A converging lens of focal length 15 cm and a converging mirror of focal length 10cm are placed 50cm apart with common principal axis. A point source is placed in between the lens and mirror at a distance of 40 cm from the lens. Find the distance (in cm) between the two images formed .

To solve this problem, we need to analyze the behavior of light as it passes through the converging lens and reflects off the converging mirror. We will use the lens formula and the mirror formula to find the positions of the images formed by both optical elements and then calculate the distance between these two images.

Step 1: Understanding the Setup

We have a converging lens with a focal length (f_lens) of 15 cm and a converging mirror with a focal length (f_mirror) of 10 cm. The distance between the lens and the mirror is 50 cm. A point source is located 40 cm from the lens, which means it is 10 cm away from the mirror (since 50 cm - 40 cm = 10 cm).

Step 2: Finding the Image Formed by the Lens

We will use the lens formula, which is given by:

1/f = 1/v - 1/u

Where:

• f = focal length of the lens
• v = image distance from the lens
• u = object distance from the lens (taken as negative in the lens formula)

For our lens:

• f = 15 cm
• u = -40 cm (since the object is on the same side as the incoming light)

Plugging in the values:

1/15 = 1/v - 1/(-40)

Rearranging gives:

1/v = 1/15 + 1/40

Finding a common denominator (120):

1/v = 8/120 + 3/120 = 11/120

Thus, we find:

v = 120/11 ≈ 10.91 cm

This means the image formed by the lens is approximately 10.91 cm on the opposite side of the lens.

Step 3: Finding the Object Distance for the Mirror

Now, we need to determine the object distance for the mirror. Since the image formed by the lens acts as the object for the mirror, we calculate the distance from the mirror:

Distance from the lens to the mirror = 50 cm

Distance from the lens to the image = 10.91 cm

Thus, the object distance for the mirror (u_mirror) is:

u_mirror = 50 cm - 10.91 cm = 39.09 cm

Since this is on the same side as the incoming light, we take it as negative:

u_mirror = -39.09 cm

Step 4: Finding the Image Formed by the Mirror

Using the mirror formula:

1/f = 1/v' + 1/u'

Where:

• f = focal length of the mirror (10 cm)
• v' = image distance from the mirror
• u' = object distance from the mirror (-39.09 cm)

Plugging in the values:

1/10 = 1/v' - 1/39.09

Rearranging gives:

1/v' = 1/10 + 1/39.09

Finding a common denominator (390.9):

1/v' = 39.09/390.9 + 10/390.9 = 49.09/390.9

Thus, we find:

v' = 390.9/49.09 ≈ 7.96 cm

This means the image formed by the mirror is approximately 7.96 cm on the same side as the object (the original point source).

Step 5: Calculating the Distance Between the Two Images

Now, we have the positions of both images:

• Image from the lens: 10.91 cm from the lens (to the right)
• Image from the mirror: 7.96 cm from the mirror (to the left)

To find the distance between these two images, we need to consider their positions relative to a common reference point, which we can take as the lens:

Distance from the lens to the mirror = 50 cm

Distance from the lens to the image formed by the lens = 10.91 cm

Distance from the lens to the image formed by the mirror = 50 cm - 7.96 cm = 42.04 cm

Now, the distance between the two images is:

Distance = 42.04 cm - 10.91 cm = 31.13 cm

Final Result

The distance between the two images formed by the converging lens and the converging mirror is approximately 31.13 cm.