Derive mirror formula.
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||u||v||f||R||h||h of the image|
|Convex mirror||-||No real image||+||+||+||+||No real image||+|
DerivationBack to Top
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