Reflection in curved mirrors

Key ideas

A concave (converging) mirror curves inward and reflects parallel rays to a single focal point in front of it, a distance $f$ away. It obeys the same equations as a lens — only now the image forms on the same side as the object (the reflecting side).

The mirror equation

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

• $f$ — focal length of the mirror
• $d_o$ — object distance (mirror to object)
• $d_i$ — image distance (mirror to image)

For a concave mirror the focal length is half the radius of curvature: $f = \tfrac{R}{2}$.

Magnification and image type

$m = \frac{h_i}{h_o} = \frac{d_i}{d_o}$

• Object beyond the focal point ($d_o > f$): image is real, inverted, in front of the mirror ($d_i$ positive).
• Object inside the focal point ($d_o < f$): image is virtual, upright, enlarged, behind the mirror ($d_i$ negative) — like a shaving/make-up mirror.

:::tip The mirror and lens equations are identical — same formula, same reciprocal trap. The only thing to keep straight is where the image sits: a real image from a mirror is on the same side as the object (in front), whereas a real image from a lens is on the far side. :::

Practice question

An object is $12$ cm in front of a concave mirror of focal length $6.0$ cm. Find the image distance.

Worked solution: $\tfrac{1}{d_i} = \tfrac{1}{6} - \tfrac{1}{12} = \tfrac{2-1}{12} = \tfrac{1}{12}$, so $d_i = 12$ cm (object at $2f$ gives an image at $2f$, same size, inverted).