Derivation of Lens Formula

View Notes

What is Lens?

A transmissive optical device that focuses on or disperses a light beam using refraction is known as a lens. Lenses are created from components such as plastic or glass and are ground and moulded or polished as per the desired pattern. Unlike a prism, a lens can focus light to form an image. Similarly, devices that disperse or focus radiation and waves other than visible light are also known as lenses, such as electron lenses, microwave lenses, acoustic lenses, or explosive lenses.

What is the Lens Equation?

The relationship among the image distance (v), object distance (u), and the focal length (f) of a lens is given by the Lens formula. Lens formula is relevant for convex as well as concave lenses. These lenses have negligible density. The Lens formula is given below.

1/v  -  1/u  =   1/f

Here, "v" is the distance of the image from the optical center of the lens, "u" is the distance of the object from the optical center of the lens and "f" is the focal length of the lens.

Derivation of Lens Formula

The lens formula establishes the relation between the object, and image distances from the optical center, and the focal length of a lens.

Concave Lens: A concave lens diverges the parallel beam of light at the focal point and bends the light rays outward. It is also known as a diverging lens. A concave lens is thinner in the center and thicker at the edges and is used to correct myopia or short-sightedness.

• Derivation of Lens maker Formula for Concave Lens

Let us consider an object OO’ located in front of the focal length of the concave lens " f " on the principal axis of the lens. The concave lens creates an erect and virtual image II’ at a range of "q" from the optical center of the lens, as shown in the below diagram.

Consider  ⃤  OÓP and  ⃤  IÍP similar triangle

$\frac{OO'}{II'}$ = $\frac{O'P}{I'P}$

$\frac{OO'}{II'}$ = $\frac{p}{q}$ ⟶ 1

⃤  EFP and  ⃤  IÍF

$\frac{EP}{II'}$ = $\frac{PF}{I’F}$ ∵  EP = OO’

$\frac{OO'}{II'}$ = $\frac{PF}{I’F - PI’}$  =  I’F = PF - PI’

$\frac{OO'}{II'}$ = $\frac{f}{f-q}$ ⟶ 2

Comparing equation 1 and 2,

$\frac{p}{q}$ = $\frac{f}{f-q}$

p (f - q) = fp

pf - pq = fp

Dividing both sides by ‘pqf’,

$\frac{pf}{pqf}$ - $\frac{pq}{pqf}$ = $\frac{fp}{pqf}$

1/q - 1/f = 1/p

-1/q + 1/f = 1/p

For a concave lens:

f = Negative

q = Negative

∴ f = -f  and q = -q

Thus, $\frac{1}{f}$ = $\frac{1}{p}$ + $\frac{1}{q}$

• Derivation of Lens Formula for Convex Lens

Convex Lens: A convex lens converges a parallel beam of light towards the principal axis. It is thicker at the center and is thinner at the edges. A convex lens is used to correct Hypermetropia or long-sightedness. It is also used in cameras as it produces a clear and crisp picture and focuses light.

Let us consider a convex lens with O as an optical center as shown in the diagram.

F = Principle focus

f = Focal length

AB = Perpendicular to the principal axis at a distance beyond the focal length of the lens.

A real, inverted magnified image A’B’ is formed, as shown in the diagram.

In the given diagram △ABO = △A’B’O

Therefore,

A’B’/AB = OB’/OB ………….1

Similarly, △A’B’F = △OCF , hence

A’B/OC = FB’/OF

But OC = AB

A’B/AB = FB’OF ………….2

Equating eq 1 and 2,

OB’/OB = FB’/OF = $\frac{OB'- OF}{OF}$

Substituting the convention sign, we get:

OB =  -u

OB’ = v

OF = f

$\frac{v}{-u}$ = $\frac{v-f}{f}$

Vf = -uv + uv or uf = uf - vf

Dividing uvf by both the sides, we get:

uv/uvf = uf/uvf - vf/uvf

⇒ $\frac{1}{f}$ = $\frac{1}{v}$ - $\frac{1}{u}$

∴ This is the lens equation formula.