A convex lens forms a real image of a point object at a distance of 50 cm from the convex lens. A concave lens is placed 10 cm behind the convex lens on the image side. On placing a plane mirror on the image side and facing the concave lens, it is observed that the final image now coincides with the object itself. The focal length of the concave lens is

Options

A. 50 cm
B. 20 cm
C. 40 cm
D. 25 cm

Answer

To find the focal length of the concave lens in the given scenario, we can follow these steps:

### Step 1: Understand the setup
A convex lens forms a real image of a point object at a distance of 50 cm. A concave lens is placed 10 cm behind the convex lens on the image side. A plane mirror is placed facing the concave lens, and it is observed that the final image coincides with the object itself.

### Step 2: Identify the distances
- The distance from the convex lens to the object (U) is -50 cm (the object distance is negative in lens formula).
- The distance from the convex lens to the concave lens is 10 cm, so the distance from the concave lens to the image formed by the convex lens is:
\[ U_{concave} = 50 \text{ cm} - 10 \text{ cm} = 40 \text{ cm} \]
Since the image formed by the convex lens acts as a virtual object for the concave lens, we take this distance as positive. Thus, \( U_{concave} = +40 \text{ cm} \).

### Step 3: Determine the image distance for the concave lens
When the plane mirror is placed, it reflects the light back through the concave lens. The image formed by the concave lens is at infinity because the final image coincides with the object itself. Therefore, we can say:
\[ V_{concave} = \infty \]

### Step 4: Use the lens formula for the concave lens
The lens formula is given by:
\[ \frac{1}{F} = \frac{1}{V} - \frac{1}{U} \]
Substituting the values we have:
\[ \frac{1}{F} = \frac{1}{\infty} - \frac{1}{40} \]
Since \( \frac{1}{\infty} = 0 \), we can simplify this to:
\[ \frac{1}{F} = 0 - \frac{1}{40} = -\frac{1}{40} \]

### Step 5: Calculate the focal length
From the equation above, we can find the focal length \( F \):
\[ F = -40 \text{ cm} \]
The negative sign indicates that it is a concave lens.

### Conclusion
The focal length of the concave lens is \( 40 \text{ cm} \).

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