Lens

aLensLens is a piece of a refracting material bounded by at least a spherical surface. There are two types of lens i) Converging lensii) Diversing lens

Magnification (Linear)The linear magnification of a lens is defined as the ration of the soze of image produced by it to the size of the object . magnification (m) = size of image (I)

size of object (o) = image distance (V)

object distgance (u) Angular Magnification Visual angle: The angle subtended by the object at the eye is known as visual angle. The size of image formed by eye depends upon the visual angle. Greater the visual angle, greater is the image formed.

Angular MagnificationAngular Magnification is defined as the ratio of the angle subtended at the eye by the image formed by an optical instrument to that subtended at the eye by the object when not viewed through the instrument.angular magnification = visual angle with instrument (𝛽)

visual angle without lens (𝛼) Lens FormulaThe relation of focal length of a lens with the object distance and image distance is known as lens formula.1

f=1

u+1

vLens formula for a convex lens forming real image

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.

	∴AB A’B’	=CA CA’…(i)
	Similarly 𝛥‘sCDF and A’B’F are similar,
	∴CD A’B’	=CF FA’
	But CD=AB
	∴AB A’B’	=CF FA’…(ii)
	From equations (i) and (ii),we have
	CA CA’	=CF FA’
	or, CA CA’	=CF CA’-CF
	or, u v	=f v-f
	where, CA= u is object distance,
	CA'= v is image distance
	CF= f is focal length
	or, uv–uf	=vf
	or, uv	=uf+fv
	Dividing both sides by uvf
	uv uvf	=uf uvf+vf uvf
	1 f	=1 u+1 v which is lens formula.

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

	∴AB A’B’	=CA CA’…(i)
	Similarly 𝛥‘sCDF and A’B’F are similar,
	∴CD A’B’	=CF A’F
	But CD=AB
	∴AB A’B’	=CF A’F…(ii)
	From equations (i) and (ii),we have
	CA CA’	=CF A’F
	or, CA CA’	=CF CA’+CF
	or, u -v	=f -v+f
	where, CA= u is object distance,
	CA’=- v is image distance
	CF= f is focal length
	or, -uv+uf	=-vf
	or, uv	=uf+vf
	Dividing both sides by uv
	1	=f v+f u
	1 f	=1 u+ 1 v which is lens formula.

Lens formula in 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

	∴AB A’B’	=CA CA’…(i)
	Similarly 𝛥‘sCDF and A’B’F are similar,
	∴CD A’B’	=CF A’F
	But CD=AB
	∴ AB A’B’	=CF A’F…(ii)
	From equations (i) and (ii),we have
	CA CA’	=CF A’F
	or, CA CA’	=CF CF-CA’
	or, u -v	=-f -f+v
	where, CA= u is object distance,
	CA’=- v is image distance
	CF=- f is focal length
	or, uv-uf	=vf
	or, uv	=uf+vf
	Dividing both sides by uv
	1	=f v+f u
	1 f	=1 u+1 v which is lens formula.

Lens maker FormulaThe relation of focal length of a lens with the refractive index of the material it is made with and the radii of curvatures is known as lens maker formula.1

f=(𝜇-1) (1

R1 +1

R2)Proof:

Consider a thin convex lens of focal length f and refractive index µ. Suppose a ray OP parallel to the principle axis incident on the lens at a small height, h above it. After refraction, the ray will pass through the focus F as shown in the figure and deviates through an angled. The angle of deviation is given by

	tan𝛿	=h f
	Since𝛿 is small,tan𝛿=𝛿,and
	𝛿	=h f…(i)

The portion PQ of the lens is a small angle prism which is formed by two tangent planes to the lens surfaces at P and Q. Since the angle of deviation in small angle prism is independent of the angle of incident, equation (i) is equal to the angle of deviation in such prism,

	𝛿=A(𝜇-1)𝛿…(ii)
	where A is the small angle of the prism.
	From equation (ii)and (iii),we have
	h f	=A(𝜇-1 )
	or, 1 f	=A h(𝜇-1)…(iii)

Let C1 and C2 be the centre of curvature of the two spherical surfaces. In figure PC1 = R1 is normal at P and OC2 = R2 normal at Q where C1 and C2 are centre of curvature of lens surfaces. Let ϴ and Ф be the angles made by R1 and R2 with the principal axis. Since the angle between two tangents forming a prism is equal to the angle between two radii, so we have

	∠PKC=∠QKC1=A
	From the geometry, we have
	A	=𝜃+𝜙
	and for small angles,𝜃=h R2 and 𝜙=h R1,and then
	A	=h R2+h R1
	or, A h	=1 R1+1 R2…(iv)
	Substituting equation (iv)in equation (II), we get
	1 f	=A h(𝜇-1)
	or, 1 f	=(1 R1+1 R2)(𝜇–1)
	or, 1 f	=(𝜇–1)(1 R1+1 R2)…(v)

This is lens maker’s formula. Here µ is refractive index of lens material to the medium outside.Combination of thin lenses in contactLet us consider two lenses A and B of focal length f1 and f2 placed in contact with each other. An object is placed at O beyond the focus of the first lens A on the common principal axis. The lens A produces an image at I1 . This image I1 acts as the object for the second lens B. The final image is produced at I as shown in Fig. Since the lenses are thin, a common optical centre P is chosen.

Let PO = u, object distance for the first lens (A), PI = v, final image distance and PI1 = v1, image distance for the first lens (A) and also object distance for second lens (B). For the image I1 produced by the first lens A, 1

f1=1

u+1

v1.............(i) For the final image I, produced by the second lens B, 1

f2=1

-v1+1

v............. (ii) Adding equations (i) and (ii), 1

f1+1

f2=1

u+1

v1-1

v1+1

v1

f1+1

f2=1

u+1

v.................(iii) If we consider the cobination of lens as a single lens, u is object distance and v is final image distance. If F is focal length of the combination, then. 1

F= 1

u+1

v..............(iv) From equations (iii) and (iv) 1

F= 1

f1+1

f2 This F is the focal length of the equivalent lens for the combination. The derivation can be extended for several thin lenses of focal lengths f1, f2, f3 ... in contact. The effective focal length of the combination is given by 1

F=1

f1+1

f2+1

f3+............. so, the power of the combination will be P = P1 + P2 + P3 + .... i.eThe power of a combination of lenses in contact is the algebraic sum of the powers of individual lenses.The combination of lenses is generally used in the design of objectives of microscopes, cameras, telescopes and other optical instruments.Power of lensPower of lens is a measure of its abilitu to converse of diverse the ray of light. It is defined as the reciporcal of the focal length.power(P) = 1

f Its unit is Diopter (D).For a combination of lenses combined power is given byP = 1

f1+1

f2+1

f3+.............=P1+P2+P3+..........