derivation of mirror formula as well as lens formula ?

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Mirror Formula (Concave Mirror)

 

Mirror formula is the relationship between object distance (u), image distance (v) and focal length.

cancave mirror formula

Derivation

The figure shows an object AB at a distance u from the pole of a concave mirror. The image A1B1 is formed at a distance v from the mirror. The position of the image is obtained by drawing a ray diagram.

Consider the D A1CB1 and D ACB

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

But ED = AB

From equations (1) and (2)

If D is very close to P then EF = PF

But PC = R, PB = u, PB1 = v, PF = f

By sign convention

PC = -R, PB = -u, PF = -f and PB1 = -v

 Equation (3) can be written as

Dividing equation (4) throughout by uvf we get

Equation (5) gives the mirror formula

Derivation of 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.

 

lens formula

 

OF1 = Focal length = f

OA = Object distance = u

OA1 = Image distance = v

But from the ray diagram we see that OC = AB

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

Dividing throughout by uvf

Derivation of Lens Formula (Convex Lens)

 
convex lens formula

 

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

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

HOPE THIS HELPS U.........

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