State the relationship between object distance, image distance, and focal length of a spherical mirror.

Relationship Between Object Distance, Image Distance, and Focal Length of a Spherical Mirror
The relationship between the object distance uu, the image distance vv, and the focal length ff of a spherical mirror (whether concave or convex) is given by the mirror equation:
1f=1v+1u\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
Where:
• ff is the focal length of the mirror,
• vv is the image distance, which is the distance from the mirror to the image formed,
• uu is the object distance, which is the distance from the mirror to the object being reflected.
Sign Convention for Spherical Mirrors:
In order to apply this formula, it's important to follow the sign conventions:
1. Object Distance (u):
o For a real object, uu is taken as negative (since it is on the same side as the incoming light, in front of the mirror).
o For a virtual object, uu is taken as positive.
2. Image Distance (v):
o For a real image, vv is positive (the image is formed on the same side as the reflected light, in front of the mirror).
o For a virtual image, vv is negative (the image is formed on the opposite side of the reflected light).
3. Focal Length (f):
o For a concave mirror (converging mirror), the focal length ff is positive.
o For a convex mirror (diverging mirror), the focal length ff is negative.
Derivation of the Mirror Equation:
The mirror equation can be derived from the geometry of spherical mirrors. It relates the object, image, and focal points based on the mirror's curvature. This equation is applicable for both concave and convex mirrors.
Summary of the Mirror Equation:
1f=1v+1u\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
• This equation helps in determining the position of the image formed by a spherical mirror for any given object distance.
• It also helps in understanding the nature of the image (whether it is real or virtual, inverted or upright, magnified or diminished), based on the sign and magnitude of vv and uu.
By knowing any two of the quantities (object distance, image distance, and focal length), you can calculate the third using this equation.