Magnification Formula
Calculate optical magnification for lenses and mirrors.
Find image size, object distance, and magnification ratio for microscopes, cameras, and telescopes.
The Formula
M = -dᵢ / dₒ = hᵢ / hₒ
Magnification tells you how much larger or smaller the image is compared to the object. A negative magnification means the image is inverted (flipped upside down).
Variables
| Symbol | Meaning |
|---|---|
| M | Magnification (unitless ratio) |
| dᵢ | Image distance from the lens or mirror |
| dₒ | Object distance from the lens or mirror |
| hᵢ | Height of the image |
| hₒ | Height of the object |
Example 1
Object is 20 cm from a lens, image forms at 40 cm. Find the magnification.
M = -dᵢ / dₒ = -40 / 20
M = -2 (image is twice as large and inverted)
Example 2
A 5 cm tall object produces a 2 cm tall image. Find the magnification.
M = hᵢ / hₒ = 2 / 5
M = 0.4 (image is 40% of the original size, upright)
When to Use It
Use the magnification formula when:
- Determining how much a microscope or telescope enlarges an image
- Calculating image size in camera and projector systems
- Finding whether an image will be upright or inverted
- Designing optical instruments with specific zoom levels
Key Notes
- Negative M means the image is inverted (real image, formed through the lens); positive M means the image is upright (virtual image, on the same side as the object)
- |M| > 1 means enlarged; |M| < 1 means the image is smaller than the object (reduced or "minified")
- In a compound microscope, total magnification = objective magnification × eyepiece magnification; a 40× objective with a 10× eyepiece gives 400× overall — each stage multiplies, not adds
- This formula applies to thin lenses and mirrors; real optical systems use principal planes and equivalent focal lengths for accurate magnification calculation
Key Notes
- Linear magnification: m = h_i / h_o = −d_i / d_o: h_i and h_o are image and object heights; d_i and d_o are image and object distances. Negative m: image is inverted. |m| > 1: magnified. |m| < 1: reduced. m = −1: same size, inverted (e.g., object at 2f for a converging lens).
- Angular magnification: M = α_instrument / α_naked eye: Used for optical instruments. For a simple magnifier held at the near point (25 cm): M = 25/f + 1 (in cm). For a telescope: M = −f_objective / f_eyepiece. Negative sign indicates image inversion.
- Microscope total magnification: M_total = M_objective × M_eyepiece: The objective forms a real, magnified intermediate image; the eyepiece re-magnifies it. A 40× objective with a 10× eyepiece gives 400× total. The objective numerical aperture (NA) limits resolution — magnifying beyond resolution is "empty magnification."
- Resolution limit — not magnification: The Rayleigh criterion sets the minimum resolvable feature: d = 0.61λ/NA. For a light microscope with NA = 1.4 and λ = 500 nm: d ≈ 220 nm. No amount of magnification can reveal finer detail than this diffraction limit.
- Applications: Magnification calculations are used in microscope and telescope design, camera focal length selection (field of view and subject size), projector throw distance calculation, endoscope design, and calibrating measurement eyepieces for scientific and industrial microscopy.