Optics

Convex and concave mirrors

A spherical mirror is a reflecting part of a sphere. A concave mirror can converge parallel rays; a convex mirror diverges them and provides a wide field of view.

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \qquad f = \frac{R}{2}

Definition: Key concepts

For a spherical mirror near the principal axis, focal length f is half the radius of curvature R. The mirror equation relates f, object distance do, and image distance di; a virtual image has negative di in the usual convention.

Switch between concave and convex mirrors; vary object position and curvature to inspect images.

Law and application

Example: Worked example

A concave mirror has focal length 12 cm; an object is 36 cm away. Find the image using the mirror equation.

Solution

1/di=1/12−1/36=1/18 cm⁻¹, so di=18 cm. The real image is in front of the mirror; m=−di/do=−0.5, so it is inverted and half-size.

Quick check

Two principal rays help locate an image: a ray parallel to the principal axis reflects through the focus of a concave mirror (or appears to come from the focus of a convex mirror); a ray aimed through the center of curvature retraces its path. Where reflected rays actually meet is a real image; where their extensions meet is a virtual image. A concave mirror magnifies a nearby face when it lies inside the focal length. A convex vehicle mirror gives a wider field of view, but objects appear smaller.

Before substituting numbers, state the sign convention: for a concave mirror, a real object in front usually has $d_o>0$ ; a real image has $d_i>0$ , and a virtual image has $d_i<0$ . The sign of magnification gives orientation: $m<0$ means inverted, while $m>0$ means upright. The mirror equation works well for a small-aperture spherical mirror, where rays near the axis are paraxial. Far-off-axis rays may focus at different positions, producing spherical aberration.

What image does a convex mirror usually form of a real object?

In the paraxial approximation, how is a spherical mirror’s focal length related to radius R?

References

Young, Freedman (2019). University Physics