Draw a ray diagram showing the image formation when a concave mirror produces a real, inverted, and magnified image of an object and hence obtain the mirror formula.
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When the object is placed between the focal point and the mirror, the image formed is real, inverted, and magnified. The image is located beyond the center of curvature.
To derive the mirror formula, let's first draw the ray diagram for the given case.
The diagram shows the concave mirror forming a real, inverted, and magnified image. In this case, the object is placed between the focal point and the mirror (closer than the focal point but farther than the mirror’s pole).
Derivation of Mirror Formula:
Let:
- \( u \) be the object distance,
- \( v \) be the image distance,
- \( f \) be the focal length of the mirror.
For a concave mirror, the mirror formula is derived from the geometry of light reflection:
1. The first ray (Ray 1) parallel to the principal axis reflects through the focal point \( F \).
2. The second ray (Ray 2) passing through the focal point \( F \) reflects parallel to the principal axis.
These two reflected rays meet at a point, forming the image.
Using the relation for curvature and geometry of the mirror:
\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\]
This is the mirror formula, where:
- \( u \) is the object distance (measured from the mirror to the object),
- \( v \) is the image distance (measured from the mirror to the image),
- \( f \) is the focal length (distance from the mirror’s pole to the focus).
