Mirrors

Image formation by reflection from flat and spherical surfaces, including the mirror equation and lateral magnification.

Conceptual context

A mirror is a polished surface that reflects light specularly, following the law of reflection. Within geometrical optics, mirrors are the first device capable of forming images of objects: luminous copies whose position, size and orientation depend on the surface geometry and the object distance.

Mirror optics complements lens optics by using reflection instead of refraction. Understanding how a spherical mirror forms images is essential for interpreting reflecting telescopes, vehicle rear-view mirrors, dental mirrors and solar concentrators, all of which appear in everyday life and engineering.

🟢 Essential level

A flat mirror produces an image the same size as the object, at the same distance behind the mirror and laterally reversed (left-right inversion). The image is virtual: it cannot be captured on a screen because the reflected rays do not actually converge; rather, their extensions appear to cross behind the mirror.

A concave mirror has its reflecting surface on the inner side of a sphere. Rays parallel to the optical axis converge at a point called the focus (F), located halfway between the vertex and the center of curvature. If the object is beyond the focus the image is real and inverted; if the object is between the focus and the mirror the image is virtual, upright and enlarged.

A convex mirror has its reflecting surface on the outer side of the sphere. Parallel rays diverge after reflection as though they came from a virtual focus behind the mirror. The image is always virtual, upright and reduced, providing a wide field of view useful in rear-view and security mirrors.

🔵 Formal level

Basic formulas

The quantitative relationship between the object distance $d_o$, the image distance $d_i$ and the focal length $f$ is given by the mirror equation:

Mirror equation

Here $d_o$ is the distance from the object to the mirror vertex, $d_i$ the distance from the vertex to the image and $f$ the focal length. In the standard sign convention, real distances (in front of the mirror) are positive and virtual distances (behind the mirror) are negative. For a concave mirror $f$ is positive; for a convex mirror $f$ is negative.

The relative size of the image is described by the lateral magnification:

Lateral magnification

The sign of $M$ indicates orientation: negative means an inverted image and positive means an upright image. The absolute value gives the scale factor.

Real student questions

Why are car rear-view mirrors convex rather than concave? A convex mirror always produces a reduced image, covering a wider field of view than a flat mirror of the same size. The drawback is that objects appear farther away than they actually are.

Can a concave mirror really start a fire? Yes. When a large-diameter concave mirror faces the Sun, it concentrates sunlight at the focus. The energy density at that point can be sufficient to ignite flammable materials. This principle is used in solar furnaces.

What happens if I place an object exactly at the focus of a concave mirror? No image forms at any finite position: the reflected rays emerge parallel and "meet at infinity". In practice, the image is projected as a collimated beam.

Is a flat mirror a special case of a spherical mirror? Yes. A flat mirror is equivalent to a spherical mirror with infinite radius of curvature, giving infinite focal length. The mirror equation then reduces to $d_i$ equal to minus $d_o$, a virtual image at equal distance behind the mirror.

Final synthesis

Spherical mirrors form images whose position and size are determined by the mirror equation and lateral magnification. The sign convention distinguishes real from virtual images and inverted from upright orientation. Mastering these relationships enables interpretation of everything from a simple rear-view mirror to a reflecting telescope, and forms the foundation for the subsequent study of lenses and compound optical systems.