Thin lenses — lens equation, lens maker's equation

1. Setup, conventions, and WHAT we are computing

WHAT we want: given object position and lens shape, find where the image forms. Two equations do this:

Lens equation relates $u,v,f$ (uses the lens as a black box).
Lens maker's equation computes $f$ from the glass shape ( $R_1,R_2,n$ ).

2. Refraction at a SINGLE spherical surface (the building block)

For a single surface of radius $R$ separating media $n_1$ (incoming) and $n_2$ (outgoing), the object–image relation is the single-surface refraction formula:

$\boxed{\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}}$

Worked example HOW we derive it (paraxial, first principles)

Why this step? Use Snell's law for small angles so $\sin\theta\approx\theta$ . Take a ray from axial object $O$ hitting the surface at height $h$ , at angle $i$ to the normal (normal passes through centre $C$ ). Snell: $n_1 i = n_2 r$ (paraxial).

Using the exterior-angle geometry (angles small, so $\tan\theta\approx\theta\approx h/\text{distance}$ ):

$i = \alpha + \phi$ , where $\alpha=\dfrac{h}{-u}$ (object) and $\phi=\dfrac{h}{R}$ (radius).
$r = \phi - \beta$ , where $\beta=\dfrac{h}{v}$ (image).

Why this step? Substitute into $n_1 i=n_2 r$ and cancel the common $h$ : $n_1\!\left(\frac{h}{-u}+\frac{h}{R}\right)=n_2\!\left(\frac{h}{R}-\frac{h}{v}\right)$ $-\frac{n_1}{u}+\frac{n_1}{R}=\frac{n_2}{R}-\frac{n_2}{v}$ Rearranged: $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ . ∎

3. Deriving the LENS MAKER'S equation (two surfaces)

Formula Apply the single-surface rule twice

Lens material $n$ , surrounding medium air ( $n=1$ ). Surfaces $R_1$ (first) and $R_2$ (second).

Surface 1 (air → glass): object at $u$ , image at intermediate $v_1$ : $\frac{n}{v_1}-\frac{1}{u}=\frac{n-1}{R_1} \quad(1)$

Why this step? The image of surface 1 becomes the object for surface 2. Thin lens ⇒ no gap, so its object distance is also $v_1$ , and now glass → air: $\frac{1}{v}-\frac{n}{v_1}=\frac{1-n}{R_2}\quad(2)$

Why this step? Add (1)+(2): the $n/v_1$ terms cancel — that is the whole trick. $\frac{1}{v}-\frac{1}{u}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

By definition $f$ is where parallel rays ( $u\to-\infty$ ) focus, giving $\frac1v=\frac1f$ . So:

$\boxed{\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)}\quad\textbf{Lens maker's equation}$

And comparing, the thin-lens (Gaussian) equation:

$\boxed{\frac{1}{v}-\frac{1}{u}=\frac{1}{f}}$

Figure — Thin lenses — lens equation, lens maker's equation

4. Magnification

5. Worked examples

6. Common mistakes (Steel-man + fix)

7. Active recall

Single spherical surface formula

\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}

Lens maker's equation

\dfrac1f=(n-1)\left(\dfrac1{R_1}-\dfrac1{R_2}\right)

Thin-lens (Gaussian) equation

\dfrac1v-\dfrac1u=\dfrac1f

Why does the intermediate image vanish in the derivation?

Adding the two single-surface equations cancels the

n/v_1

term

Sign of

R

rule

R>0

if centre of curvature lies on the outgoing (transmission) side

Magnification formula

m=\dfrac{h'}{h}=\dfrac{v}{u}

What is

f

for a flat plate (

R\to\infty

Infinite power zero —

1/f=0

, no focusing

Converging lens sign of

f

Positive (

f>0

)

Object 30 cm before

f=20

cm lens, find

v

v=+60

cm, real,

m=-2

inverted

Why must light "slow" in glass for bending?

(n-1)

factor; if

n=1

no speed change, no bending

Recall Feynman: explain to a 12-year-old

Imagine a marching band walking off a road onto muddy grass at an angle. The side that hits mud first slows down, so the whole line swings. Glass is the mud for light. A lens has two curved muddy edges; light swings a bit at each one. If both swings push light toward one spot, all the rays meet there — that's the image! Fatter, more curved glass = more swing = the meeting point is closer.

Connections

Refraction at a single spherical surface
Snell's law & paraxial approximation
Spherical mirrors — mirror equation
Magnification and image formation
Power of a lens (dioptres)
Lens combinations & equivalent focal length
Lens aberrations (chromatic, spherical)

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek lens basically do curved glass surfaces ka combo hai. Light jab air se glass me jaati hai to slow ho jaati hai (kyunki refractive index $n$ alag hai), aur isi wajah se bend hoti hai — yehi Snell's law hai. "Thin lens" ka matlab simple: glass itna patla hai ki andar ka travel ignore kar do, dono surfaces ko ek hi point pe maano.

Pehle hum ek single curved surface ke liye formula nikaalte hain: $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$ . Phir yahi formula do baar lagao — pehli surface (air→glass) aur doosri surface (glass→air). Jab dono equations ko add karte ho, to beech wala intermediate image term ( $n/v_1$ ) cancel ho jaata hai, aur seedha mil jaata hai Lens Maker's equation: $\frac1f=(n-1)\left(\frac1{R_1}-\frac1{R_2}\right)$ . Aur lens equation banti hai $\frac1v-\frac1u=\frac1f$ .

Sabse important baat: sign convention. Real object ke liye $u$ negative, converging lens ke liye $f$ positive, aur $R$ positive tab jab uska centre outgoing side pe ho. Yaad rakhna lens me minus ( $v-u$ ) aur mirror me plus — yahin pe zyada students galti karte hain. $R_2$ ka sign bhar mat bhoolna, warna pura $f$ galat aa jaata hai.

Practical use? Chashma, camera, microscope, telescope — sab isi do equation pe chalte hain. Ek baar derivation samajh gaye to ratne ki zaroorat hi nahi; tum khud derive kar sakte ho exam me.

Thin lenses — lens equation, lens maker's equation

1. Setup, conventions, and WHAT we are computing

2. Refraction at a SINGLE spherical surface (the building block)

3. Deriving the LENS MAKER'S equation (two surfaces)

4. Magnification

5. Worked examples

6. Common mistakes (Steel-man + fix)

7. Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Optics

Connections