To solve this problem, we need to analyze the setup involving a concave mirror and a plane mirror placed along the optical axis. The goal is to find the location of the plane mirror such that the image formed by the concave mirror coincides with the original point source.
Understanding the Concave Mirror's Image Formation
Firstly, let's establish some essential parameters:
- Object distance (u): The distance from the point source to the concave mirror, which is -35 cm (the negative sign indicates that the object is in front of the mirror).
- Focal length (f): Given as +25 cm for the concave mirror.
Using the Mirror Formula
The mirror formula relates the object distance (u), image distance (v), and the focal length (f) of the mirror:
1/f = 1/v + 1/u
Substituting the known values into the formula:
1/25 = 1/v + 1/(-35)
Calculating the Image Distance
Rearranging the equation gives:
1/v = 1/25 + 1/35
To perform this addition, we find a common denominator, which is 175:
1/v = (7/175) + (5/175) = 12/175
Now, taking the reciprocal gives:
v = 175/12 ≈ 14.58 cm
Positioning the Plane Mirror
The image formed by the concave mirror is at approximately 14.58 cm from the mirror. To have the plane mirror reflect this image back to the point source, it must be placed at a distance that equals the distance from the concave mirror to the image, which in this case is 14.58 cm.
Calculating the Total Distance from the Point Source
The total distance from the point source to the plane mirror can be determined by adding the distance from the point source to the concave mirror (35 cm) and the distance from the concave mirror to the plane mirror (14.58 cm):
Total distance = 35 cm + 14.58 cm ≈ 49.58 cm
Conclusion
Therefore, the plane mirror should be placed approximately 49.58 cm along the optical axis from the point source for the image formed by the concave mirror to coincide with the original point source.
