Maharashtra State Board · Class 10 · Physics · Chapter 7

Lenses — Formula Sheet

Board Formulas

10 formulas

1.Cartesian Sign Convention for Lenses
$u < 0 (real object), f_{convex} > 0, f_{concave} < 0$
Optical centre is the origin. Distances in the direction of incident light (left to right) are positive, against it negative. Heights above the principal axis are positive, below negative.
2.Lens Formula★
$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$
$u$ : Object distance from the optical centre (cm or m) · $v$ : Image distance from the optical centre (cm or m) · $f$ : Focal length (cm or m)
Valid for both convex and concave lenses with Cartesian signs. Note the MINUS sign — this is not the mirror formula.
3.Magnification★
$M = \frac{h _{2}}{h _{1}} = \frac{v}{u}$
$M$ : Magnification (no unit) · $h_{1}$ : Height of the object (cm) · $h_{2}$ : Height of the image (cm)
M negative ⇒ real and inverted image; M positive ⇒ virtual and erect. |M| > 1 magnified, |M| < 1 diminished. M has no unit.
4.Power of a Lens★
$P = \frac{1}{f ( in m )}$
$P$ : Power (dioptre, D) · $f$ : Focal length in metres
SI unit: dioptre (D); 1 D = 1 m⁻¹. Convex lens ⇒ positive power, concave lens ⇒ negative power. A shorter focal length means a more powerful lens.
5.Power with Focal Length in cm
$P = \frac{100}{f ( in cm )}$
Same formula, with the conversion built in. f = −40 cm ⇒ P = 100/(−40) = −2.5 D.
6.Lenses in Contact: Focal Length
$\frac{1}{f} = \frac{1}{f _{1}} + \frac{1}{f _{2}}$
$f_{1}, f_{2}$ : Focal lengths of the individual lenses · $f$ : Equivalent focal length of the combination
For thin lenses kept touching each other. Use the sign of each focal length.
7.Lenses in Contact: Power★
$P = P_{1} + P_{2} + P_{3} + \dots$
Powers add (with signs). Focal lengths do NOT add. +5 D and −2 D in contact give +3 D.
8.Near Point and Far Point of a Normal Eye
$near point = 25 cm, far point = \infty$
25 cm is the least distance of distinct vision. The eye adjusts the focal length of its lens (accommodation) using the ciliary muscles to see objects anywhere in this range.
9.Correction of Myopia (Near-sightedness)
$f = - x (x = far point distance)$
$x$ : Distance of the defective eye's far point
A myopic eye sees near objects but not distant ones. A concave lens forms a virtual image of a distant object at the far point x. From the lens formula with u = −∞, v = −x (spectacle lens taken to be at the eye; the textbook treats this qualitatively).
10.Correction of Hypermetropia (Far-sightedness)
$\frac{1}{f} = \frac{1}{- N} - \frac{1}{- 25} \Rightarrow f = \frac{25 N}{N - 25} (cm)$
$N$ : Near point of the defective eye (cm), N > 25 cm
A hypermetropic eye cannot see near objects clearly; its near point N is beyond 25 cm. A convex lens forms a virtual image, at N, of an object kept at 25 cm. Lens formula with u = −25 cm, v = −N (spectacle lens taken to be at the eye).