The object distance (u), image distance (v) and focal length (f).
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.
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