25.7 Image Formation by Mirrors

Mirrors are fascinating optical devices that manipulate light to create images. From flat mirrors to curved ones, they play with reflection to show us the world in new ways. Understanding how mirrors work is key to grasping the basics of optics.

This topic dives into the nitty-gritty of image formation by mirrors. We'll explore how different mirror shapes affect light rays, create various types of images, and learn the math behind it all. It's a reflection of the broader principles of light and optics.

Image Formation by Mirrors

Image formation in flat mirrors

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• Light rays reflect off the mirror's surface maintaining equal angles between the incident ray and the reflected ray relative to the surface normal (perpendicular line to the surface)
• The incident angle is equal to the reflection angle
• The reflected rays appear to originate from behind the mirror, creating a virtual image that cannot be projected onto a screen as light does not actually pass through the image location
• The image formed by a flat mirror exhibits the following characteristics:
  • Upright orientation matching the object's vertical orientation
  • Equal size to the object with a 1:1 ratio
  • Laterally inverted appearance with left and right sides reversed (as if looking at your reflection)
  • Equidistant location behind the mirror compared to the object's distance in front of the mirror

Spherical mirror image diagrams

• Concave mirrors have a reflecting surface that curves inward, acting as converging mirrors
  • Parallel light rays incident on a concave mirror converge at the focal point after reflection
  • Concave mirrors can form both real images (inverted and projectable on a screen) and virtual images (upright and not projectable) based on the object's position relative to the focal point
    • Objects beyond the focal point form real, inverted, and smaller or larger images
    • Objects between the focal point and the mirror form virtual, upright, and magnified images
• Convex mirrors have a reflecting surface that curves outward, acting as diverging mirrors
  • Parallel light rays incident on a convex mirror diverge after reflection, appearing to originate from a virtual focal point located behind the mirror
  • Convex mirrors always form virtual, upright, and smaller (diminished) images compared to the object, regardless of the object's distance from the mirror (rearview mirrors in vehicles)

Calculations for spherical mirrors

• Focal length ($f$) represents the distance between the mirror's surface and the focal point
  • Concave mirrors have a positive focal length, with the focal point located in front of the mirror
  • Convex mirrors have a negative focal length, with a virtual focal point located behind the mirror
• The mirror equation relates the focal length ($f$), object distance ($d_o$), and image distance ($d_i$): $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$
  • Solving this equation allows for the calculation of the image distance or object distance when the other values are known
• The radius of curvature ($R$) is twice the focal length: $R = 2f$, representing the radius of the imaginary sphere that the mirror's surface is a part of
• Magnification ($m$) quantifies the size relationship between the image and the object, calculated as the ratio of the image height ($h_i$) to the object height ($h_o$) or the negative ratio of the image distance ($d_i$) to the object distance ($d_o$): $m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$
  • A negative magnification value indicates an inverted (upside-down) image
  • Magnification greater than 1 ($|m| > 1$) signifies an enlarged image compared to the object
  • Magnification less than 1 ($|m| < 1$) signifies a reduced image compared to the object (makeup mirrors)

Key concepts in optics for mirrors

• The principal axis is an imaginary line passing through the center of curvature and the mirror's vertex
• The center of curvature is the center of the sphere of which the mirror is a part
• Spherical aberration occurs when light rays reflecting from different parts of a spherical mirror do not converge at a single focal point, affecting image quality