Curved mirrors Knowledge-Based Study

Study notes

1. Fundamentals of Spherical Mirrors

   A curved mirror is a reflecting surface formed by a section of a sphere. Curved mirrors are classified into two primary types based on which side of the spherical surface reflects light:

   • Concave mirror (converging): The reflecting surface curves inward toward the center of the sphere.
   • Convex mirror (diverging): The reflecting surface curves outward away from the center of the sphere.

   Key Geometric Parameters

   Every spherical mirror is defined by key reference points along its optical geometry:

   • Center of Curvature ($C$): The center of the hollow sphere from which the mirror surface is cut.
   • Vertex ($V$): The geometric center or pole of the mirror's reflecting surface.
   • Principal Axis: The straight reference line passing through both the center of curvature and the vertex.
   • Radius of Curvature ($R$): The distance from the vertex to the center of curvature ($R = CV$).
   • Focal Point ($F$): The point on the principal axis where rays incident parallel to the principal axis converge (in a concave mirror) or appear to diverge from (in a convex mirror) after reflection.
   • Focal Length ($f$): The distance between the vertex and the focal point. Under the paraxial approximation (rays close to the principal axis), the focal length is exactly half of the radius of curvature: $$f = \\frac{R}{2}$$

   Rules of Reflection for Ray Tracing

   Ray tracing allows us to graphically determine the position, orientation, and size of an image formed by a curved mirror. Under the law of reflection, the angle of incidence equals the angle of reflection relative to the normal at the point of incidence (which always passes through $C$).

   Standard Incident Rays

   To locate an image, we track at least two of the following principal rays originating from the tip of an object:

   1. Parallel Ray: A ray travelling parallel to the principal axis reflects through the focal point $F$ (for a concave mirror) or reflects such that its extension passes through $F$ behind the mirror (for a convex mirror).
   2. Focal Ray: A ray passing through the focal point $F$ (or directed toward $F$) reflects parallel to the principal axis.
   3. Center of Curvature Ray: A ray passing through (or directed toward) the center of curvature $C$ strikes the mirror along the normal and reflects back along its original path.
   4. Vertex Ray: A ray striking the vertex $V$ reflects symmetrically across the principal axis, maintaining equal angles of incidence and reflection relative to the axis.

   Image Formation by Concave and Convex Mirrors

   Images are described by three attributes: nature (real or virtual), orientation (upright or inverted), and size (magnified, diminished, or same size).

   • Real Image: Formed by the actual intersection of reflected light rays. A real image can be projected onto a screen and is always located in front of the reflecting surface.
   • Virtual Image: Formed by the apparent intersection of backward-extended reflected rays. A virtual image cannot be captured on a screen and appears behind the mirror.

   Concave Mirror Scenarios

   • Object beyond $C$: The image is real, inverted, diminished, and located between $C$ and $F$.
   • Object at $C$: The image is real, inverted, equal in size to the object, and located at $C$.
   • Object between $C$ and $F$: The image is real, inverted, magnified, and located beyond $C$.
   • Object at $F$: Reflected rays emerge parallel; no image is formed (the image is at infinity).
   • Object between $F$ and $V$: The image is virtual, upright, magnified, and located behind the mirror.

   Convex Mirror Behavior

   For any real object placed in front of a convex mirror, the reflected rays diverge. The image is always virtual, upright, diminished, and located between the vertex $V$ and the focal point $F$ behind the mirror. This property provides convex mirrors with a wide field of view, making them ideal for vehicle side mirrors and security surveillance.

   Quantitative Analysis of Curved Mirrors

   The positions and dimensions of objects and images are related quantitatively by the spherical mirror equation and the magnification formula.

   The Mirror Equation

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

   Where:

   • $f$ is the focal length of the mirror.
   • $d_o$ is the object distance from the vertex.
   • $d_i$ is the image distance from the vertex.

   Linear Magnification

   Magnification ($m$) describes the ratio of image height ($h_i$) to object height ($h_o$):

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

   Sign Conventions (Cartesian System)

   • Focal length ($f$): Positive ($+$) for concave mirrors; negative ($-$) for convex mirrors.
   • Object distance ($d_o$): Positive ($+$) for real objects in front of the mirror.
   • Image distance ($d_i$): Positive ($+$) for real images (in front of the mirror); negative ($-$) for virtual images (behind the mirror).
   • Magnification ($m$): Positive ($+$) indicates an upright image; negative ($-$) indicates an inverted image.
   • $|m| > 1$ means the image is enlarged, whereas $|m| < 1$ means the image is diminished.