Lens Formula

Lens Formula

1. Convex lens, when real image is formed
   Consider a convex lens of focal length f. let AB be an object placed normally on the principle axis of the lens figure. The ray of light from the object AB after refracting through the convex lens meets at point B’. so A’B’ is real image of the object AB.
   
   
   Since\(\Delta\) ‘s ABC and A’B’C’ are similar to their corresponding sides are proportional. \begin{align*}\therefore \frac {AB}{A’B’} &= \frac {CA}{CA’} \dots (i) \\ \text {Similarly} \: \Delta ‘s \text {CDF and A’B’F are similar,} \\ \therefore \frac {CD}{A’B’} &= \frac {CF}{FA’} \\ \text {But CD} = \text {AB} \\ \therefore \frac {AB}{A’B’} &= \frac {CF}{FA’} \dots (ii) \\ \text {From equations} \: (i) \: \text {and} \: (ii), \text {we have} \\ \frac {CA}{CA’} &= \frac {CF}{FA’} \\ \text {or,} \: \frac {CA}{CA’} &= \frac {CF}{CA’ - CF} \\ \text {or,} \: \frac {u}{v} &= \frac {f}{v - f} \:\\ \text {where, CA} = \: \text {u is object distance}, \:\\ \text {CA'} = \: \text { v is image distance} \: \\ \text {CF} = \: \text {f is focal length} \\ \text {or,} \: uv – uf &= vf \\ \text {or,} \: uv &= uf + fv \\ \text {Dividing both sides by uvf} \\ \frac {uv}{uvf} &= \frac {uf}{uvf} + \frac {vf}{uvf} \\ \frac 1f &= \frac 1u + \frac 1v \: \text {which is lens formula.} \\ \end{align*}
   
   
2. Convex lens: when virtual image is formed.
   Consider a convex lens of focal length f. Let AB be an object on the principle axis between the optical centre (C) and focus (F) of the lens. The rays of light from the object after refraction through the lens appear to come from point B’. So A’B’ is the virtual, erect and magnified image of the object in figure.
   
   
   Since\(\Delta\) ‘s ABC and A’B’C’ are similar to their corresponding sides are proportional \begin{align*}\therefore \frac {AB}{A’B’} &= \frac {CA}{CA’} \dots (i) \\ \text {Similarly} \: \Delta ‘s \text {CDF and A’B’F are similar,} \\ \therefore \frac {CD}{A’B’} &= \frac {CF}{A’F} \\ \text {But CD} = \text {AB} \\ \therefore \frac {AB}{A’B’} &= \frac {CF}{A’F} \dots (ii) \\ \text {From equations} \: (i) \: \text {and}\: (ii), \text {we have} \\ \frac {CA}{CA’} &= \frac {CF}{A’F} \\ \text {or,} \: \frac {CA}{CA’} &= \frac {CF}{CA’ + CF} \\ \text {or,} \: \frac {u}{-v} &= \frac {f}{- v + f} \: \\ \text {where, CA} = \: \text {u is object distance}, \:\\ \text {CA’} = - \: \text { v is image distance} \: \\ \text {CF} = \: \text {f is focal length} \\ \text {or,} \: - uv + uf &= - vf \\ \text {or,} \: uv &= uf + vf \\ \text {Dividing both sides by uv} \\ 1&= \frac {f}{v} + \frac {f}{u} \\ \frac 1f &= \frac 1u + \frac 1v \: \text {which is lens formula.} \\ \end{align*}
   
   
3. Concave lens
   Consider a concave lens of focal length f. Let AB be an object placed normally on the principle axis of the lens. A’B’ is the virtual image of the object AB formed by concave lens.
   
   Since\(\Delta\) ‘s ABC and A’B’C’ are similar to their corresponding sides are proportional \begin{align*}\therefore \frac {AB}{A’B’} &= \frac {CA}{CA’} \dots (i) \\ \text {Similarly} \: \Delta ‘s \text {CDF and A’B’F are similar,} \\ \therefore \frac {CD}{A’B’} &= \frac {CF}{A’F} \\ \text {But CD} = \text {AB} \\ \therefore \frac {AB}{A’B’} &= \frac {CF}{A’F} \dots (ii) \\ \text {From equations} \: (i) \: \text {and}\: (ii), \text {we have} \\ \frac {CA}{CA’} &= \frac {CF}{A’F} \\ \text {or,} \: \frac {CA}{CA’} &= \frac {CF}{CF - CA’} \\ \text {or,} \: \frac {u}{-v} &= \frac {-f}{-f + v} \: \\ \text {where, CA} = \: \text {u is object distance}, \: \\ \text {CA’} = - \: \text { v is image distance} \: \\ \text {CF} = - \: \text {f is focal length} \\ \text {or,} \: uv - uf &= vf \\ \text {or,} \: uv &= uf + vf \\ \text {Dividing both sides by uv} \\ 1&= \frac {f}{v} + \frac {f}{u} \\ \frac 1f &= \frac 1u + \frac 1v \: \text {which is lens formula.} \\ \end{align*}
   
   
4. Lens formula from refraction of light through lens
   
   Suppose a convex lens of focal length f. A point object O, is placed on the principal axis at a distance to from the lens which is greater than its focal length. As shown in figure, a ray OA from O, is incident on the lens at a small height ‘h’ from the principal axis and after refraction at P and Q, it passes through a point I on the axis. Another ray OC, along the axis passes without deviation and two refracted rays QI and CI cross each other at I. So the point I, at distance v from the lens, is the real image of the object O.
   When the tangents at two points P and Q on the surfaces of the lens are drawn, these tangents planes from a small-angle prism as shown in the figure. Hence, the portion PQ of the lens may be considered as s small angle prism enclosed by tangent planes to the lens surfaces.
   
   Let α and β be the angles made by the rays OP and IQ on the principal axis respectively. Then angle of deviation, d is given by
   \begin{align*} d &= \alpha + \beta \\ \text {Sine angles are small, we have} \\ \alpha &= \frac hu , \: \beta = \frac hv \\ \text {So} \: d &= \frac hu + \frac hv \\ \end{align*}
   
   Consider a ray LP, parallel to the principal axis, incident on the lens at the same height h, as shown in the figure. After refraction, the ray passes through the focus F, at a distance f from the lens. The angle of deviation in this case is
   $$ d = \frac hf $$
   Since the angle of deviation, d is independent of the angles of incidence for small angle prism; the two angles of deviation must be equal. Therefore,
   \begin{align*} \frac hf &= \frac hu + \frac hv \\ \text {or,} \: \frac 1f &= \frac 1u + \frac 1v \\ \end{align*}
   This equation holds for both convex and concave lenses and for real and virtual images. The sign convention used in the derivation of formula is

1. All the distances are measured from the optical centre of the lens.
2. The real object and real image distances are taken as positive distances.
3. The virtual object and virtual image distances are taken as positive distances. For real focus, focal length is positive and for virtual focus, focal length is negative. That is, + f for concave lens and – f for convex lens.