Similarity and optics

INTERDISCIPLINARY PROJECT

APPLICATION OF MATHEMATICS

IN

SCIENCE (OPTICS)ALFARHAN ZAHEDI

X-C (05)

ACKNOWLEDGEMENT

I would like to express my special thanks of gratitude to

my teacher Mrs. Tina and Mr. Sunit Mali as well as our principal Mrs. Swagata Banarjee who gave me the golden opportunity to do this wonderful project which

helped me in doing a lot of Research and I came to know about so many new things.I am really thankful to them.

Secondly I would also like to thank my parents who helped me a lot in finishing this project within the limited time. I

would also like to thank the creator of the websites- wikipedia.org, google.com and yahoo.com.

I am making this project not only for marks but to also increase my knowledge.

THANKS AGAIN TO ALL WHO HELPED ME!!!!

APPLICATOIN

OF

SIMILARITY IN OPTICS

Similarity i.e. Similar Triangles has a very special and close link with optics – a branch of science.

From the very beginning, the usage of Similarity is seen. Mirror formula has been derived by using the cocept of Similar Triangles. So has

been derived the formula for the magnification in curved mirrors (both concave and convex),

the lens formula and the formula for the magnification in lens (both concave and convex).

Similarity, thus, serves as a foundation stone of optics

MIRROR FORMULA

The distance of the position of the object on the principle axis from the pole of a

spherical mirror is known as the object distance. It is denoted by u.

The distance of the position of the image on the principle axis from the pole of a

spherical mirror is known as the image distance. It is denoted by v.

The relation between u, v and focal length (f) of a spherical mirror is known as mirror formula.

It is given by:

1/v+1/u = 1/fMAGNIFICATION

Linear magnification produced by a mirror is defined as the ratio of the height of the

image to the height of the object. It is denoted by m.

Linear magnification has no units

If h2 is the height of the image and h1 is the height of the object, then -

m = h2/h1

LENS FORMULADistance of an object from the optical centre of a lens is known as the abject distance.

It is denoted by u.

Distance of an image from the optical centre of a lens is known as the image distance.

It is denoted by v.

The relation between object distance (u), image distance (v) and focal length (f) of a lens is called lens formula.

It is given by:

1/v – 1/u = 1/fMAGNIFICATION

The ratio of the size of the image to the size of the object is known as the magnification produced by the lens.

Therefore, magnification (m) = Size of image (h2)/Size of object (h1) = h2/h1

m = h2/h1

SIMILAR TRIANGLES

Two triangles are similar if(i) Their corresponding angles are equal, and(ii) Their corresponding sides are proportional.

The symbol ‘~’ stands for the phrase “is similar to”

If ΔABC ~ ΔDEF, then by definition

∠∠A = ∠D, ∠B = ∠E,∠C = ∠F and FD/CA = EF/BC = DE/AB.

CRITERION FOR THE SIMILARITY OF TRIANGLES

AAA – CRITERION FOR SIMILARITYIf in two triangles, the corresponding angles are equal the triangles are similar. The triangles shown in the fig. are similar by the above criterion.

SSS – CRITERION FOR SIMILARITYIf the corresponding sides of two triangles are proportional the triangles are similar. The triangles shown in the fig.

are similar by the above criterion.

SAS - CRITERION FOR SIMILARITYIf one angle of a triangle is equal to one angle of the other triangle and the sides containing these angles are proportional, the triangles are similar. The triangles shown in the fig. are similar by the above criterion.

Similar Triangles can have shared partsTwo triangles can be similar, even if they share some elements. In the figure below, the larger triangle PQR is similar to the smaller one STR. S and T is the midpoints of PR and QR respectively. They share the vertex R and part of the sides PR and QR. They are similar on the basis of AAA, since the corresponding angles in each triangle are the same.

SIGN CONVENTIONS

The following sign convention is used for measuring various distances in the ray diagrams of spherical mirrors:

All distances are measured from the pole of the mirror.

Distances measured in the direction of the incident ray are positive and the distances measured in the direction opposite to that of the incident rays are negative.

Distances measured above the principal axis are positive and that measured below the principal axis are negative.

MIRROR FORMULA

(CONCAVE MIRROR)

Mirror formula is the relationship between object distance (u), image distance (v)

and focal length.

The mirror formula for a cincave mirror is 1/v+1/u = 1/f.

Derivation

The figure shows an object AB at a distance u from the pole of a concave mirror. The image A 1 B 1 is formed at a distance v from the mirror. The position of the image is obtained by drawing a ray diagram.

Consider the  A1CB1 and ACB

[When two angles of D A1CB1 and D ACB are equal then the third

angle 

(AAA – similarity criterion)

But ED = AB

From equations (1) and (2)

If D is very close to P then EF = PF

But PC = R, PB = u, PB 1 = v, PF = f

By sign conventionPC = -R, PB = -u, PF = -f and PB 1 = -v

 Equation (3) can be written as

Dividing equation (4) throughout by uvf we get

Equation (5) gives the mirror formula. MIRROR FORMULA

(CONVEX MIRROR)Let AB be an object placed on the

principal axis of a convex mirror of

focal length f. u is the distance

between the object and the mirror and

v is the distance between the image

and the mirror.

(AAA – similarity criterion)

But DE = AB and when the aperture is very small EF = PF.

 Equation (2) becomes

 From equations (1) and (3) we get

[PF = f, PB1 = v, PB = u, PC = 2f]

Dividing both sides of the equation (4) by uvf we get

The above equation gives the mirror formula. LENS FORMULA(CONVEX LENS)

Let AB represent an object placed at right angles to the principal axis at a distance

greater than the focal length f of the convex lens. The image A1B1 is formed beyond

2F2 and is real and inverted.

OA = Object distance = u

OA1 = Image distance = v

OF2 = Focal length = f

OAB and  OA1B1 are similar (AAA – similarity criterion)

But we know that OC = AB The above equation can be written as

From equation (1) and (2), we get

Dividing equation (3) throughout by uvf

The above equation is the lens formula.

LENS FORMULA(CONCAVE LENS)

Let AB represent an object placed at right angles to the principal axis at a distance

greater than the focal length f of the convex lens. The image A1B1 is formed

between O and F1 on the same side as the object is kept and the image is erect and

virtual.

OF1 = Focal length = f

OA = Object distance = u

OA1 = Image distance = v

(AAA – similarity criterion)

Similarily,

But from the ray diagram we see that OC = AB

From equation (1) and equation (2), we get

Dividing throughout by uvf

The above equation is the lens formula.

MAGNIFICATION IN MIRROR

Let AB be an object placed perpendicular to the principle axis in front of concave mirror. A ray AD parallel to the principle axis passes through the focus after reflection from the mirror. A ray AP making ∠i with the principle axis after reflection

makes an angle ∠i = ∠r with the principle axis. These two reflected rays intersect

each other at A1. So A1B1 is the real, inverted and magnified image of the object

Now,

Between APB andA1PB1, we have –

∠ABP = ∠A1B1P

∠APB = ∠A1PB1

’s APB and A1PB1 are similar (AA – similarity criterion)

AB/A1B1 = BP/B1P

Height of object (h1)/height of image (h2) =object distance (u)/image dist. (v).

Applying sign conventions we get –

h1/-h2 = -u/-v

Or, h1/h2 =-u/v

Or, m =-u/v

Since (h1/h2 = m).

This the formula for the magnification produced by a spherical mirror.

Note: The formula for the magnification produced by both convex and concave mirror is the same.

MAGNIFICATION IN LENS

In the above figure, AB is the size of the object and A’B’ is the size or height of the

image. Now,

Between’s AOB and A’OB’, we have-

∠AOB = ∠A’OB’ (vertically opposite angle),

∠BAO = ∠B’A’O (900 each)

’s AOB and A’OB’ are similar. (AA – similarity criterion)

A’B’/AB = A’O/AO (sides are proportional)

height of the image (h’)/ height of the object (h) = image dist.(v)/object dist.(u)

Applying sign conventions we get –

-h’/h = -v/u

Or, h’/h = v/u

Or, m = v/u

This the formula for the magnification produced by a lens.