A spherical concave mirror of focal length 10 cm and a double convex lens of focal length 5 cm are arranged on the common principal axis as shown in the figure. A small object is placed on the principal axis between the focal points $F_1$ and $F_2$ of the mirror and the lens, respectively. If two real and mutually inverted images are formed by the lens at the same location on the principal axis, what is the distance of the object from the mirror on the principal axis?
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An object placed at the center of curvature of a concave mirror ($u = 2f$) forms an image at the exact same location.
Using this property is the most common way to make two independent optical paths coincide in position.
Step 1: Understanding the Concept:
For two optical paths to produce images at the same final location, the rays must either follow the same path or be projected by the lens from the same virtual or real point. Step 2: Detailed Explanation:
1. Path 1: Rays go from object through the lens directly.
2. Path 2: Rays go from object, reflect off the mirror, and then through the lens.
3. Condition for Coincidence: If the object is placed at the center of curvature of the concave mirror, the rays striking the mirror reflect directly back along their own path.
4. Verification: If rays reflect back to the object position, then for the lens, the "reflected image" and the "original object" are at the exact same location. The lens will then project both to the same final image point on the other side.
5. Calculation: Focal length of mirror \( f_{m} = 10 \text{ cm} \). Center of curvature \( C = 2f_{m} = 20 \text{ cm} \). Step 3: Final Answer:
The object must be 20 cm from the mirror.
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