A convex lens of focal length 20 cm is placed in front of convex mirror with principal axis coinciding each other. The distance between the lens and mirror is 10 cm. A point object is placed on principal axis at a distance of 60 cm from the convex lens. The image formed by combination coincides the object itself. The focal length of the convex mirror is _______ cm.

Let us analyze the given setup step by step. A convex lens of focal length 20 cm is placed in front of a convex mirror, with their principal axes coinciding. The distance between the convex lens and the convex mirror is given as 10 cm, and the point object is placed at a distance of 60 cm from the convex lens on the principal axis.

Step 1: Refraction through the Convex Lens

For the convex lens of focal length 20 cm, the lens formula is:

1/v - 1/u = 1/f

Here, u = -60 cm (negative as the object is on the left side of the lens), f = 20 cm, and we need to calculate v:

1/v - 1/(-60) = 1/20

Simplifying:

1/v + 1/60 = 1/20

Taking LCM and solving for v:

1/v = 1/20 - 1/60

1/v = (3 - 1)/60

1/v = 2/60

v = 30 cm

Thus, the image formed by the convex lens of focal length 20 cm is at a distance of 30 cm from the lens.

Step 2: Reflection from the Convex Mirror

The distance between the convex lens and the convex mirror is 10 cm. The image formed by the lens acts as an object for the convex mirror. For the final image to coincide with the original object, the light rays must strike the mirror perpendicularly. This means the image formed by the lens must be at the center of curvature of the convex mirror.

The distance to the center of curvature R can be calculated as:

R = v - distance between the lens and mirror

Substituting the values:

R = 30 cm - 10 cm = 20 cm

Step 3: Calculating the Focal Length of the Convex Mirror

The focal length f of the convex mirror is related to the radius of curvature R by the formula:

f = R/2

Substituting R = 20 cm:

f = 20/2 = 10 cm

Final Answer

The focal length of the convex mirror is 10 cm.

This solution demonstrates how the convex lens of focal length 20 cm and the convex mirror work together to ensure that the final image coincides with the object itself.