Power of a lens, combination of lenses

1. What is Power?

WHY define it this way? Imagine a ray parallel to the axis hitting a lens at height $h$ from the axis. After refraction it crosses the axis at the focus, a distance $f$ away. The angle of bending $\delta$ satisfies (for small angles): $\tan\delta \approx \delta = \frac{h}{f}$

So the bending angle per unit height $= \dfrac{\delta}{h} = \dfrac{1}{f}$ . This is exactly the deviation-producing strength of the lens — and that is precisely what we christen power. Hence $P = 1/f$ is not arbitrary; it is the natural measure of "deviation per unit ray height."

2. Combination of Thin Lenses in Contact

Derivation from scratch (two thin lenses in contact):

Place two thin lenses of focal lengths $f_1, f_2$ touching each other. An object at distance $u$ forms an image through lens 1.

Step 1 — Lens 1 alone: $\frac{1}{v_1} - \frac{1}{u} = \frac{1}{f_1}$ Why this step? This is the thin-lens equation applied to the first lens; $v_1$ is its image, which becomes the object for lens 2.

Step 2 — Lens 2 alone: The image $I_1$ (at $v_1$ ) acts as the object for lens 2. Since the lenses are in contact, the object distance for lens 2 is also $v_1$ : $\frac{1}{v} - \frac{1}{v_1} = \frac{1}{f_2}$ Why this step? Lens 2 takes whatever lens 1 produced and refracts again; final image is at $v$ .

Step 3 — Add the equations: Add them so $v_1$ cancels: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f_1} + \frac{1}{f_2}$ Why this step? The intermediate $v_1$ is internal bookkeeping — it must vanish, leaving an equation in object/image only.

Step 4 — Define the equivalent lens: Compare with $\dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{F}$ : $\boxed{\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}} \quad\Longleftrightarrow\quad \boxed{P = P_1 + P_2}$

Figure — Power of a lens, combination of lenses

3. Lenses Separated by a Distance $d$

WHY the extra term? When the lenses are apart, the ray bent by lens 1 travels a distance $d$ and lands at a different height on lens 2 than it entered lens 1. Lens 2's bending depends on that height, so the bends no longer add cleanly — the correction term $-dP_1P_2$ accounts for this. Note when $d=0$ it reduces to $P_1+P_2$ ✓ (a good sanity check).

4. Worked Examples

5. Common Mistakes

Recall Feynman: explain to a 12-year-old

A lens is like a slide that pushes light toward one spot. A "strong" lens pushes the light really hard, so it meets quickly — that strong push is its power. A "weak" lens pushes gently, light meets far away. We measure the push with a number called dioptres. If you put two lenses right against each other, you get both pushes one after another — so you just add their power numbers. A concave lens pushes light outward instead, so we give it a minus push that cancels some of the plus.

Flashcards

What is the power of a lens, and its unit?

P = 1/f

(f in metres); unit = dioptre (D) = m⁻¹.

A lens has

f=-25

cm. What is its power?

P = 1/(-0.25) = -4\,\text{D}

(concave, negative).

Why do powers add for lenses in contact?

Light is bent successively; adding the two thin-lens equations cancels the intermediate image distance, giving

1/F=1/f_1+1/f_2

Combined power of two lenses separated by distance

d

P = P_1+P_2 - dP_1P_2

Lenses in contact behave like which resistor configuration?

Parallel resistors (reciprocals add).

Net magnification of a lens combination?

m = m_1 \times m_2 \times \dots

Two lenses

+5\,

D and

-2\,

D in contact: net power and focal length?

+3\,

F=+33.3

cm.

Why does small focal length mean high power?

Bending angle per unit ray height

=1/f

, so smaller

f

bends light more strongly.

Connections

Thin Lens Equation — the foundation we added together to get $1/F$ .
Lensmaker's Equation — gives $P=(n-1)(1/R_1-1/R_2)$ .
Magnification of Lenses — net $m$ multiplies.
Microscope and Telescope — use lens combinations (objective + eyepiece).
Resistors in Parallel — same reciprocal-addition mathematics.
Defects of Vision — spectacle power in dioptres prescribed by opticians.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, lens ka kaam hai light ko mod-na (bend karna). Jo lens light ko zyada zor se modta hai, woh "strong" hai aur uska focal length chhota hota hai. Is strength ko hum power kehte hain, aur formula simple hai: $P = 1/f$ , jahan $f$ metre mein hona chahiye. Unit hai dioptre (D). Convex lens ka power positive, concave ka negative. Yaad rakho — chhota $f$ matlab bada power, kyunki bending angle $= h/f$ hota hai.

Jab do lens ek doosre se chipke (in contact) hote hain, toh light pehle lens se bend hoti hai, phir turant doosre se. Do bending ek saath add ho jaati hain, isliye $P = P_1 + P_2$ . Isko derive karne ke liye dono lens ka thin-lens equation likho aur add kar do — beech ka image distance $v_1$ cancel ho jaata hai, aur seedha $1/F = 1/f_1 + 1/f_2$ mil jaata hai. Ye bilkul parallel resistors jaisa hai, series jaisa NAHI — yahi sabse common galti hai.

Agar lenses ke beech gap $d$ ho, toh ek extra term aata hai: $P = P_1 + P_2 - dP_1P_2$ . Kyun? Kyunki gap mein ray thodi spread ho jaati hai aur doosre lens pe alag height pe girti hai, isliye correction chahiye. Check: agar $d=0$ ho toh wapas $P_1+P_2$ aa jaata hai — perfect.

Exam tip: hamesha pehle cm ko metre mein convert karo, sign dhyan se lagao (concave = minus), aur "Power Inverts, Contact Adds" yaad rakho. Bas itna pakka kar lo, ye topic full marks ka hai!