Convex Mirror Image Formation

Physics of Convex Mirrors

A convex mirror, also known as a diverging mirror, is a curved mirror in which the reflective surface bulges outward toward the light source. Because of this shape, when parallel rays of light strike a convex mirror, they reflect outward and spread apart (diverge).

Principal Components and Definitions

• Pole (P): The geometric center of the reflecting spherical surface of the mirror.
• Center of Curvature (C): The center of the hollow sphere of glass from which the mirror was cut. It lies behind the reflecting surface of a convex mirror.
• Radius of Curvature (R): The radius of the sphere of which the mirror is a part. Under the Cartesian sign convention, $R$ is positive for convex mirrors.
• Principal Axis: The straight line passing through the Pole (P) and the Center of Curvature (C).
• Principal Focus (F): The point behind the mirror on the principal axis where all rays parallel to the axis appear to diverge from after reflection.
• Focal Length (f): The distance between the Pole (P) and the Principal Focus (F). It is positive for convex mirrors and satisfies:

Rules for Drawing Ray Diagrams

To trace and construct the image formed by a convex mirror, we utilize at least two of the following key rays originating from the tip of the object:

1. Ray Parallel to Principal Axis: A ray starting parallel to the principal axis reflects such that its back-extension passes through the virtual focus (F) behind the mirror.
2. Ray Directed Toward Focus: A ray traveling toward the virtual focus (F) reflects parallel to the principal axis.
3. Ray Directed Toward Center of Curvature: A ray directed toward the Center of Curvature (C) hits the mirror surface perpendicularly (along the normal line) and reflects directly back along its own path.
4. Ray Striking the Pole: A ray incident at the Pole (P) reflects symmetrically at an equal angle relative to the principal axis (obeying the Law of Reflection, $\theta_i = \theta_r$).

Image Formation and Properties

Unlike concave mirrors, which form different types of images depending on distance, a convex mirror **always** forms the same class of image for any real object position:

The Spherical Mirror Equations

The quantitative relationship between the object distance ($u$), image distance ($v$), and focal length ($f$) is governed by the **mirror formula**:

The ratio of the image height ($h_i$) to the object height ($h_o$) is defined as the **linear magnification** ($m$):

Cartesian Sign Convention for Convex Mirrors:

• The object is placed in front of the mirror (against the direction of light), so the object distance **$u$ is always negative** ($u < 0$).
• The focus and center of curvature lie behind the mirror, so **$f$ and $R$ are always positive** ($f > 0$, $R > 0$).
• Consequently, the image distance **$v$ is always positive** ($v > 0$), meaning the image is always formed behind the mirror.
• Since $v$ is positive and $u$ is negative, $m = -v/u$ is **always positive** ($m > 0$), representing an upright, virtual image.

Real-world Applications

• Automobile Side Mirrors: Used to provide a wide-angle view of the lanes behind, helping drivers change lanes safely.
• Store Security Mirrors: Dome convex mirrors mounted high in supermarket aisles let clerks monitor hidden corners.
• Blind Corner Safety: Placed at sharp street turns, parking garage exits, and driveways to show oncoming vehicle headlights.
• ATM Safety: Small convex mirrors on ATMs allow users to see if someone is standing too close behind them.