Question

What is the magnification formula?

StartFraction h Subscript I Baseline over h Subscript o Baseline EndFraction equals StartFraction d Subscript I Baseline over d Subscript o Baseline EndFraction.
The negative of StartFraction h Subscript I Baseline over h Subscript o Baseline EndFraction minus StartFraction d Subscript I Baseline over d Subscript o Baseline EndFraction. = 1
StartFraction h Subscript o Baseline over h Subscript i Baseline EndFraction equals the negative of StartFraction d Subscript I Baseline over d Subscript o Baseline EndFraction.
The negative of StartFraction h Subscript I Baseline over h Subscript o Baseline EndFraction equals StartFraction d Subscript o Baseline over d Subscript i Baseline EndFraction.

Answer 1

Magnification (lateral magnification) relates image size to object size and to their distances. Using common symbols:
- hi = image height
- ho = object height
- di = image distance (from lens/mirror)
- do = object distance

Standard formula:
m = hi / ho = − di / do

Equivalently:
hi/ho = − di/do
−hi/ho = di/do
ho/hi = − do/di

Notes:
- The leading minus sign comes from the usual sign convention: a negative m means the image is inverted relative to the object; a positive m means it is upright.
- The absolute value |m| gives the size ratio (e.g., |m| = 2 means the image is twice as large).
- For thin lenses or mirrors you use this together with the thin-lens/mirror equation 1/do + 1/di = 1/f to find di and then m.

Example:
ho = 4 cm, do = 20 cm, di = −10 cm (virtual, upright image) → m = −(−10)/20 = +0.5, so hi = m·ho = 2 cm (upright, half-size).