7. Lenses
Topics: Types of Lenses Key Terms Ray Diagrams Lens Formula & Magnification Power & Lens Combinations Human Eye Vision Defects Uses of Lenses
What is a Lens?
A lens is a transparent optical medium bounded by (usually) two spherical surfaces. Light refracts twice as it passes through a lens—on entering and on emerging—so its direction changes.
Types (by shape)
- Convex (Double/Plano/Positive Meniscus) – thicker at centre; converging lens.
- Concave (Double/Plano/Negative Meniscus) – thinner at centre; diverging lens.
Key Terms
- Centres of Curvature \(C_1, C_2\): centres of the spheres forming the two surfaces.
- Radii of Curvature \(R_1, R_2\).
- Principal Axis: line through \(C_1\) and \(C_2\).
- Optical Centre \(O\): point on the principal axis through which a ray passes undeviated.
- Principal Foci \(F_1, F_2\): special points related to parallel rays (converge/diverge).
- Focal length \(f\): \(OF\) (distance from optical centre to principal focus).
Ray Diagrams & Rules
Convex Lens (Converging)
- Ray parallel to principal axis → refracts through \(F_2\).
- Ray through \(F_1\) → emerges parallel to principal axis.
- Ray through \(O\) → passes undeviated.
Concave Lens (Diverging)
- Ray parallel to principal axis → appears to emerge from \(F_1\) (on object side).
- Ray directed towards \(F_2\) → emerges parallel.
- Ray through \(O\) → passes undeviated.
Images by a Convex Lens (Quick Table)
| # | Object Position | Image Position | Size | Nature |
|---|---|---|---|---|
| 1 | At ∞ | At \(F_2\) | Point-like | Real, Inverted |
| 2 | Beyond \(2F_1\) | Between \(F_2\) & \(2F_2\) | Smaller | Real, Inverted |
| 3 | At \(2F_1\) | At \(2F_2\) | Same size | Real, Inverted |
| 4 | Between \(F_1\) & \(2F_1\) | Beyond \(2F_2\) | Larger | Real, Inverted |
| 5 | At \(F_1\) | At ∞ | Very large | Real, Inverted |
| 6 | Between \(F_1\) & \(O\) | Same side of lens | Larger | Virtual, Erect |
Images by a Concave Lens
| # | Object Position | Image Position | Size | Nature |
|---|---|---|---|---|
| 1 | At ∞ | At \(F_1\) | Point-like | Virtual, Erect |
| 2 | Anywhere (between \(O\) & ∞) | Between \(O\) & \(F_1\) | Smaller | Virtual, Erect |
Cartesian Sign Convention
- Object placed to the left of lens; origin at \(O\) on principal axis.
- Distances to the right of \(O\): \(+\)ve; to the left: \(-\)ve.
- Heights above axis: \(+\)ve; below: \(-\)ve.
- Convex lens: \(f>0\); Concave lens: \(f<0\).
Lens Formula
\[
\boxed{\;\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\;}
\]
Magnification
\[
M=\frac{h_2}{h_1}=\frac{v}{u}
\]
Tip \(M<0\) → image inverted (real); \(M>0\) → image erect (virtual).
Power of a Lens
\[
P=\frac{1}{f(\text{in metres})}\quad\text{Unit: Dioptre (D)}
\]
Combination of Lenses (in contact)
\[
\frac{1}{f_{\text{eq}}}=\frac{1}{f_1}+\frac{1}{f_2}
\quad\Longleftrightarrow\quad
P_{\text{eq}}=P_1+P_2
\]
Solved Examples
1) Image by a Convex Lens
Given: \(h_1=5\,\text{cm},\ f=+10\,\text{cm},\ u=-20\,\text{cm}\).
\[
\frac{1}{f}=\frac{1}{v}-\frac{1}{u}
\;\Rightarrow\;
\frac{1}{10}=\frac{1}{v}-\frac{1}{-20}
=\frac{1}{v}+\frac{1}{20}
\Rightarrow
\frac{1}{v}=\frac{1}{10}-\frac{1}{20}=\frac{1}{20}
\Rightarrow v=+20\,\text{cm}
\]
\[
M=\frac{v}{u}=\frac{+20}{-20}=-1
\quad\Rightarrow\quad
h_2=M\,h_1=-1\times 5=-5\,\text{cm}
\]
Result: Image at \(+20\) cm (other side), real, inverted, same size (5 cm).
2) Power from Focal Length
Given: Convex lens \(f=+20\,\text{cm}=0.20\,\text{m}\).
\[
P=\frac{1}{f(\text{m})}=\frac{1}{0.20}=+5\,\text{D}
\]
Human Eye: Working of the Lens
- Cornea refracts most of the incoming light first.
- Iris controls the pupil size → regulates light entry.
- Crystalline (double convex) lens fine-tunes focus; forms real, inverted image on the retina.
- Optic nerve carries signals to the brain.
Accommodation: Ciliary muscles change lens curvature → focal length varies: flatter for distant objects (larger \(f\)), more rounded for nearer objects (smaller \(f\)).
- Near point (least distance of distinct vision): ~25 cm.
- Far point of a normal eye: infinity.
Defects of Vision & Corrections
1) Nearsightedness (Myopia)
- Sees near clearly; far objects blur (image forms in front of retina).
- Causes: excessive curvature / elongated eyeball.
- Correction: Concave lens (negative power) diverges rays so eye lens focuses on retina.
2) Farsightedness (Hypermetropia)
- Sees far clearly; near objects blur (image forms behind retina).
- Causes: reduced curvature / shortened eyeball.
- Correction: Convex lens (positive power) pre-converges rays.
3) Presbyopia
- With age, accommodation reduces; near point shifts farther away.
- Correction: Appropriate reading lenses; if combined with myopia → bifocals (upper concave for distance, lower convex for near).
Apparent Size of an Object
An object appears larger when it subtends a larger angle at the eye. Bringing an object closer increases the angle, but closer than ~25 cm, the eye cannot focus comfortably without strain.
Uses of Lenses
Concave Lenses
- Correct myopia (spectacles).
- “Peep holes” in doors for a wider field.
- Optical assemblies in scanners/CD players/medical equipment (beam shaping).
- Torches (to spread light from a small bulb).
Convex Lenses
- Simple microscope (magnifying glass): virtual, erect, magnified image; large \(M\) for small \(f\).
- Compound microscope: short-\(f\) objective + longer-\(f\) eyepiece; image by objective acts as object for eyepiece.
- Telescopes (refracting / reflecting): large-\(f\), large-aperture objective for light collection; eyepiece for viewing.
- Camera, projector, spectrograph; correcting hypermetropia (spectacles).
Persistence of Vision & Colour Perception
- Persistence of vision: retinal impression lasts ≈ \(1/16\) s after stimulus removal → basis of motion pictures, spinning illusions.
- Retina cells: Rods (brightness, work in dim light) and Cones (colour; sensitive to red/green/blue).
- Colour blindness: deficiency in certain cone responses; visual acuity otherwise normal.
Practice ideas: Measure the focal length of a convex lens by imaging a distant tree on a screen; verify lens formula with different object positions; explore the effect of adding two lenses (sum of powers).
Exercise – Solutions (Chapter 7: Lenses)
Match the Columns Figures & Definitions Ray Diagrams Scientific Reasons Telescope Distinguish Between Eye: Iris & Ciliary Muscles Numericals
1) Match the columns and explain
| Column 1 | Column 2 | Column 3 | Correct Match & Explanation |
|---|---|---|---|
| Farsightedness (Hypermetropia) | Far away object can be seen clearly | Convex lens | Farsightedness → Far-away clear → Convex lens Hypermetropic eye focuses near-object images behind the retina. A convex lens pre-converges rays so the eye lens forms the image on the retina. |
| Presbyopia | Problem of old age | Bifocal lens | Presbyopia → Age-related accommodation loss → Bifocal Near point drifts away; many also have distance error. Bifocal: upper concave (distance), lower convex (near). |
| Note: “Nearby object can be seen clearly” + “Concave lens” describe Nearsightedness (Myopia), which is not in Column 1 here (distractor). | |||
2) Figure: terms related to a lens
Labels to remember: Principal axis, optical centre \(O\), principal foci \(F_1,F_2\), points \(2F_1,2F_2\), centres of curvature \(C_1,C_2\), radii \(R_1,R_2\).
3) Object position for same-sized real image (Convex lens)
Keep the object at \(2F_1\). The image forms at \(2F_2\), is real, inverted and same size.
Lens formula check: \(\displaystyle \frac{1}{f}=\frac{1}{v}-\frac{1}{u}\) with \(u=-2f\Rightarrow v=+2f\), and \(M=\dfrac{v}{u}=-1\).
4) Give scientific reasons
- Simple microscope is used for watch repairs.
A convex lens of small focal length gives a virtual, erect, magnified image at a comfortable viewing distance (≈ 25 cm), enlarging tiny parts and improving precision. - One can sense colours only in bright light.
Colour perception is by cone cells which need bright (photopic) light. In dim light, rods function (scotopic vision) but are insensitive to colour. - We cannot clearly see an object kept closer than 25 cm.
25 cm is the least distance of distinct vision. The ciliary muscles cannot increase the lens curvature enough to focus light from nearer objects on the retina without strain; image blurs.
5) Working of an Astronomical Telescope (Refraction)
An astronomical (refracting) telescope uses two convex lenses aligned on one axis:
- Objective (large \(f_o\), large aperture) forms a real, inverted, diminished image of a distant object at its focal plane.
- Eyepiece (small \(f_e\)) acts as a simple microscope and provides an enlarged virtual image for the eye (normal adjustment: final image at ∞).
Angular magnification (normal adjustment): \(\displaystyle M=\frac{f_o}{f_e}\).
For normal adjustment, separation ≈ \(f_o+f_e\).
For normal adjustment, separation ≈ \(f_o+f_e\).
Large objective collects more light (faint celestial objects). Eyepieces of different \(f_e\) give different magnifications.
6) Distinguish between
a) Farsightedness vs Nearsightedness
| Point | Farsightedness (Hypermetropia) | Nearsightedness (Myopia) |
|---|---|---|
| Clear vision | Distant objects | Near objects |
| Image forms | Behind the retina (for near objects) | In front of the retina (for far objects) |
| Main cause | Reduced curvature / shortened eyeball | Excess curvature / elongated eyeball |
| Correction | Convex lens (\(P>0\)) | Concave lens (\(P<0\)) |
| Near/Far point | Near point > 25 cm (farther away) | Far point < ∞ (closer than infinity) |
b) Concave lens vs Convex lens
| Point | Concave Lens (Diverging) | Convex Lens (Converging) |
|---|---|---|
| Thickness | Thinner at centre | Thicker at centre |
| Parallel rays | Diverge; appear from \(F_1\) | Converge at \(F_2\) |
| Focal length sign | Negative | Positive |
| Common images | Always virtual, erect, diminished | Real/inverted (most cases), virtual/erect when object within \(F_1\) |
| Use in spectacles | Corrects myopia | Corrects hypermetropia |
7) Functions of the iris and ciliary muscles
- Iris: Controls pupil size and hence the amount of light entering the eye (contracts in bright light, dilates in dim light).
- Ciliary muscles (connected to lens): Alter lens curvature for accommodation. Tighten → lens more rounded (shorter \(f\)) for near; relax → lens flatter (longer \(f\)) for distant objects.
8) Numericals
i) Power and focal length
Prescription: \(P=+1.5\,\text{D}\).
\[
f=\frac{1}{P}=\frac{1}{1.5}\,\text{m}=0.666\ldots \text{ m}\approx \boxed{+0.67\,\text{m}}
\]
Type: Convex lens (\(P>0\)). Defect: Farsightedness (hypermetropia). Ans matches
ii) Image by a converging lens
\(h_1=5\,\text{cm},\; u=-25\,\text{cm},\; f=+10\,\text{cm}\).
\[
\frac{1}{f}=\frac{1}{v}-\frac{1}{u}
\Rightarrow
\frac{1}{10}=\frac{1}{v}+\frac{1}{25}
\Rightarrow
\frac{1}{v}=\frac{1}{10}-\frac{1}{25}=\frac{3}{50}
\Rightarrow
v=\boxed{\,\frac{50}{3}\,\text{cm}\approx 16.7\,\text{cm}\,}
\]
\[
M=\frac{v}{u}=\frac{50/3}{-25}=-\frac{2}{3}
\Rightarrow
h_2=Mh_1=-\frac{2}{3}\times 5=\boxed{-3.33\,\text{cm}}
\]
Image: Real, inverted; position \(\approx 16.7\) cm, size \(\approx 3.3\) cm. Ans matches
iii) Power of lenses in contact
\(P_1=2\,\text{D},\; P_2=2.5\,\text{D},\; P_3=1.7\,\text{D}\).
\[
P_{\text{eq}}=P_1+P_2+P_3=2+2.5+1.7=\boxed{6.2\,\text{D}}
\]
Ans matches
iv) Virtual image in front of lens
Object \(u=-60\,\text{cm}\). Virtual image “in front” (same side as object) ⇒ \(v=-20\,\text{cm}\).
\[
\frac{1}{f}=\frac{1}{v}-\frac{1}{u}=\frac{1}{-20}-\frac{1}{-60}
=-\frac{1}{20}+\frac{1}{60}=-\frac{2}{60}=-\frac{1}{30}
\Rightarrow f=\boxed{-30\,\text{cm}}
\]
Lens type: Diverging (concave). Ans matches
Handy Formulae
\[
\boxed{\;\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\;}\quad
\boxed{\;M=\frac{h_2}{h_1}=\frac{v}{u}\;}\quad
\boxed{\;P=\frac{1}{f(\text{m})}\;}\quad
\boxed{\;P_{\text{eq}}=P_1+P_2(+\cdots)\;}
\]