Combination of LensesJEE Main

Concept

Lenses in contact simply add their powers: $P = P_1 + P_2$. Separate them by d and a cross-term appears — the second lens receives converging or diverging light depending on where the first lens's image lands. Set $d = f_1 + f_2$ and the pair becomes afocal: a telescope.

Key formula

Derivation

Send in a parallel ray: lens 1 directs it toward its focus at $f_1$. That point is a virtual object at $f_1 - d$ for lens 2; apply the thin-lens equation and define F by where the ray finally crosses the axis.

In contact ($d=0$) the cross-term dies: powers just add. This is why opticians stack trial lenses.

Scenarios to explore

• Combination of Lenses — Two thin lenses: P = P₁ + P₂ − dP₁P₂, telescopes at d = f₁+f₂.

Real-world applications

• Telescopes (d = f₁ + f₂) and microscopes (small f, larger d).
• Camera zoom lenses move elements to vary d, hence F.
• Achromatic doublets cancel dispersion: ω₁/f₁ + ω₂/f₂ = 0.

JEE exam tips

• Convert to powers immediately — addition beats reciprocal juggling.
• d in the power formula must be in METRES when P is in dioptres.
• Equal & opposite lenses in contact (f, −f): P = 0, plane glass.

Common mistakes

• Adding focal lengths instead of powers.
• Forgetting the −dP₁P₂ term for separated lenses.
• Sign slips passing the intermediate image as the object for lens 2 (u₂ = v₁ − d).

Exam traps to avoid

• The equivalent lens does NOT sit at either lens — its principal planes shift.
• Total magnification is the PRODUCT m₁·m₂, never the sum.