Spherical Mirrors: Ray Diagrams, Image Formation, and Mirror Equation (4)

Geometric Optics

Spherical Mirrors

Spherical mirrors are mirrors whose surfaces form part of a sphere. They are classified as concave or convex depending on which side is reflective. Concave mirrors have a reflective inner surface, while convex mirrors have a reflective outer surface.

• Concave Mirror: Reflecting surface is on the inside of the sphere.

• Convex Mirror: Reflecting surface is on the outside of the sphere.

• Principal Axis: The straight line passing through the center of curvature and the mirror's vertex.

• Center of Curvature (C): The center of the sphere from which the mirror segment is taken.

• Focal Point (F): The point where parallel rays converge (concave) or appear to diverge from (convex).

Example: Makeup mirrors (concave) and security mirrors (convex) are common applications.

Ray Diagrams for Spherical Mirrors

Ray diagrams are used to determine the position, size, and nature of images formed by spherical mirrors. Three principal rays are typically used:

• P ray (Parallel Ray): Travels parallel to the principal axis and reflects through the focal point (concave) or appears to diverge from the focal point (convex).

• F ray (Focal Ray): Passes through the focal point and reflects parallel to the principal axis.

• C ray (Center Ray): Passes through the center of curvature and reflects back along its path.

Additional info: For convex mirrors, all reflected rays appear to diverge from the focal point behind the mirror.

Image Formation with Concave and Convex Mirrors

The location and nature of the image depend on the object's position relative to the mirror's focal point and center of curvature.

• Concave Mirror:

  • Object beyond C: Image is real, inverted, and smaller.

  • Object between C and F: Image is real, inverted, and larger.

  • Object inside F: Image is virtual, upright, and larger.

• Convex Mirror:

  • Image is always virtual, upright, and smaller than the object.

  • As the object moves farther from the mirror, the image size decreases and approaches the focal point.

Example: Car side mirrors use convex mirrors to provide a wide field of view with smaller, upright images.

Using Rays to Locate the Image

To locate the image, draw the three principal rays from the top of the object. The intersection of the reflected rays gives the image position. If the rays do not physically intersect, extend them backward to find the virtual image.

• Real Image: Formed where reflected rays actually converge.

• Virtual Image: Formed where extensions of reflected rays appear to converge.

Mirror Equation and Magnification

The mirror equation relates the object distance ($d_o$), image distance ($d_i$), and focal length ($f$) of a spherical mirror:

• Mirror Equation:

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

• Magnification Equation:

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

Where $h_i$ and $h_o$ are the image and object heights, respectively.

Sign Conventions for Mirrors

Quantity	Sign Convention
Focal Length ($f$)	Positive for concave mirrors, negative for convex mirrors
Image Distance ($d_i$)	Positive for images in front of the mirror (real), negative for images behind the mirror (virtual)
Object Distance ($d_o$)	Positive for objects in front of the mirror (real), negative for objects behind the mirror (virtual)
Magnification ($m$)	Positive for upright images, negative for inverted images

Special Cases and Applications

• Upright Image from Concave Mirror: Occurs when the object is placed inside the focal point; the image is virtual, upright, and magnified.

• Convex Mirror: Always produces a virtual, upright, and diminished image regardless of object position.

Example: Makeup mirrors (concave) allow for magnified upright images when the face is close to the mirror.

Additional info:

• Historical note: Leonardo da Vinci used mirror writing, which is readable only in a mirror, illustrating the concept of image reversal in plane mirrors.

• Mirror tracing tasks in psychology use plane mirrors to study visual-motor coordination, demonstrating the reversal of images.