NDA Physics · Light and Optics

Lenses and the Lens Formula

A lens refracts light through two curved surfaces: a convex (converging) lens bends rays together, a concave (diverging) lens spreads them apart. The lens formula 1/v − 1/u = 1/f, power P = 1/f in dioptres, the lens maker's equation, and magnification together fix every image.

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Why this matters

Twelve PYQs, and the bank's cleanest source of numeric marks — power calculations and combinations appear almost every year. The recurring tests are: P = 1/f (with f in metres), adding powers for lenses in contact, the lens maker's equation, the New Cartesian sign convention, and the fixed fact that a concave lens only ever gives a virtual, erect, diminished image. All three HARD questions in the chapter involving lenses sit here.

Concept 1 of 6: Convex and concave lenses

A convex lens is thick in the middle and squeezes parallel rays to a point — it converges light and can make real images. A concave lens is thin in the middle and fans rays apart — it diverges light and can only ever make a virtual, smaller, upright image. That single fact about the concave lens settles a surprising number of questions.

Definition

Convex (converging) lens: thicker at the centre; brings parallel rays to a real focus. It can form real or virtual images, and images larger, equal, or smaller than the object, depending on object position.
Concave (diverging) lens: thinner at the centre; spreads parallel rays so they appear to come from a virtual focus. It always forms a virtual, erect, diminished image — never a real one.

A convex lens can spread out a beam of sunlight (it refracts the rays toward a focus and beyond); a plane mirror cannot.

A convex lens converges light. The parallel ray bends through the far focus; the ray through the optic centre goes straight. Where they cross is the image.

Worked example

State which statement is NOT correct: (i) a convex lens can produce both real and virtual images; (ii) a concave lens can produce both real and virtual images; (iii) a concave lens always produces a diminished image.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Sep 2019 · Q71Moderate

Example 1 · Light and Optics · Lenses and Lens Formula

Which one of the following statements regarding lenses is

\textbf{\text{not}}

correct?

A concave lens has no real-image setting

Unlike a concave MIRROR (which gives real images for most positions), a concave LENS gives a virtual, erect, diminished image for EVERY object position. Any claim that a concave lens makes real images is the wrong statement.

Drill 1 more on convex and concave lenses

Concept 2 of 6: Lens formula and the sign convention

The lens formula links object distance, image distance and focal length, just like the mirror formula but with a minus sign in a different place. The New Cartesian convention does the bookkeeping: measure everything from the optic centre, take the incident-light direction as positive, and a convex lens gets a positive focal length while a concave lens gets a negative one.

Definition

Lens formula: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$ (New Cartesian sign convention; distances measured from the optic centre).

Convex lens: $f$ is positive. Concave lens: $f$ is negative.
The same general FORM $1/v \pm 1/u = 1/f$ underlies both mirror and lens — the convention differs because light is transmitted (lens) versus reflected (mirror), so the formula 'applies to spherical mirrors as well as spherical lenses'.

Lens formula

\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}

uobject distance from optic centre (−ve for a real object)
vimage distance from optic centre
ffocal length (+ve convex, −ve concave)

Worked example

An object is placed 30 cm in front of a convex lens of focal length 20 cm. Where is the image formed?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Apr 2021 · Q68Moderate

Example 2 · Light and Optics · Lenses and Lens Formula

According to the New Cartesian Sign Convention, which one of the following is correct in respect of the formula

\frac{1}{f} = \frac{1}{v} + \frac{1}{u}

, where symbols have their usual meanings ?

Lens uses 1/v − 1/u; mirror uses 1/v + 1/u

The two formulae look almost identical — a sign on the 1/u term is the only difference. Mixing them up flips your image distance. Memorise: lens is MINUS, mirror is PLUS.

Concept 3 of 6: Power of a lens — the dioptre

Power tells you how strongly a lens bends light: a short focal length means a powerful lens. Power is just one over the focal length, with the focal length in metres — and it is measured in dioptres. A convex lens has positive power; a concave lens has negative power.

Definition

Power $P = \dfrac{1}{f}$ with $f$ in metres; unit: dioptre (D).

Convex lens: $P > 0$ . Concave lens: $P < 0$ .
A short focal length ⟹ large power (strongly bending).
Watch the units: f given in cm must be converted to metres first ( $f\,\text{cm} = f/100\,\text{m}$ ).

Power of a lens

P = \dfrac{1}{f\,(\text{in metres})}

Ppower (dioptre, D)
ffocal length in METRES

Worked example

A convex lens has a focal length of 25 cm. What is its power?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Sep 2021 · Q132Easy

Example 3 · Light and Optics · Lenses and Lens Formula

Power of a lens of focal length 25 cm is

Convert cm to metres before computing power

P = 1/f needs f in METRES. f = 25 cm gives P = 1/0.25 = 4 D, not 1/25 = 0.04 D. Forgetting the conversion is the standard power-question trap.

Drill 2 more on power of a lens — the dioptre

Concept 4 of 6: Lens maker's equation

How do you build a lens of a chosen focal length? The lens maker's equation says it depends on two things: how much the glass slows light (its refractive index) and how sharply the two faces are curved (their radii). Make the surfaces more curved or the glass more refractive, and you get a more powerful lens.

Definition

Lens maker's equation: $\dfrac{1}{f} = (n - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ , for a thin lens in air.

$n$ is the refractive index of the lens material; $R_1, R_2$ are the radii of curvature of the two faces (signed by convention).
For a double convex lens, $R_1 > 0$ and $R_2 < 0$ , so the two terms add.
Higher $n$ or smaller radii ⟹ shorter focal length ⟹ greater power.

Lens maker's equation

\dfrac{1}{f} = (n - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)

nrefractive index of the lens material
R_1, R_2radii of curvature of the two faces (signed)
ffocal length

Worked example

A double convex lens has faces of radii 20 cm and 20 cm, made of glass with n = 1.5. Find its power.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Apr 2017 · Q106Hard

Example 4 · Light and Optics · Lenses and Lens Formula

The radii of curvature of the faces of a double convex lens are 10 cm and 20 cm. The refractive index of the glass is 1·5. What is the power of this lens (in units of dioptre) ?

Sign the radii — convex faces are not both positive

For a double convex lens R₁ is positive but R₂ is negative, so 1/R₁ − 1/R₂ becomes 1/R₁ + 1/|R₂| — the terms ADD. Treating both as the same sign halves your answer.

Drill 1 more on lens maker's equation

Concept 5 of 6: Lenses in contact — powers add

Put two thin lenses against each other and they behave like one lens whose strength is just the sum of the two. Because power adds so cleanly, combination problems are almost always faster in power than in focal length — convert each f to a power, add, then convert back if needed.

Definition

For thin lenses in contact, the powers add:

P = P_1 + P_2 + \dots = \dfrac{1}{f_1} + \dfrac{1}{f_2} + \dots

and the combined focal length is

f = 1/P

A convex (+) and a concave (−) lens partly cancel; the sign of the net power tells you whether the combination converges (+) or diverges (−).
Always work in dioptres (f in metres) — adding focal lengths directly is wrong.

Combination of thin lenses in contact

P = P_1 + P_2, \qquad f = \dfrac{1}{P}

P_1, P_2powers of the individual lenses (D)
Ppower of the combination (D)
ffocal length of the combination (m)

Worked example

A convex lens of power +3 D is placed in contact with a concave lens of power −1 D. Find the focal length of the combination.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Apr 2018 · Q105Moderate

Example 5 · Light and Optics · Lenses and Lens Formula

Two convex lenses with power 2 dioptre are kept in contact with each other. The focal length of the combined lens system is

Add powers, not focal lengths

For lenses in contact the POWERS add. Two +2 D lenses give +4 D ⟹ f = 0.25 m, NOT 1 m. Never add focal lengths directly.

Drill 2 more on lenses in contact — powers add

Concept 6 of 6: Lens magnification and image formation

Magnification for a lens is image distance over object distance — its sign tells you whether the image is upright or inverted, its size tells you how much bigger or smaller. As with mirrors, get the signs from the convention first, then the arithmetic is straightforward.

Definition

Magnification: $m = \dfrac{h'}{h} = \dfrac{v}{u}$ for a lens.

$m > 0$ : virtual, erect image. $m < 0$ : real, inverted image.
$|m| > 1$ : magnified; $|m| < 1$ : diminished.

Convex-lens image table (object moving in): beyond 2F → real, inverted, diminished (between F and 2F); at 2F → real, inverted, same size; between F and 2F → real, inverted, magnified; at F → at infinity; inside F → virtual, erect, magnified (the magnifying glass).

Lens magnification

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

mmagnification
vimage distance
uobject distance

Worked example

A convex lens of focal length 15 cm forms an image of an object placed 30 cm away. Find the image distance and magnification.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Apr 2022 · Q98Hard

Example 6 · Light and Optics · Lenses and Lens Formula

What is the magnification produced by a concave lens of focal length 10 cm, when an image is formed at a distance of 5 cm from the lens?

Lens m = v/u (no minus); mirror m = −v/u

The magnification formula differs by a sign between lens and mirror. For a lens m = v/u; for a mirror m = −v/u. The sign convention then makes both give the right orientation — but use the correct one for the device.

Drill 1 more on lens magnification and image formation

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (5)

Lens formula and the sign convention
Lens formula
$\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$
Power of a lens — the dioptre
Power of a lens
$P = \dfrac{1}{f\,(\text{in metres})}$
Lens maker's equation
Lens maker's equation
$\dfrac{1}{f} = (n - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$
Lenses in contact — powers add
Combination of thin lenses in contact
$P = P_1 + P_2, \qquad f = \dfrac{1}{P}$
Lens magnification and image formation
Lens magnification
$m = \dfrac{h'}{h} = \dfrac{v}{u}$

Watch out for (6)

A concave lens has no real-image setting
→ Convex and concave lenses
Lens uses 1/v − 1/u; mirror uses 1/v + 1/u
→ Lens formula and the sign convention
Convert cm to metres before computing power
→ Power of a lens — the dioptre
Sign the radii — convex faces are not both positive
→ Lens maker's equation
Add powers, not focal lengths
→ Lenses in contact — powers add
Lens m = v/u (no minus); mirror m = −v/u
→ Lens magnification and image formation

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