Question

Prove for a concave mirror \[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]

Hint:

The mirror formula \( \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \) is valid for all spherical mirrors, with sign conventions applied for different cases.

Answer 1

Step 1: Consider a concave mirror with an object placed at a distance \( u \) from the mirror, forming an image at \( v \). The focal length of the mirror is \( f \). Using the geometry of the concave mirror and the laws of reflection, we apply the similar triangles method:
Step 2: Using the sign convention, we consider the two triangles formed: \[ \triangle ABP \sim \triangle A'B'F \] \[ \frac{A'B'}{AB} = \frac{B'F}{BP} \] Similarly, using another set of similar triangles: \[ \triangle A'B'P \sim \triangle OFB' \] \[ \frac{A'B'}{AB} = \frac{OF}{OP} \]
Step 3: From these geometric relations, we derive the mirror formula: \[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \] Conclusion: The above derivation proves the mirror formula for a concave mirror.