Question

Write the assumptions, sign conventions and then derive the mirror formula for concave mirror.

Hint:

For a concave mirror, the object and image distances are related by the mirror formula \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \), where \( f \) is the focal length, \( u \) is the object distance, and \( v \) is the image distance.

Answer 1

Step 1: Assumptions for the concave mirror.
1. The object is placed in front of the mirror, and the light rays are parallel to the principal axis.
2. The mirror is spherical, with its center of curvature at \( C \) and focal point at \( F \).
3. The rays are reflected according to the laws of reflection.
Step 2: Sign conventions.
For a concave mirror, the following sign conventions are used:
1. The focal length \( f \) is considered positive for a concave mirror.
2. The object distance \( u \) is measured from the mirror along the principal axis, and it is negative if the object is in front of the mirror (real object).
3. The image distance \( v \) is positive if the image is formed in front of the mirror (real image), and negative if the image is formed behind the mirror (virtual image).
4. The radius of curvature \( R \) is positive for a concave mirror, as the center of curvature lies in front of the mirror.
Step 3: Deriving the mirror formula.
The mirror formula relates the object distance \( u \), image distance \( v \), and focal length \( f \) of a mirror. Using the law of reflection and geometry of the mirror, we can derive the mirror formula. The relationship is given by: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] This formula allows us to calculate the image distance or object distance when the other two quantities are known.
Step 4: Conclusion.
Thus, the mirror formula for a concave mirror is: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] which is a fundamental equation for spherical mirrors.