Calculate Mirror Equation

For spherical mirrors, the same base equation as the lens equation applies, but with sign conventions for concave and convex mirrors.

What is the Mirror Equation?

The mirror equation (or mirror formula) describes the relationship between the object distance (g), the image distance (b), and the focal length (f) for spherical mirrors. It is one of the most important tools in geometric optics and applies to both concave (converging) and convex (diverging) mirrors. The mirror equation allows calculation of the size and position of a mirror image.

The Mirror Equation: Fundamental Formula

The fundamental mirror equation is:

• f – Focal length of the mirror (in cm or m)
• g – Object distance (distance from object to mirror)
• b – Image distance (distance from image to mirror)

This equation can also be rearranged to calculate each individual quantity:

Spherical Mirrors: Concave vs. Convex

Sign Convention in Optics

In geometric optics, the sign convention is crucial:

• Focal Length f:
  • f > 0 for concave mirrors (converging)
  • f < 0 for convex mirrors (diverging)
• Image Distance b:
  • b > 0 when image is real (in front of mirror)
  • b < 0 when image is virtual (behind mirror)
• Object Distance g: Normally always g > 0 (object is always in front of mirror)
• Magnification m:
  • m > 0 for upright images
  • m < 0 for inverted images

Focal Length and Radius of Curvature

The focal length is related to the radius of curvature by a simple relationship:

• R – Radius of curvature of the mirror
• The focal point is located at f = R/2
• For a flat mirror: R = ∞, therefore f = ∞

Magnification

The magnification (or lateral magnification) describes the size ratio between image and object:

• m – Magnification (dimensionless)
• h' – Image height
• h – Object height
• m < 0: Image is inverted
• m > 0: Image is upright
• |m| > 1: Image is magnified
• |m| < 1: Image is minified
• |m| = 1: Image is same size

Image Construction with Rays

To understand mirror images, we use ray diagrams. Important rays are:

• 1. Axial Parallel Ray: Comes parallel to optical axis → reflects through focal point (concave) or as extension from focal point (convex)
• 2. Focal Ray: Passes through focal point (or its extension) → reflects parallel to optical axis
• 3. Central Ray: Passes to center of curvature C → reflects back on itself (perpendicular to surface)

Concave Mirror – Image Types by Position

Convex Mirror – Always Virtual

A convex mirror always produces a virtual, upright, minified image, regardless of object position:

• Image distance: b < 0 (virtual)
• Magnification: 0 < m < 1 (upright and minified)
• Location: Image always behind the mirror
• Practical application: Traffic mirrors, security mirrors (wide field of view)

Focal Length of Different Mirror Types

Practical Applications of Mirrors

• Concave Mirrors:
  • Bathroom magnifying mirrors
  • Dental mirrors for tooth examination
  • Telescope mirrors (reflecting telescopes like Hubble)
  • Flashlights and headlights (parallel rays)
  • Solar cookers (heat focusing)
  • Headlamps (e.g., surgical headlamps)
• Convex Mirrors:
  • Traffic mirrors at street corners
  • Rear-view mirrors in cars (wide field of view)
  • Security mirrors in stores
  • Wide-angle surveillance cameras

Numerical Examples

Example 1: Concave Mirror – Magnified Image

Given: Focal length f = 10 cm, object distance g = 15 cm

• Using mirror equation: 1/10 = 1/15 + 1/b
• 1/b = 1/10 - 1/15 = 3/30 - 2/30 = 1/30
• Image distance: b = 30 cm (real image in front of mirror)
• Magnification: m = -b/g = -30/15 = -2 (twice as large, inverted)

Example 2: Concave Mirror – Focal Point

Given: f = 20 cm, g = 60 cm

• 1/20 = 1/60 + 1/b
• 1/b = 1/20 - 1/60 = 3/60 - 1/60 = 2/60 = 1/30
• Image distance: b = 30 cm
• Magnification: m = -30/60 = -0.5 (half size, inverted)

Example 3: Convex Mirror

Given: f = -15 cm (convex!), g = 30 cm

• 1/(-15) = 1/30 + 1/b
• 1/b = -1/15 - 1/30 = -2/30 - 1/30 = -3/30 = -1/10
• Image distance: b = -10 cm (virtual, behind mirror)
• Magnification: m = -(-10)/30 = 0.33 (upright, 1/3 size)

Frequently Asked Questions

Q: Why is the focal length negative for convex mirrors?

A: By optical sign convention, the focal point of a convex mirror is behind the mirror. Points behind the mirror are counted as negative, so f < 0.

Q: Can you magnify with a convex mirror?

A: No, convex mirrors always produce smaller, virtual images (|m| < 1). They are designed for wide-angle applications, not magnification.

Q: What is the best object position for strong magnification with a concave mirror?

A: Just in front of the focal point (g < f). The closer the object is to the focal point, the larger the virtual, upright image becomes behind the mirror.

Q: What happens when the object is exactly at the focal point?

A: The reflected rays travel parallel. According to the mirror equation, 1/b = 0, so b = ∞. No focused image is formed – rays go to infinity.

Summary