Mirror Formula
An expression showing the relation between object distance, image distance and focal length of a mirror is called mirror formula.
Assumptions and Sign conventions
To derive the mirror formula following assumptions and sign conventions are made;
- The aperture of the mirror is small
- Object should be placed on the principal axis in the form of point object
- All distances are measured from the pole of the mirror
- The distances of the real object and real images are taken as positive whereas that of virtual objects and images are taken negative.
- Focal length and radius of curvature of a concave mirror are positive whereas convex mirror negative.
Mirror formula for concave mirror when real image is formed
Let us take a concave mirror of aperture mirror of aperture XY where a light ray AC is travelling parallel to principle axis from object AB to mirror at C and reflect through focus F and pass through A'. Let another light ray pass directly through focus from A to D at mirror and reflect through A'.
Here A'B' is the image and B'P is image distance and, BP is object distance. Let us drop CN perpendicular to principle axis and DN' perpendicular to principal axis.From figure,
ΔABF ~ ΔFN'D
AB / DN' = BF/ FN'
AB / A'B' = (BN' - FN')/ FN' [.: DN' = A'B']
Since N' is very close to point p, FN' is similar to FP and BN' is similar to BP
AB / A'B' = BP – FP / FP ………. (i)
Again,
ΔA'B'F ~ ΔFCN
CN / A'B' = FN / B'F
AB / A'B' = FN / B'N – FN [CN = AB]
Since, N is very close to P, B'N is similar to B'P and FN is similar to FP
AB / A'B' = FP / B'P – FP …… (ii)
Equating equation (i) and (ii)
U – f/ f = f/ v-f
(U – f). (v-f) = f2
Uv – uf – vf + f2 = f2
Dividing both sides by uvf we get,
1/f = 1/u + 1/v
In the case of convex mirror
Let a convex mirror of a small aperture where light ray AD is striking in the mirror and is diverged appearing to pass through the principal focus F of the mirror. Let another ray from the top of the object AB pass normally from the center of curvature. Let us drop a perpendicular DN to principle axis, so thatΔDNF ~ ΔA'B'F
DN / A'B' = NF / B'F
Since N is very closed to C, NF is similar to PF,
= f / f – v
For virtual image,
= - f/ v – f ……. (i)
Similarly, In ΔABC & ΔA'B'N,
AB / A'B' = BC / B'N
= u / -v …. (ii)
Equating equation (i) and (ii),
u / -v = -f / v -f
on solving and dividing by uvf, we get,
1/f = 1/u + 1/v
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