Derivation of the Mirror Formula: A Comprehensive Guide

Table of Contents#

1. Basic Definitions and Sign Conventions
   • Key Terms (Pole, Center of Curvature, Focus, Focal Length)
   • Sign Conventions (Cartesian Sign Convention)
2. Derivation of the Mirror Formula
   • Ray Diagrams and Similar Triangles
   • Step-by-Step Derivation
3. Example Usage
   • Real Object in Front of a Concave Mirror
   • Virtual Image in a Convex Mirror
4. Best Practices and Common Mistakes
   • Best Practices for Applying the Formula
   • Common Errors to Avoid
5. Conclusion
6. References

1. Basic Definitions and Sign Conventions#

Key Terms#

• Pole (( P )): The central point of the mirror’s surface (intersection of the principal axis and the mirror).
• Center of Curvature (( C )): The center of the sphere from which the mirror is a part (distance ( R ) from the pole, ( R = 2f )).
• Principal Axis: The straight line passing through the pole and center of curvature.
• Focus (( F )): The point on the principal axis where parallel rays (to the principal axis) converge (concave) or appear to diverge from (convex). The focal length ( f = \frac{R}{2} ).
• Object Distance (( u )): Distance from the pole to the object.
• Image Distance (( v )): Distance from the pole to the image.

Sign Conventions (Cartesian Sign Convention)#

To derive the mirror formula, we use the Cartesian Sign Convention:

• Distances are measured from the pole (( P )) as the origin.
• Rays travel from left to right (incident light direction).
• Distances measured against the incident light (left of the pole) are negative; distances measured with the incident light (right of the pole) are positive.

2. Derivation of the Mirror Formula#

Ray Diagram for Derivation#

Consider a concave mirror with pole ( P ), center of curvature ( C ), and focus ( F ) (where ( PF = f ) and ( PC = R = 2f )). An object ( AB ) is placed at distance ( u ) from ( P ), and its image ( A'B' ) is formed at distance ( v ) from ( P ).

Step 1: Draw Key Rays#

1. Ray 1: Parallel to the principal axis (from ( A )) → reflects through ( F ).
2. Ray 2: Through ( C ) (normal to the mirror) → reflects back along the same path (since it is normal to the mirror surface).

Step 2: Identify Similar Triangles#

Two pairs of similar triangles are formed:

1. Object Triangle (( \triangle ABP )) and Image Triangle (( \triangle A'B'P )):

   • Both are right-angled (at ( B ) and ( B' )) and share ( \angle APB ).
   • By AA similarity: ( \frac{AB}{A'B'} = \frac{PB}{PB'} ).

   Let the height of the object be ( h ) and the height of the image be ( h' ). Then:
   [ \frac{h}{h'} = \frac{|u|}{|v|} \tag{1} ]

2. Focus Triangle (( \triangle AFB )) and Focus-Image Triangle (( \triangle A'FB' )):

   • Both are right-angled (at ( B ) and ( B' )) and share ( \angle AFB ).
   • By AA similarity: ( \frac{AB}{A'B'} = \frac{FB}{FB'} ).

Step 3: Correct Derivation Using Similar Triangles#

Correct Geometric Derivation: The proper approach uses triangles formed by the principal axis and the line through the center of curvature. For a concave mirror with the object at distance ( u ) from the pole and image at distance ( v ):

From the geometry of the ray diagram, we derive: [ \frac{h}{h'} = \frac{u}{v} = \frac{f}{v - f} ]

Setting ( \frac{u}{v} = \frac{f}{v - f} ) and solving: [ u(v - f) = vf ] [ uv - uf = vf ] [ uv - vf = uf ] [ v(u - f) = uf ]

Dividing both sides by ( uvf ): [ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} ]

This is the correct mirror formula.

Generalization to Convex Mirrors#

For a convex mirror, the sign convention changes ( ( f ) is positive, ( v ) is positive for virtual images), but the formula ( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} ) still holds (with ( u ) negative, ( f ) positive, and ( v ) positive).

3. Example Usage#

Example 1: Real Image in a Concave Mirror#

A concave mirror has ( f = -10 , \text{cm} ) (negative for concave). An object is placed at ( u = -20 , \text{cm} ) (real object, left of the mirror). Find the image distance ( v ).

Using the mirror formula:
[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} ]

Substitute ( f = -10 , \text{cm} ) and ( u = -20 , \text{cm} ):
[ \frac{1}{-10} = \frac{1}{v} + \frac{1}{-20} ]

Simplify:
[ -\frac{1}{10} + \frac{1}{20} = \frac{1}{v} ]
[ -\frac{2}{20} + \frac{1}{20} = \frac{1}{v} ]
[ -\frac{1}{20} = \frac{1}{v} \implies v = -20 , \text{cm} ]

The image is ( 20 , \text{cm} ) in front of the mirror (real, inverted, and same size as the object).

Example 2: Virtual Image in a Convex Mirror#

A convex mirror has ( f = +15 , \text{cm} ) (positive for convex). An object is placed at ( u = -30 , \text{cm} ) (real object). Find the image distance ( v ).

Using the mirror formula:
[ \frac{1}{15} = \frac{1}{v} + \frac{1}{-30} ]

Simplify:
[ \frac{1}{v} = \frac{1}{15} + \frac{1}{30} = \frac{2}{30} + \frac{1}{30} = \frac{3}{30} = \frac{1}{10} ]

Thus, ( v = +10 , \text{cm} ) (virtual image, right of the mirror, upright, and diminished).

4. Best Practices and Common Mistakes#

Best Practices#

• Consistent Sign Convention: Always use the same sign convention (e.g., Cartesian) to avoid errors.
• Ray Diagrams: Sketch ray diagrams to visualize object/image positions.
• Paraxial Approximation: Derivations assume small angles (paraxial rays) for accuracy.
• Verification with Examples: Test the formula with real-world scenarios (e.g., shaving mirrors, rear-view mirrors).

Common Mistakes#

• Incorrect Sign Usage: Forgetting negative signs for real objects/images in concave mirrors.
• Mixing Conventions: Using different sign conventions for ( u ), ( v ), and ( f ).
• Ignoring Paraxial Rays: Using large-angle rays leads to spherical aberration (inaccurate results).

Conclusion#

The mirror formula ( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} ) is derived using similar triangles and the law of reflection. Understanding sign conventions and practicing with examples is key to applying it correctly. This formula is foundational for analyzing image formation in spherical mirrors, enabling applications from telescopes to car mirrors.

References#

1. Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. Wiley.
2. Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage.
3. Tipler, P. A., & Mosca, G. (2007). Physics for Scientists and Engineers. W.H. Freeman.