To solve the problem of finding the locations of the images formed by the two parts of a concave mirror that has been broken and separated, we first need to understand how concave mirrors work and how they form images. Given that the focal length of the mirror is 10 cm, we can use the mirror formula and the properties of concave mirrors to determine the image locations.
Understanding the Basics of Concave Mirrors
A concave mirror is a spherical mirror that curves inward. It has a focal point (F) where parallel rays of light converge after reflecting off the mirror. The focal length (f) is the distance from the mirror's surface to the focal point. In this case, the focal length is given as 10 cm, which means the focal point is located 10 cm in front of the mirror.
Setting Up the Problem
When the mirror is broken into two parts and separated by a distance of 1 cm, we can consider each part as a separate concave mirror. The object is located midway between the two principal axes, which means it is positioned at a distance of 5 cm from each part of the mirror (since the total distance between the two mirrors is 10 cm).
Calculating Image Locations
We can use the mirror formula to find the image location for each part of the mirror. The mirror formula is given by:
1/f = 1/v + 1/u
Where:
- f = focal length of the mirror
- v = image distance from the mirror
- u = object distance from the mirror
For each part of the mirror:
For the Left Part of the Mirror
Let’s denote the left part of the mirror as Mirror 1. The object distance (u) for Mirror 1 is -5 cm (negative because the object is in front of the mirror). The focal length (f) is -10 cm (negative for concave mirrors). Plugging these values into the mirror formula:
1/(-10) = 1/v + 1/(-5)
Rearranging gives:
1/v = 1/(-10) + 1/5
1/v = -1/10 + 2/10 = 1/10
Thus, v = 10 cm. This means the image formed by Mirror 1 is located 10 cm in front of it.
For the Right Part of the Mirror
Now, let’s consider the right part of the mirror, which we’ll call Mirror 2. The object distance (u) for Mirror 2 is also -5 cm. Using the same focal length of -10 cm:
1/(-10) = 1/v + 1/(-5)
Rearranging gives:
1/v = 1/(-10) + 1/5
1/v = -1/10 + 2/10 = 1/10
Thus, v = 10 cm. This means the image formed by Mirror 2 is also located 10 cm in front of it.
Final Image Locations
Now, let’s summarize the locations of the images:
- The image formed by Mirror 1 is located 10 cm in front of it, which is at a distance of 10 cm from the left mirror.
- The image formed by Mirror 2 is also located 10 cm in front of it, which is at a distance of 10 cm from the right mirror.
Since the two mirrors are separated by 1 cm, the images will be located 10 cm from each mirror, but since they are separate, the actual distance from the original object position will be 10 cm from the left mirror and 9 cm from the right mirror.
In conclusion, both images are formed at the same distance from their respective mirrors, but they are positioned relative to the separation between the two mirrors. This illustrates how the properties of concave mirrors remain consistent, even when altered in configuration.
