Plane mirrors Knowledge-Based Study

Study notes

1. Reflection of Light on Flat Surfaces

   A plane mirror is a flat, highly polished reflective surface, typically formed by coating one side of a smooth glass sheet with a thin metal film such as silver or aluminum.

   When light strikes a plane mirror, it undergoes specular reflection. The behavior of light rays is governed by the two fundamental laws of reflection:

   • The incident ray, the reflected ray, and the normal to the mirror surface at the point of incidence all lie in the same plane.
   • The angle of incidence ($i$) is equal to the angle of reflection ($r$): $i = r$. Both angles are measured relative to the normal line, which is perpendicular to the mirror surface.

   How Images Are Formed

   An image is formed when diverging light rays originating from an object point reflect off the mirror and reach an observer's eye. The eye and brain trace these reflected rays straight back along their lines of sight.

   Because the reflected rays diverge after striking the mirror, they never actually intersect in front of the mirror. Instead, their straight-line backward extensions intersect at a point behind the reflective surface.

   To locate the image of an object point using ray tracing, at least two distinct rays are drawn from the point:

   • A ray incident perpendicularly to the mirror ($i = 0^\circ$), which reflects back along its original path ($r = 0^\circ$).
   • An oblique ray striking the mirror at an arbitrary angle $i$, which reflects at an angle $r = i$.

   The intersection of the virtual extensions of these reflected rays defines the position of the image point.

   Properties of Images in Plane Mirrors

   An image formed by an ideal plane mirror consistently exhibits four primary characteristics:

   • Virtual: The image is formed by the apparent intersection of backward-projected rays, not real light rays. Consequently, it cannot be projected onto a physical screen.
   • Upright: The image maintains the same vertical orientation as the object; it is not inverted top-to-bottom.
   • Equal in Size: The magnification $m$ of a plane mirror is exactly $+1$. The image height ($h_i$) equals the object height ($h_o$).
   • Equal Distance: The perpendicular distance from the object to the mirror surface ($d_o$) equals the perpendicular distance from the mirror surface to the image ($d_i$), such that $d_i = d_o$.

   Additionally, images in plane mirrors exhibit lateral inversion (perceived left-right reversal), which is physically an inversion along the axis perpendicular to the mirror surface (front-to-back reversal).

   Field of View and Mirror Systems

   The field of view of an observer looking into a plane mirror is the spatial region from which light rays can reflect off the mirror's edges and enter the observer's pupils. To determine the visible region:

   1. Locate the virtual position of the observer's eye behind the mirror plane.
   2. Draw lines from this virtual eye position passing through the physical boundaries of the mirror and extending into the space in front of the mirror.
   3. Any object located between these boundary lines lies within the observer's field of view.

   Multiple Reflections

   When two plane mirrors are placed at an angle $\theta$ relative to each other, light undergoes successive reflections, producing multiple images. For an object positioned symmetrically between two mirrors separated by an angle $\theta$ (in degrees), the total number of visible images $n$ is given by:

   $$n = \frac{360^\circ}{\theta} - 1$$

   When $\theta = 90^\circ$, three distinct images are formed. If the mirrors are parallel ($\theta = 0^\circ$), an infinite theoretical sequence of images is produced.