MHT-CET Physics · Optics (Ray)

Lenses: the Lens Formula, the Lensmaker's Equation and Combinations

A thin lens forms an image where 1/v − 1/u = 1/f with magnification v/u; its focal length comes from its surfaces through the lensmaker's equation 1/f = (μ − 1)(1/R₁ − 1/R₂); and lenses in contact add their powers P = 1/f in dioptres.

All Optics (Ray) notes MHT-CET Physics strategy

Why this matters

24 PYQs, 6 of them HARD. Nine use the lens formula — object and image distances for a given magnification, the least object–image distance, an air bubble in water. Fifteen use the lensmaker's equation and powers: a lens immersed in a liquid, cut or ground flat, lenses in contact or apart, and a plano-convex lens fitted into a plano-concave one. Two cards.

Concept 1 of 2: The Lens Formula and Magnification

With the Cartesian convention, 1/v − 1/u = 1/f and m = v/u. For a real image n times the size of the object, v = −nu (opposite sides), so the object distance is f(1 + 1/n) and the image distance f(n + 1). The object and its real image are closest, 4f apart, when both sit at 2f. An air bubble in water is a double-convex lens of the rarer medium inside the denser, so it diverges. Chromatic aberration — colours focusing at different points — comes from dispersion in the lens. At a single spherical surface, μ₂/v − μ₁/u = (μ₂ − μ₁)/R.

Definition

$\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$ , $m = \dfrac{v}{u}$ , power $P = \dfrac{1}{f}$ (m).
Real image n times larger: $|u| = f\left(1 + \dfrac{1}{n}\right)$ , $v = f(n + 1)$ (twice, f = 1/3 m ⇒ u = 0.5 m).
Least object–image distance for a real image: 4f.
Air bubble in water: diverging. Colours not meeting: chromatic aberration.
One surface: $\dfrac{\mu_2}{v} - \dfrac{\mu_1}{u} = \dfrac{\mu_2 - \mu_1}{R}$ .

Lens formula

\frac{1}{v} - \frac{1}{u} = \frac{1}{f}, \qquad m = \frac{v}{u}

Worked example

An object is 30 cm from a convex lens of focal length 20 cm. Image distance and magnification?

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 3rd May Shift 2 · Q40Hard

Example 1 · Optics (Ray) · Lenses — Lens Formula, Power, and Lensmaker

A convex lens of focal length '

f

' produces a real image whose size is '

n

' times the size of an object. The distance of the object from the lens is

Taking v = nu for a real image in a lens

A lens's real image is on the OTHER side, so v and u have opposite signs: v = −nu. Using v = nu gives (n − 1) where the answer has (n + 1).

Drill 8 more on the lens formula and magnification

Concept 2 of 2: Lensmaker's Equation and Combining Lenses

1/f = (μ − 1)(1/R₁ − 1/R₂): a biconvex lens with equal radii R has 1/f = 2(μ − 1)/R. Grinding one face flat, or cutting the lens along its axis into two plano-convex halves, removes one term, so the focal length doubles and the power halves. In a liquid the lens works with the relative index μ/μ_l: a 1.5 lens in a 1.25 liquid keeps only 0.2/0.5 of its power, and in a liquid denser than the glass a converging lens diverges. Lenses in contact add powers; separated by d, P = P₁ + P₂ − dP₁P₂, which with the contact value fixes both powers. A plano-convex lens of index n₁ fitted into a plano-concave one of n₂ with the same R has 1/f = (n₁ − n₂)/R.

Definition

$\dfrac{1}{f} = (\mu - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ .
One face made plane, or cut along the axis: $f \to 2f$ , $P \to \dfrac{P}{2}$ .
In a liquid: $\dfrac{P'}{P} = \dfrac{\mu/\mu_l - 1}{\mu - 1}$ (1.5 in 1.25 ⇒ 2 : 5; 1.5 in 2 ⇒ $-\tfrac{1}{2}$ ).
Contact: $P = P_1 + P_2$ ; apart by d: $P = P_1 + P_2 - dP_1P_2$ (+10 D, +6 D at 0.25 m ⇒ 8 D and 2 D).
Fitted pair: $f = \dfrac{R}{n_1 - n_2}$ .

Lensmaker and combinations

\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right), \qquad P = P_1 + P_2 - dP_1P_2

Worked example

A biconvex lens (μ = 1.5) has both radii 30 cm. Its focal length, and its focal length in water (μ = 4/3)?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 14th May Shift 1 · Q1Moderate

Example 2 · Optics (Ray) · Lenses — Lens Formula, Power, and Lensmaker

A combination of two thin lenses in contact have power +10 D. The power reduces to +6 D when the lenses are 0.25 m apart. The power of individual lens is

Adding focal lengths instead of powers

Lenses in contact add POWERS, 1/f. A 40 cm convex with a 25 cm concave gives 2.5 − 4 = −1.5 D, not 15 cm.

Using centimetres in the power

Power in dioptres is 1/f with f in METRES: 100/f with f in cm.

Drill 14 more on lensmaker's equation and combining lenses

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 (2)

The Lens Formula and Magnification
Lens formula
$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}, \qquad m = \frac{v}{u}$
Lensmaker's Equation and Combining Lenses
Lensmaker and combinations
$\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right), \qquad P = P_1 + P_2 - dP_1P_2$

Watch out for (3)

Taking v = nu for a real image in a lens
→ The Lens Formula and Magnification
Adding focal lengths instead of powers
→ Lensmaker's Equation and Combining Lenses
Using centimetres in the power
→ Lensmaker's Equation and Combining Lenses

Test yourself on Optics (Ray)

20 past MHT-CET questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Optics (Ray) chapter test Drill all 24 questions