Formula & Calculator
Magnification (Lens or Mirror)
Calculates the linear magnification of an image formed by a lens or mirror, including whether the image is upright or inverted.
Interpretation
m = −d_i/d_o. Ratio of image height to object height. Negative sign indicates inverted image. Used in imaging systems.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| m | Magnification | |
| d_i | Image distance | cm |
| d_o | Object distance | cm |
What it means
Magnification is the ratio of the image size to the object size. For lenses and mirrors, it is the negative of the ratio of image distance to object distance (using sign conventions). A positive m indicates an upright image; negative indicates inverted. This formula is used in microscopy, telescopy, photography, and any imaging system to predict image size. Understanding magnification helps in designing optical instruments and in interpreting images.
Worked example
Magnification – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| di (cm) | 30 |
| do (cm) | 15 |
| Parameter | Value |
|---|---|
| di | −20 |
| do | 20 |
Common mistakes
- Magnification (lens or mirror): m = −d_i / d_o – the negative sign indicates image orientation.
- Sign: |m| > 1 means magnified; |m| < 1 means reduced. A negative m means inverted image; positive m means upright.
- d_i and d_o: Use signed distances according to the sign convention.
- Units: d_i and d_o in the same units – m is dimensionless.
- Lateral vs. angular magnification: This is lateral (transverse) magnification – different from angular magnification (e.g., magnifying glass).
Applications
Magnification (lens or mirror), m = −d_i/d_o, gives the lateral magnification, where d_i and d_o are image and object distances. The negative sign indicates inverted images. Engineers and opticians use this to calculate the size of images, to design optical systems for desired magnification, and to specify lenses for cameras and microscopes. By understanding magnification, they can ensure that the final image meets resolution and size requirements. This formula is also used in machine vision to calibrate systems. It is a key parameter in both geometrical and optical design, enabling the prediction of how objects will appear after passing through a lens or reflecting off a mirror.
- Design of magnifiers, microscopes, and telescopes
- Camera lens selection and system calibration
- Optical projection systems (slides, cinema)
- Image size calculations in machine vision
- Educational understanding of image scaling and inversion
Frequently Asked Questions
It calculates the linear magnification of an image formed by a lens or mirror: m = −dᵢ/dₒ. It gives the image size relative to the object and its orientation.
m = linear magnification (dimensionless).
dᵢ = image distance (with sign).
dₒ = object distance (positive for real objects).
A negative m means the image is inverted relative to the object. A positive m means the image is upright.
It comes from the geometry of similar triangles in the ray diagram: height ratio equals distance ratio, with a sign to indicate orientation.
Linear magnification (m) relates to image size; angular magnification relates to the apparent size as seen by the eye (used in magnifiers and telescopes).
An object is 20 cm from a lens, and the image is formed at 30 cm on the other side. dₒ = 20, dᵢ = 30 (positive). m = −30/20 = −1.5. The image is 1.5 times larger and inverted.
- Forgetting the negative sign, which determines orientation.
- Using distances without proper sign conventions.
- Confusing magnification with power.
As the object moves closer to the focal point, magnification increases in magnitude, becoming infinite when the object is at the focal plane.
For a given lens, m is proportional to dᵢ; if the image is farther away, the magnification is larger.
It determines the size of the subject on the sensor, affecting composition and depth of field.