Lens Formula Numericals
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Direct answer
Two distances and one focal length decide every thin-lens numerical: with NCERT's Cartesian convention (distances measured from the optical centre, positive along the direction of incident light), the lens equation is 1/v − 1/u = 1/f, magnification m = v/u = h′/h, and power P = 1/f in dioptres when f is in metres. Converging lenses carry positive f and diverging lenses negative; a real image has v positive, a virtual image v negative, and the sign of m tells erect (+) from inverted (−). NEET numericals almost always chain the formula with m = v/u, so solve for v first and read m off it. For thin lenses in contact, 1/f = 1/f1 + 1/f2 and P = P1 + P2, with signs carried faithfully.
What you must remember
- Master equations: 1/v − 1/u = 1/f and m = v/u; lenses in contact: 1/f = 1/f1 + 1/f2, P = P1 + P2 (dioptres).
- Sign discipline: u negative for a real object; f positive for convex, negative for concave; v positive for real images (opposite side), negative for virtual (same side as object).
- Magnification reading: m negative means real and inverted, positive means virtual and erect; a single convex lens gives virtual images only when the object sits inside f.
- Anchor positions: object at 2f → image at 2f with m = −1; object at f → image at infinity; object between f and lens → magnified erect virtual image (the magnifier).
- Power convention: P = 1/f(m) in dioptres; a diverging lens of f = −25 cm has P = −4 D; +10 D and −4 D in contact act as +6 D.
- Simple magnifier formulas: m = D/f for the relaxed eye (image at infinity) and m = 1 + D/f for the image at the near point, D = 25 cm.
- Exam hygiene: fix one length unit before substituting; the options always include the answer spoiled by one sign error.
Working through a typical numerical
An object stands 20 cm from a convex lens of f = 15 cm. Substitute with signs: 1/v = 1/f + 1/u = 1/15 − 1/20 = 1/60, so v = +60 cm — a real image 60 cm behind the lens — and m = v/u = 60/(−20) = −3, inverted and thrice the object's height. Now a concave lens, f = −20 cm, with the object at 30 cm: 1/v = −1/20 − 1/30 = −1/12, giving v = −12 cm and m = +0.4, a virtual, erect, diminished image on the object's side. Finish with a combination: lenses of +10 D and −4 D pressed together give P = +6 D, hence f = 1/6 m ≈ 16.7 cm converging. Every step is the same ritual — signs in, v out, m from v.
How NEET frames it
The paper's favourite error is the dropped sign: plugging u = +20 in the first problem yields v = 12 cm, and 12 sits right there among the options as bait. Magnification questions exploit the same soft spot — "find the magnification" offers both −3 and 3, and only the sign separates an error from a mark, since NCERT counts the negative sign as the statement "inverted". Position traps ask for the image distance from the object (60 + 20 = 80 cm, not 60) or from the other side of a two-lens system. Finally, power questions borrow the human eye: a myope's corrective lens carries negative power, and NEET enjoys asking for the power of a combination in which one term is negative — the addition must be algebraic, never a sum of magnitudes.
Frequently asked questions
What is the lens formula and its sign convention?
1/v − 1/u = 1/f, with all distances measured from the optical centre: u negative for real objects, f positive for converging and negative for diverging lenses.
Where should an object be placed before a convex lens for a same-size image?
At 2f; the image forms at 2f on the far side, real and inverted, with magnification exactly −1.
What magnification does a concave lens always produce for a real object?
A virtual, erect and diminished image with m between 0 and 1 — for f = −20 cm and u = −30 cm, v = −12 cm and m = +0.4.
How do powers of thin lenses in contact combine?
Algebraically: P = P1 + P2, so +10 D with −4 D behaves as a single +6 D lens of focal length about 16.7 cm.
When does a convex lens form a virtual image?
Only when the object lies within the focal length; the image then sits on the same side, erect and magnified — the principle of the simple magnifier.