Image formation from a spherical mirror

[latex] \begin{flushleft} \textbf{Definition:} \\ The laws of reflection for a spherical mirror are similar to those for a plane mirror. However, due to the curved surface of spherical mirrors (concave or convex), the behavior of reflected light changes, leading to image formation at specific locations depending on the mirror’s geometry. \\ \\ \textbf{Theory:} \\ 1. **Laws of Reflection:** \begin{itemize} \item The angle of incidence (\(\theta_i\)) is equal to the angle of reflection (\(\theta_r\)). \item The incident ray, the reflected ray, and the normal to the mirror’s surface at the point of incidence lie in the same plane. \end{itemize} 2. **Spherical Mirror Terminology:** \begin{itemize} \item **Pole (P):** The geometric center of the mirror’s surface. \item **Center of Curvature (C):** The center of the sphere from which the mirror is a part. \item **Principal Axis:** The straight line passing through the center of curvature and the pole. \item **Focus (F):** The point where parallel rays converge (concave mirror) or appear to diverge (convex mirror) after reflection. \item **Focal Length (\(f\)):** The distance between the focus and the pole (\(f = R/2\), where \(R\) is the radius of curvature). \end{itemize} 3. **Ray Diagrams and Behavior of Reflected Rays:** \begin{itemize} \item A ray parallel to the principal axis passes through the focus after reflection (concave) or appears to diverge from the focus (convex). \item A ray passing through the focus becomes parallel to the principal axis after reflection. \item A ray directed toward the center of curvature reflects back along the same path. \item A ray incident at the pole reflects symmetrically about the principal axis. \end{itemize} 4. **Mirror Equation:** The relationship between object distance (\(u\)), image distance (\(v\)), and focal length (\(f\)) is: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] Magnification (\(M\)) is given by: \[ M = -\frac{v}{u} \] \\ \\ \textbf{Applications of Spherical Mirrors:} \\ 1. **Concave Mirrors:** \begin{itemize} \item Used in reflecting telescopes to gather and focus light. \item Utilized in shaving mirrors for magnified images. \item Employed in headlamps and torches to focus light into a beam. \end{itemize} 2. **Convex Mirrors:** \begin{itemize} \item Commonly used as rearview mirrors in vehicles for a wider field of view. \item Installed in hallways and stores for surveillance. \item Used in diverging lenses to create virtual images. \end{itemize} \\ \\ \textbf{Examples:} \\ 1. **Concave Mirror Example:** \\ An object placed beyond the center of curvature of a concave mirror produces a real, inverted, and smaller image between the center of curvature and the focus. 2. **Convex Mirror Example:** \\ An object placed in front of a convex mirror always produces a virtual, upright, and smaller image, regardless of the distance. 3. **Focus Adjustment in Telescopes:** \\ In reflecting telescopes, the concave mirror is adjusted to ensure parallel rays converge at the focus for clear observation. \\ \\ \textbf{Real-Life Uses:} \\ \begin{itemize} \item Designing optical instruments like telescopes, cameras, and microscopes. \item Enhancing vehicle safety with convex rearview mirrors. \item Focusing sunlight in solar concentrators for energy generation. \item Creating magnified images in dental and makeup mirrors. \end{itemize} \\ \\ \textbf{Observations:} \\ \begin{itemize} \item For concave mirrors, image characteristics depend on the object’s position relative to the focus and center of curvature. \item Convex mirrors always produce virtual, upright, and diminished images. \item Reflection from spherical mirrors adheres to the laws of reflection but results in image formation due to the curved surface. \item The focal length of a spherical mirror is half of its radius of curvature. \end{itemize} \end{flushleft} [/latex]