Question

With the help of a suitable ray diagram, derive the formula \( \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \) for a concave mirror.

Hint:

The mirror equation holds true for both concave and convex mirrors, where the distances are measured according to sign conventions. Always check the direction of the object and image to apply the correct signs.

Answer 1

Step 1: Sign Convention and Ray Diagram.
In the case of a concave mirror, the following sign conventions are used: - Focal length \( f \) is considered positive for a concave mirror.
- The object distance \( u \) is negative when the object is in front of the mirror.
- The image distance \( v \) is positive when the image is formed on the same side as the object.
Now, consider the concave mirror and an object placed in front of it. The image is formed by the reflection of light rays. The ray diagram for a concave mirror looks like this: \[ \begin{array}{c} \text{Ray diagram showing the reflection of rays from a concave mirror} \\ \text{The object is placed at a distance } u \text{ from the mirror.} \\ \text{The image is formed at a distance } v \text{ from the mirror.} \end{array} \] Step 2: Mirror Equation.
From the geometry of the situation and the laws of reflection, the relationship between the object distance \( u \), image distance \( v \), and focal length \( f \) for a concave mirror is given by the mirror equation: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u}. \] Step 3: Derivation.
We can derive this equation from the reflection laws and the geometry of the concave mirror. The relationship between the angles of incidence and reflection, along with the distances \( u \), \( v \), and \( f \), gives the above formula. By considering the geometry of the ray diagram and applying trigonometry, the mirror equation is obtained. Final Answer:
The formula relating the object distance \( u \), the image distance \( v \), and the focal length \( f \) for a concave mirror is: \[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f}. \]