Class 10 > Unit # 13: Geometrical Optics > Mirror Equation

Mirror Equation Derivation — Chapter 13 Geometrical Optics

Welcome to Talha's Physics Academy. In this tutorial, we will learn how to derive the essential mirror equation used for solving problems and long questions related to spherical and concave mirrors in Class 10 Physics.

Video Lecture

1. Considerations & Terminology

Before jumping into the geometric proof, let's establish our variables based on the ray diagram:

• Focal Length (\(f\)): Distance from pole \(P\) to focus \(F\).
• Object Height (\(h_o\)): Represented by height \(AB\).
• Image Height (\(h_i\)): Represented by height \(A'B'\).
• Object Distance (\(p\)): Distance from pole \(P\) to object \(B\).
• Image Distance (\(q\)): Distance from pole \(P\) to image \(B'\).

2. Step-by-Step Derivation

Step 1: Similarity of Triangles \(ABP\) and \(A'B'P\)

Using the geometric theorem that ratios of corresponding sides of similar triangles are equal:

$$\frac{AB}{A'B'} = \frac{PB}{PB'}$$

Substituting with our defined variables (\(h_o, h_i, p, q\)):

$$\frac{h_o}{h_i} = \frac{p}{q} \quad \text{--- (Equation 1)}$$

Step 2: Similarity of Triangles \(ABF\) and \(PDF\)

Comparing triangles \(ABF\) and \(PDF\), where segment \(PD\) is equal in height to \(A'B'\):

$$\frac{AB}{PD} = \frac{BF}{PF} \implies \frac{h_o}{h_i} = \frac{p - f}{f} \quad \text{--- (Equation 2)}$$

Step 3: Combining Equations

Equating Equation 1 and Equation 2 since their left-hand sides are identical:

$$\frac{p}{q} = \frac{p - f}{f}$$

Rearranging and breaking down fractions yields the final relation:

$$\frac{1}{f} = \frac{1}{p} + \frac{1}{q}$$

This finalized formula is known as the Mirror Equation, serving as the foundational tool for solving numerical problems in geometrical optics.