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Lens Formula For Convex Lens (Real and Virtual)

Lens formula


Lens Formula

It is a relationship between the focal length of a lens and distances of object and image from the optical center of the lens. To derive this formula we use the following sign conventions. 

1. All distances are measured from the optical center of the lens. 

2. Real distances are taken as positive and virtual distances are taken as negative. 

3. For a convex lens, f is positive and for a concave lens, f is negative. 

4. For a real image, magnification M is positive and for a virtual image, magnification M is negative. 


Convex lens 

The image formed may be real or virtual. 

1. (i) Real image (extended object method) 

Let us consider a convex lens as shown in figure. Let O be the optical centre and F be the principal focus of a convex lens of focal length OF=f. 

Let AB be and extended object placed in between F and 2F of the lens as shown in figure. 

AB is held perpendicular to the principal axis of the lens. A real, inverted and magnified image A'B' is formed beyond 2F as shown in figure. 


As A'B'O and ABO are similar, 

A'B'/AB  = OB'/OB …………………… eq(n) (1)

Again A'B'F and ODF are similar, 

A'B'/OD  = FB'/OF …………………… eq(n) (2)

But OD = AB, 

A'B'/OD  = FB'/OF  …………………… eq(n) (3)

From equations (1) and (3), 

OB'/OB  = FB'/OF = (OB^'-OF)/OF  

Using sign conventions, 

OB =  u, real object distance 

0B' = v, real image distance 

OF = f, focal length of lens 

So,     v/u  = (v-f)/f

or, vf - uv = uf 

or, uv = uf+ vf 

Divide both sides by uvf,

uv/uvf  = uvf/uvf + vf/uvf  

or,  1/f  = 1/v + 1/u  is the required equation.


 (ii) Real image (point object method) 

Consider a thing convex lens as shown in Figure. Let C be the optical centre of lens. Thin lens can be supposed to be made up of large number of small angle prisms, and XYZ is one of such prisms. In small angle prism, deviation produced is independent of angle of incidence. 

Consider real point object O on the principal axis of the lens. A ray of light OX incident on the lens gets refracted and passes through point I as shown in Fig (i). Another ray of light OC passing through optical centre goes without bending. Two refracted rays intersect at I. Hence, I is the real image of O. If there were no lens, ray OX would travel along ON as shown in Fig.(i).Let M be a point at a height h above C as shown in Fig. (i) and (ii). 


In Fig.(i). 

 NMI = δ: angle of deviation. 

Let angle  MOC = α,  MIC = β. 

From Fig.(i) 

δ = α + β ……………………………………… Eq(n) 1 (Exterior angle) 

From Fig. tanα = MC/OC

Tanβ = MC/IC

For small angles α and β,

Tan α ≈ α, Tan β ≈ β,

So, α = MC/OC and   β = MC/IC

Substituting these values in equation (1), 

β = MC/OC + MC/IC

Where MC = h 

OC = u, real object distance 

IC = v, real image distance 

So, β = h/u + h/v ….........………Eq(n)2

Another ray of light PX parallel to the principal axis passed through focus after refraction as shown in Fig. (ii). If there were no lens, ray PX would travel along PQ.

In Fig. (ii) 

 QMF = δ

From Fig,  MFC =  QMF = δ  (Alternate angles) 

From Fig. (ii) 

Tanδ = MC/CF 

For small angle δ, 

Tanδ ≈ δ 

δ = MC/CF

Where, MC = h, CF = f, focal length of lens. 

δ = h/f

Substituting this value in equation (2),

 h/f = h/u + h/v

So, 1/f = 1/u + 1/v is the required equation.


Virtual image 

Consider an extended object AB placed infront of a convex lens. Let O be the optical centre of the lens. When the object AB is held close to the lens between O and F the image A'B' formed by convex lens is virtual, erect and magnified.


As A'B'O and ABO are similar, 

A'B'/AB  = OB'/CB …………………… eq(n) (1)

Again A'B'F and ODF are similar,

A'B'/OD  = FB'/OF …………………… eq(n) (2)

But OD = AB, 

A'B'/OD  = FB'/OF  …………………… eq(n) (3)

From equations (1) and (3), 

OB'/OB  = FB'/OF = (OB^'+OF)/OF  

Using sign conventions, 

OB =  u, real object distance 

OB' = -v, virtual image distance 

OF = f, focal length of lens 

So,     (-v)/u  = (-v+f)/f

or, -uv + uf = -vf 

or, uv = uf+ vf 

 Divide both sides by uvf,

uv/uvf  = uvf/uvf + vf/uvf  

or,  1/f  = 1/v + 1/u  is the required equation.