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 Question:
We are given a concave mirror ($f_m = 10$ cm) and a double convex lens ($f_l = 5$ cm) separated by $50$ cm.
An object is placed on their common principal axis.
Two real images are formed at the same final location by the lens: one directly from the lens, and one after reflecting off the mirror first.
Since one image undergoes an additional reflection at the mirror, the two images are mutually inverted. Step 2: Key Formula or Approach:
For the two images to be formed at the same location, the light reflected from the mirror must retrace its path or act as if it is originating from the same point.
If the object is placed at the center of curvature of the concave mirror, the reflected light forms an inverted intermediate image at the exact position of the object itself.
This intermediate image then acts as an object for the lens, forming the second image at the same location as the direct image. Step 3: Detailed Explanation:
• Let the distance of the object from the concave mirror be $u_m$.
• The focal length of the concave mirror is $f_m = -10$ cm.
• If the object is placed at the center of curvature of the mirror:
\[ u_m = 2 f_m = -20 \text{ cm} \]
• At this position, light rays striking the mirror reflect back along their original paths and form a real, inverted intermediate image at the same position ($20$ cm from the mirror).
• The lens is placed $50$ cm away from the mirror.
• The distance of the object (and the intermediate reflected image) from the lens is:
\[ u_l = 50 - 20 = 30 \text{ cm} \]
• For the direct light path, the object is at a distance of $30$ cm from the lens.
• For the reflected light path, the intermediate image is also at a distance of $30$ cm from the lens, but inverted.
• Both of these act as objects at the same location for the lens, so the lens projects both final images to the same location on the other side.
• Since the intermediate image was already inverted by the mirror, the two final images formed by the lens will be mutually inverted.
• This matches the physical description given in the question. Step 4: Final Answer:
The distance of the object from the mirror is 20 cm.
