derivations in optics

SIGN CONVENTIONSThe 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.