Spherical Mirrors

Step 1: Sign Convention and Ray Diagram.
In the case of a concave mirror, the following sign conventions are used: - Focal length is considered positive for a concave mirror.
- The object distance is negative when the object is in front of the mirror.
- The image distance 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:

Step 2: Mirror Equation.
From the geometry of the situation and the laws of reflection, the relationship between the object distance , image distance , and focal length for a concave mirror is given by the mirror equation:

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 , , and , 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 , the image distance , and the focal length for a concave mirror is:

To solve this question, we need to apply the mirror equation and the magnification formula. The mirror equation is given by:
Where:
- is the focal length of the concave mirror
- is the image distance (the distance from the mirror to the image)
- is the object distance (the distance from the mirror to the object)
The magnification is given by: Also, the magnification is the ratio of the image height to the object height : Now, the object is placed at a distance of from the mirror. This means the object is placed outside the center of curvature. The image formed when the object is at will be real, inverted, and reduced in size. For a concave mirror, the image formed when the object is at will be at . The image formed at this position is real and smaller than the object. The magnification is: Substituting the values and , we calculate the magnification: So, the magnification is . Since the object height cm, the image height is: Therefore, the length of the image is 4 cm. Hence, the correct answer is option (C) 4 cm.