Figure 9.16 – Refraction by a Double Convex Lens
NCERT Physics Class XII, Section 9.5.2 (p. 234). Cartesian sign convention: distances measured along the incident light are positive.
Variables and their physical meaning
- u
- Object distance of O from B (BO = -u, against the light). -40.0 cm adjustable
- n₁
- Refractive index of the medium around the lens. 1 adjustable
- n₂
- Refractive index of the lens material. 1.5 adjustable
- R₁
- Radius of curvature of first surface ABC (BC₁ = +R₁). 10 cm adjustable
- R₂
- Radius of curvature of second surface ADC (negative for a double convex lens). -10 cm adjustable
- BI₁
- Distance of image I₁ formed by the first surface alone. +60.0 cm dependent
- v
- Final image distance, DI = +v. +13.3 cm dependent
- n₂₁
- n₂/n₁, index of the lens relative to its surroundings. 1.5 dependent
- f
- Focal length of the lens. +10.0 cm dependent
- m
- Magnification, h′/h = v/u. -0.333 dependent
- P
- Power of the lens, P = 1/f (f in metres). +10.00 D dependent
Interactive ray diagram
Panels (a), (b) and (c) are shown together under “All”. Rays are paraxial and the sketch is schematic; the labelled values are exact.
Parts of the diagram (text reference)
- Lens
- Double convex lens – bends light at two spherical surfaces so rays from O meet at I. Controlled by n₁, n₂, R₁, R₂.
- Surface ABC
- First surface ABC – refracts light from n₁ into n₂ and makes image I₁ (Eq. 9.17). Controlled by u, n₁, n₂, R₁.
- Surface ADC
- Second surface ADC – refracts light from n₂ back to n₁ (Eq. 9.18); I₁ acts as its object. Controlled by BI₁, n₁, n₂, R₂.
- Object O
- Object point O – source of the rays, on the principal axis at distance u from B.
- Image I₁
- Image I₁ (first surface) – image of O made by ABC alone; used only to find the final image.
- Image I
- Final image I – where rays leaving the lens meet (real if v > 0), at distance v from D.
- Focus F
- Principal focus F – where parallel paraxial rays converge after the lens, at distance f.
- Centre C₁
- Centre of curvature C₁ – centre of surface ABC; the normal at N₁ passes through it (BC₁ = R₁).
- Centre C₂
- Centre of curvature C₂ – centre of surface ADC; the normal at N₂ passes through it.
- Outside medium
- Outside medium (n₁) – medium where the object and final image lie.
- Lens material
- Lens material (n₂) – glass of the lens; light bends towards the normal on entering when n₂ > n₁.
Mathematical equations and calculations
Eq. (9.17) – first surface
Original equation n₁ OB + n₂ BI₁ = n₂ - n₁ BC₁
Substitution 1 40.0 + 1.5 BI₁ = 0.5 10
Result BI₁ = +60.0 cm (OB = -u, BC₁ = +R₁)
Eq. (9.18) – second surface
Original equation n₁ DI - n₂ DI₁ = n₁ - n₂ DC₂
Substitution 1 DI - 1.5 +60.0 = -0.5 -10
Result DI = +13.3 cm (DI₁ = BI₁, DC₂ = R₂)
Eq. (9.19) – combining both surfaces
Original equation n₁ OB + n₁ DI = (n₂ - n₁)( 1 R₁ - 1 R₂ )
Substitution 1 40.0 + 1 +13.3 = 0.5 ( 1 10 - 1 -10 )
Result LHS = 0.1000 , RHS = 0.1000 cm⁻¹
Eq. (9.21) – lens maker's formula
Original equation 1 f = (n₂₁ - 1)( 1 R₁ - 1 R₂ ), n₂₁ = n₂ n₁
Substitution 1 f = ( 1.5 - 1)( 1 10 - 1 -10 )
Result n₂₁ = 1.5 , f = +10.0 cm
Eq. (9.23) – lens formula
Original equation 1 v - 1 u = 1 f
Substitution 1 +13.3 - 1 -40.0 = 1 +10.0
Result v = +13.3 cm ( real, inverted, diminished )
Eq. (9.24) – magnification
Original equation m = h′ h = v u
Substitution m = +13.3 -40.0
Result m = -0.333 ( real, inverted, diminished )
Eq. (9.25) – power
Original equation P = 1 f
Substitution P = 1 0.100 m
Result P = +10.00 D
Notes: Eq. (9.17) is used in magnitude form (OB = -u). Eq. (9.18) is written here with signed distances (DI₁ = BI₁, DC₂ = R₂) so that it combines exactly into Eqs. (9.19) and (9.23). Starting values are illustrative. Ranges describe a double convex lens (R₁ > 0, R₂ < 0), with the object kept beyond 1.2 f so the image stays real.