# Derive mirror formula.1/v+1/u=1/f

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

## Sign Convention for Spherical Mirrors

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.

## Table Showing the Sign Convention

 Type of mirror Concave mirror Convex mirror u v f R h h of the image Real Virtual Real Virtual - - + - - + - + - No real image + + + + No real image +

## 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 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, PB 1 = v, PF = f

By sign convention

PC = -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

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