If an object is palced between two parallel mirrors ,an infinite number of images result .Suppose that the mirror are a distance 2b apart and the object is put at the mid-point between the mirrors ,Find the distance of the image from the object.

Explanation

To solve the problem of finding the distance of the images from the object when placed between two parallel mirrors, we can follow these steps:

### Step 1: Understand the Setup
We have two parallel mirrors separated by a distance of \(2b\). The object is placed at the midpoint between these mirrors. This means the distance from the object to each mirror is \(b\).

### Step 2: Identify the First Set of Images
The first images \(I_1\) and \(I_2\) are formed by the reflections of the object \(O\) in the two mirrors.
- The distance from the object \(O\) to the first image \(I_1\) (reflected in the first mirror) is \(b\).
- The distance from the object \(O\) to the second image \(I_2\) (reflected in the second mirror) is also \(b\).

### Step 3: Calculate the Distance of \(I_1\) and \(I_2\) from \(O\)
The total distance from the object to the images \(I_1\) and \(I_2\) can be calculated as:
\[ O I_1 = O I_2 = b + b = 2b \]
Thus, both images \(I_1\) and \(I_2\) are at a distance of \(2b\) from the object.

### Step 4: Identify the Second Set of Images
Next, we consider the images formed by the reflections of \(I_1\) and \(I_2\):
- The image \(I_3\) is formed by reflecting \(I_2\) in the first mirror. The distance from \(I_2\) to the first mirror is \(b\), and since \(I_2\) is \(2b\) away from \(O\), the total distance from \(O\) to \(I_3\) is:
\[ O I_3 = O I_2 + b = 2b + b = 3b \]

- Similarly, the image \(I_4\) is formed by reflecting \(I_1\) in the second mirror. The distance from \(I_1\) to the second mirror is also \(b\), and since \(I_1\) is \(2b\) away from \(O\), the total distance from \(O\) to \(I_4\) is:
\[ O I_4 = O I_1 + b = 2b + b = 3b \]

### Step 5: Calculate the Distance of \(I_3\) and \(I_4\) from \(O\)
Now we calculate the distances:
- The distances from the object \(O\) to the images \(I_3\) and \(I_4\) are both \(3b\).

### Summary of Distances
- The distance from the object \(O\) to images \(I_1\) and \(I_2\) is \(2b\).
- The distance from the object \(O\) to images \(I_3\) and \(I_4\) is \(4b\).

### Final Answer
The distance of the images from the object is:
- For \(I_1\) and \(I_2\): \(2b\)
- For \(I_3\) and \(I_4\): \(4b\)