Thin Lens Formula

Lenses are curved, transparent objects that refract light. Convex lenses (bulging outward, converging light rays) and concave lenses (bulging inward, diverging light rays) are two types of lenses (depressed inwards, diverging light rays).

Cameras, binoculars and telescopes all employ lenses. Magnifying glasses must be known to you. A single convex lens with a frame makes up a simple magnifying glass. Lenses such as spectacles and contact lenses are used to correct vision problems such as hypermetropia (long-sightedness) and myopia (nearsightedness) (short-sightedness).

Lens formula, often known as the Thin Lens Formula, is a mathematical formula that links focal length, object distance and image distance. The lens formula essentially allows you to determine the focal length of various lenses by plugging in known object and picture distances into the calculation. That is why the formula is also known as the focal length formula.

Lens

The power of a lens is one of the most exciting concepts in ray optics. Simply put, in ray optics, a lens’s ability to bend light is its strength. The power of a lens is proportionate to its capacity to refract light that travels through it. A convex lens’ converging ability is determined by its strength.

Are you aware that focal length and light ray bending are linked? As the focal length decreases, the number of light bends rises. As a result, a lens’ focal length and intensity are inversely related. A short focal range helps to boost optical strength. So now, let’s start with some fundamental principles about lens power.

Lens Formula 

Definition: The Lens Formula is an equation that shows the relationship between focal length, object distance and image distance.

• The lens formula is for lenses with very thin thickness.
• Both convex and concave lenses are affected by the Lens Formula.
• The Lens-Maker’s Formula is used to create the Lens Formula.
• The focal length of the lens is determined using the lens formula.

Mathematically, 

The reciprocal of a lens’s focal length is equal to the difference of its image and object distance reciprocals.

It is given by,

1f=1q–1p   

(thin lens formula)

where,

p = object distance
q = image distance
f  = focal length of the lens

Consider a thin lens with optical center O, having focal length f. An object of height AB (h) is placed beyond 2F of the lens. Suppose two light rays pass through the object to form a smaller image of height A’B’ (h’) between F and 2F at the other side of the lens.

Comparing similar right-angled triangles, ABO ~ A’B’O

A’B’  =   OB’   (1)
AB        OB

Also, △A’B’F and △OCF are similar; therefore,

A’B’  =   FB’
OC OF

But, OC=AB.

Hence,

A’B’  =   FB’  (2)
AB OF

Comparing equations (1) and (2), we get:

OB’  =   FB’
OB OF

=> OB’  =   OB’- OF
    OB         OF

Substituting the sign convention, we get:

OB = u, OB’= -v and OF = f

Therefore,

–  v    =    v-f
  u            f

u(v – f) =- vf

uv – uf = -vf

Dividing both sides by uvf,

uv   –    uf   =   – vf
uvf     uvf        uvf

1   –     1     =    – 1
f          v             u

or,

1f=1v–1u

As u = object distance (p) and v = image distance (q)

So,

1f=1q–1p   

(thin lens formula)

Hence, derived.

Derivation Of Lens Formula (Method 2)

Consider a light ray is incident upon a curved surface to be refracted from a medium of refractive index 1 to a medium of refractive index  2 .

Then, the equation is given as,

 μ2/v – μ1/u = (μ2-μ1)/R

When light ray enters the lens, object distance = -OB;

Image distance = BI1;

Radius = R1.

Hence, 

μ2/BI1 – μ1/-OB = (μ2-μ1)/R1

⇒ μ2/BI1 + μ1/OB = (μ2-μ1)/R1 —(1)

When a light ray exits the lens, object distance = DI1;

Image distance = DI;

Radius = -R2.

Hence, 

μ1/DI – μ2/DI1 = -(μ1-μ2)/R2

⇒ μ1/DI – μ2/DI1 = (μ2-μ1)/R2 —(2)

As we can see clearly, point B is extremely close to point D for the thin lens. Thus, DI1 = BI1

⇒ μ1/DI – μ2/BI1 = -(μ1-μ2)/R2 —(3)

(1) + (3) ⇒ μ1(1/OB + 1/DI) = (μ2-μ1)(1/R1 – 1/R2)

But, OB = -u, DI = v

Hence, 

μ1(-1/u + 1/v) = (μ2-μ1)(1/R1 – 1/R2) —(4)

When u=∞, v=f

⇒ μ1(0 + 1/v) = (μ2-μ1)(1/R1 – 1/R2)

⇒ μ1/f = (μ2-μ1)(1/R1 – 1/R2) —(5)

Comparing equation (4) and (5), we get

1/f = 1/v – 1/u. 

This is the lens formula. Hence, derived. 

(All distances opposite to the direction of the incident ray are taken negative.We can determine the power of a lens and its magnification using lens formula.)

Determining Power of a Lens by Using Lens Formula

Definition: 

Power of a lens is defined as,

the reciprocal of focal length in meters.

Mathematically,

p=1f

In terms of thin lens formula, we can write as,

p=1v–1u

Unit: The unit of power of the lens is m-1 or diopters.

Determining Magnification of the Lens by Using Lens Formula

The ratio of the height of the image created to the height of the object is the magnification of a lens. We can deduce from the ray diagram above that,

Magnification  =  M  = A’B’
AB

But, A’B’   =   OB’
        AB         OB

So,   M  =  OB’
      OB

    M   =   v/u

or,     M  =   q/p.

Hence, another definition of magnification is the ratio between image distance and the object distance. The magnification has no unit.

Conclusion

Glasses, contact lenses, telescopes, binoculars, magnifying glasses, and pretty much any optical equipment employ lenses. It emphasises the significance of lenses and the importance of determining the focal length of various lenses for varied purposes. Optical experts devised a lens formula, which was primarily used to determine the focal length of lenses. We can deduce from the preceding explanation that:The relationship between the focal length, object distance, and image distance of the lens is shown in the lens formula. If we know the other two variables of the lens, it will be easier to figure out the unknown variable. It is only suitable for ultra-thin lenses. The good news is that all we see in our daily lives is through thin glasses. The lens formula also aids in determining the lens’ magnification and power. The thin lens formula is written as 1/f = 1/v – 1/u. The formula of a lens’s power is P = 1/ f and magnification is given by M = v/u.