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.
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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.
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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.
Difference between Concave and Convex Lens
Uses of Lenses
A concave lens is a diverging lens whereas a convex lens is a converging lens. Depending upon the application required, the lenses are chosen for the instruments. However, some materials are made by using a combination of concave and convex lenses.
Lenses are used in a variety of applications such as telescopes, binocular cameras, and glasses. They are also used in correcting eye defects such as myopia or hypermetropia.
Concave Lenses are Used in the Following Ways:
These are used in spectacles to correct the vision defect of myopia
They are used in telescopes
With the combination of a convex lens, they are used to make high-quality lenses for optical instruments
They are also used for spy holes for doors
Convex Lenses are Used Following Ways:
These are used in a simple camera
They are used in the magnifying glass used by watchmakers
Used in making microscope, telescope, and projectors
They are used to correct the vision defect of hypermetropia
FAQs on Derivation of Lens Formula
1. What is a concave and convex lens?
The distribution of the lens depends upon how the rays of light bend when they travel through the lens. There are two types of lenses, a concave lens, and a convex lens. Convex lenses are thinner at the edges and thick in the center while a concave lens is thinner in the center and thicker at the edges. A concave lens is also identified as a diverging lens because it diverges the parallel light at the focal point and bends the light rays outward. The convex lens is known as a converging lens because it converges a parallel beam of light.
2. What is the difference between Lens and Mirrors?
The difference between lenses and mirrors is, mirrors have one focal point, whereas the lens has two focal points. In a lens, light rays are refracted while mirrors reflect light rays. A mirror is a glass surface with a silvery lining that presents an image through reflection, whereas a lens is a transparent plastic or glass material, with two surfaces joined together, of which one surface is curved. Mirrors are either plane or curved while the lens is always curved. There are six types of simple lenses and three types of mirrors.
3. What images are formed by convex lenses?
A convex lens is a lens that converges the rays of light coming towards it. the edges of the lens are curved outward rather than inwards. The nature of the image formed by a convex lens depends upon the position of the object. if the object is kept between F1 and 0, the nature of the image will be virtual and erect whereas in all the other conditions the image formed by a convex lens will be real and inverted
4. What images are formed by concave lenses?
A concave lens is a lens that diverges light of rays after refracting. That is it spreads out the light that has been refracted. A concave lens has edges thicker than the middle and they are curved inwards rather than outwards. irrespective of the location of the object, the image will always be formed virtual and erect. Virtual images are formed when the light rays diverge. And since concave lens always diverging light rays coming towards it, the images formed by the concave lens will be virtual and erect irrespective of the position of the object
5. What are lenses made up of?
Technology advancement creates the production of the lens fancier. Most of the lenses are made up of high-tech plastics and do not break as easily as glass. Lenses are formed by binding the two spherical surfaces and if these surfaces are bulging outwards they are known as convex lenses and if the spherical surfaces are curved inwards they are known as concave lenses. Depending upon the nature of converging or diverging light rays, lenses are used to make devices. However, some devices use not only one lens but a combination of concave and convex.
6. Why does a concave lens always form virtual images?
A concave lens is a diverging lens, that is all the rays get diverged after refracting through a concave lens. This lens is thicker at the edges and thinner at the centre, and hence it is curved inward rather than outward. Whenever the light rays diverge, virtual images are formed. The rays falling on the concave lens after refraction always diverge. Therefore no matter where the object is placed, the image formed by a concave lens will always be a virtual image and nature will be erect. The image hence form will be smaller in size as compared to the object.
