25.7 Image Formation by Mirrors

Image Formation by Mirrors

Mirrors form images by reflecting light in predictable ways. The shape of a mirror determines where an image appears, how large it is, and whether it's real or virtual. This section covers flat mirrors, concave mirrors, convex mirrors, and the math that ties it all together.

Image Formation in Flat Mirrors

Light rays bounce off a flat mirror following the law of reflection: the angle of incidence equals the angle of reflection, both measured from the normal (a line perpendicular to the mirror's surface).

After reflecting, the rays diverge as if they came from a point behind the mirror. Your brain traces those rays backward and "sees" an image there. Since light doesn't actually pass through that point, this is called a virtual image. You can't project it onto a screen.

A flat mirror image has four key properties:

• Upright orientation (same vertical direction as the object)
• Same size as the object (magnification = 1)
• Laterally inverted (left and right are swapped, like reading text in a mirror)
• Same distance behind the mirror as the object is in front of it

Spherical Mirror Image Diagrams

Spherical mirrors are curved sections of a sphere. The two types behave very differently.

Concave mirrors curve inward and act as converging mirrors. Parallel rays hitting a concave mirror all reflect through a single focal point in front of the mirror. The type of image depends on where the object sits:

• Object beyond the focal point: the image is real, inverted, and can be larger or smaller depending on exact placement.
• Object between the focal point and the mirror: the image is virtual, upright, and magnified. This is how makeup mirrors work.

Convex mirrors curve outward and act as diverging mirrors. Parallel rays spread apart after reflecting, but they appear to come from a virtual focal point behind the mirror. No matter where the object is, a convex mirror always produces a virtual, upright, and diminished (smaller) image. That's why convex mirrors are used as vehicle side mirrors: they show a wide field of view, though objects appear smaller than they really are.

Calculations for Spherical Mirrors

Three equations handle most mirror problems. Before using them, you need to know the sign conventions.

Sign conventions (standard):

• Distances measured in front of the mirror are positive; behind the mirror are negative.
• Concave mirrors have a positive focal length ($f > 0$).
• Convex mirrors have a negative focal length ($f < 0$).
• A positive image distance ($d_i > 0$) means a real image; negative means virtual.

The mirror equation relates focal length ($f$), object distance ($d_o$), and image distance ($d_i$):

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

If you know any two of these values, you can solve for the third.

Radius of curvature connects to focal length by:

$R = 2f$

$R$ is the radius of the imaginary sphere that the mirror's surface is part of. The center of curvature sits at distance $R$ from the mirror along the principal axis.

Magnification tells you how the image size compares to the object size:

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

where $h_i$ is image height and $h_o$ is object height. Here's how to read the result:

• $m > 0$: image is upright
• $m < 0$: image is inverted
• $|m| > 1$: image is enlarged
• $|m| < 1$: image is reduced

Quick example: An object is placed 30 cm in front of a concave mirror with $f = 10$ cm. Find the image distance and magnification.

1. Plug into the mirror equation: $\frac{1}{10} = \frac{1}{30} + \frac{1}{d_i}$

2. Rearrange: $\frac{1}{d_i} = \frac{1}{10} - \frac{1}{30} = \frac{3-1}{30} = \frac{2}{30}$

3. Solve: $d_i = 15$ cm (positive, so the image is real and in front of the mirror)

4. Magnification: $m = -\frac{15}{30} = -0.5$ (negative means inverted; $|m| < 1$ means the image is smaller than the object)

Key Concepts in Optics for Mirrors

• The principal axis is an imaginary line that passes through the center of curvature and the mirror's vertex (the center point of the mirror's surface).
• The center of curvature is the center of the sphere from which the mirror is cut. It sits at a distance $R = 2f$ from the mirror.
• Spherical aberration is a defect where rays reflecting from the outer edges of a spherical mirror don't converge at exactly the same focal point as rays near the center. This blurs the image slightly. Parabolic mirrors eliminate this problem, which is why telescopes use them instead of simple spherical mirrors.