{ cm}$ from a convex lens of focal length $10\text{ cm}$. Where is the image formed?

Explanation: Using $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$, $\frac{1}{10} = \frac{1}{v} - \frac{1}{-30} \implies \frac{1}{v} = \frac{1}{10} - \frac{1}{30} = \frac{2}{30} = \frac{1}{15}$, so $v = 15\text{ cm}$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which of the following describes the paraxial ray approximation condition?

Explanation: Paraxial ray approximation assumes rays make very small angles with the principal axis, allowing the approximation $\sin \theta \approx \tan \theta \approx \theta$.

2. A convex lens is cut into two identical halves along its principal axis. The focal length of each half will:

Explanation: Cutting a lens along the principal axis does not change the radii of curvature of either half, so the focal length of each piece remains the same as the original lens (though light-gathering intensity decreases).

3. If a convex lens of focal length $f$ is cut perpendicularly to its principal axis into two equal parts, the focal length of each part becomes:

Explanation: When cut perpendicular to the principal axis, each half has one flat surface and one curved surface. The radius of curvature of the flat surface is infinity, leading to a new focal length of $2f$.

4. What is the equivalent focal length of two lenses of focal lengths $+20\text{ cm}$ and $-10\text{ cm}$ in contact?

Explanation: $\frac{1}{F} = \frac{1}{20} + \frac{1}{-10} = \frac{1 - 2}{20} = \frac{-1}{20}$, so $F = -20\text{ cm}$.

5. Which lens combination is typically used to minimize chromatic aberration of two lenses separated by a distance $d$?

Explanation: The condition for achromatic combination of two lenses separated by a distance $d$ is $d = \frac{f_1 + f_2}{2}$.

6. What is the magnification of an image formed by a thin lens when the image distance is equal to object distance in magnitude ($v = -u$)?

Explanation: Magnification $m = \frac{v}{u}$. If $v = -u$, then $m = -1$, indicating a real, inverted image of the same size as the object (at $2f$).

7. In refraction at a single spherical surface, if the medium containing the denser refractive index $n_2$ has a concave boundary facing the object, what is the sign of $R$?

Explanation: By standard Cartesian sign conventions, if the centre of curvature lies to the left of the pole (against incident light), the radius of curvature $R$ is negative.

8. A liquid of refractive index equal to that of glass is used to immerse a glass convex lens. What happens to the lens?

Explanation: When $n_{lens} = n_{medium}$, $n_{rel} = 1$, making $\frac{1}{f} = 0$, so $f = \infty$. The lens behaves as a simple transparent slab.

9. Which of the following defects is NOT an optical aberration?

Explanation: The photoelectric effect is a quantum phenomenon related to light-matter interaction, whereas chromatic aberration, spherical aberration, and astigmatism are classical optical aberrations.

10. If the power of a lens is $+2.5\text{ D}$, what is its focal length?

Explanation: $f = \frac{1}{P} = \frac{1}{2.5} = \frac{10}{25} = 0.4\text{ m}$ ($+40\text{ cm}$). Positive sign indicates a convex/converging lens.