Magnification
When you look at your reflection in a spoon, you notice something curious: your face appears smaller on the convex back and larger — even upside down — on the concave inside. This is magnification at work. In optics, magnification (symbol m) is a dimensionless ratio that tells you how the size of an image produced by a mirror compares to the size of the original object. A magnification of 2 means the image is twice as tall as the object; a magnification of 0.5 means it is half as tall. The sign of m tells you whether the image is upright or inverted. Magnification is not just an abstract number — it encodes two pieces of physical information simultaneously. Its magnitude (absolute value) tells you the size ratio, while its sign tells you orientation. By convention in the standard 'real-is-positive' sign convention, a negative magnification means the image is inverted (real image formed in front of a concave mirror), whereas a positive magnification means the image is upright (virtual image, appearing behind the mirror). Understanding magnification is essential for designing telescopes, car rear-view mirrors, dentists' mirrors, and camera systems. Magnification can be expressed in two equivalent ways: as a ratio of image height to object height, or as a ratio of image distance to object distance (with a sign change). Both expressions are derived directly from the geometry of ray diagrams using similar triangles, making magnification one of the most elegantly simple yet powerful results in geometrical optics.
- What 'magnification' actually measures
- Deriving magnification from geometry
- Combining the two expressions
- Interpreting the result
What 'magnification' actually measures
Magnification m is defined as the ratio of the height of the image (h_i) to the height of the object (h_o). Heights above the principal axis are positive; heights below are negative. So if an image forms below the axis (inverted), h_i is negative and m turns out negative. This is consistent: a real image formed by a concave mirror is always inverted, giving m < 0.
Deriving magnification from geometry
Consider a ray diagram for a concave mirror. The ray from the top of the object that travels parallel to the principal axis reflects through the focal point. The ray through the centre of curvature reflects straight back. Where these rays meet is the image tip. The triangles formed by the object, image, and mirror centre are similar triangles (same angles). From these triangles: h_i / h_o = -v / u, where v is the image distance and u is the object distance, both measured from the pole of the mirror. The negative sign ensures the sign convention is honoured: a real image (v positive, u positive in real-is-positive) gives negative m, indicating inversion.
Combining the two expressions
The two expressions for magnification — height ratio and distance ratio — are equivalent and can be used interchangeably. If you know the object and image distances (perhaps from the mirror formula 1/v + 1/u = 1/f), you can find m = -v/u immediately without needing to know the actual heights. Conversely, if you know m and the object height, you can find the image height as h_i = m × h_o. This flexibility makes magnification a powerful bridge between distances and sizes.
Interpreting the result
If |m| > 1, the image is enlarged (magnified). If |m| < 1, the image is diminished (minified). If |m| = 1, the image is the same size as the object — this happens when the object is placed at the centre of curvature of a concave mirror (u = 2f). A positive m means the image is upright (same orientation as object); a negative m means inverted. A virtual image in a concave mirror (object inside focal length) has m > 1 and positive, meaning it is magnified and upright — exactly how a shaving or make-up mirror works.
The Key Equation
- m — Magnification (dimensionless): Ratio of image size to object size. Positive = upright image; negative = inverted image. |m| > 1 means enlarged, |m| < 1 means diminished.
- h_i — Image height (metres (m)): Height of the image measured perpendicular to the principal axis. Negative if the image is inverted (below the axis).
- h_o — Object height (metres (m)): Height of the object measured perpendicular to the principal axis. Usually taken as positive.
- v — Image distance (metres (m)): Distance from the pole of the mirror to the image, measured along the principal axis. Positive for real images in front of the mirror.
- u — Object distance (metres (m)): Distance from the pole of the mirror to the object. Positive when the object is in front of the mirror (real object).
Worked Example
Problem. An object 4 cm tall is placed 30 cm in front of a concave mirror of focal length 10 cm. (a) Find the image distance using the mirror formula. (b) Calculate the magnification. (c) Find the image height and state whether the image is real or virtual, and upright or inverted.
Answer. Image distance = 15 cm (real). Magnification = −0.5. Image height = −2 cm (inverted, diminished).
Origins · 1 minIbn al-Haytham (Alhazen) and how this was discovered
Ibn al-Haytham wrote the monumental 'Book of Optics' (Kitāb al-Manāẓir), in which he systematically analysed the geometry of reflection in curved mirrors, laying the mathematical groundwork for understanding how mirrors form images of different sizes — the foundation on which magnification formulae were later formalised. He correctly described the law of reflection and used similar triangles to relate image and object positions, methods that are still used today. Ibn al-Haytham was the first person to argue, based on experiment rather than philosophy alone, that vision works by light entering the eye — not by rays emitted from the eye, as ancient Greek thinkers had believed. He carried out his experiments under house arrest in Cairo!
Related Concepts
- Mirror Formula — Magnification is calculated using image and object distances, which are found directly from the mirror formula 1/v + 1/u = 1/f.
- Sign Convention for Mirrors — The sign of the magnification depends entirely on the sign convention used for distances; understanding this convention is prerequisite to interpreting whether an image is real or virtual, upright or inverted.
- Reflection of Light — Magnification arises from the geometry of reflected rays; the law of reflection underpins every ray diagram used to derive the magnification formula.
- Magnification in Lenses — The same magnification formula m = −v/u applies to thin lenses, making this concept directly transferable to the study of refracting optical instruments.