an object is placed in front of convex mirror at a distance of 50cm . a plane mirror is introduced covering the lower half of the convex mirror. if the distance between the object and the plane mirror is 30cm .it is found that there is no gap between the images formed by two mirrors. the radius of convex mirror is ..

To solve this problem, we need to analyze the situation involving both the convex mirror and the plane mirror. Let's break it down step by step to find the radius of the convex mirror.

Understanding the Setup

We have an object placed 50 cm in front of a convex mirror. When a plane mirror is introduced, it covers the lower half of the convex mirror, and the distance between the object and the plane mirror is 30 cm. The key detail is that there is no gap between the images formed by the two mirrors.

Image Formation by the Convex Mirror

First, we need to determine the image formed by the convex mirror. The formula for the image distance (v) in a convex mirror is given by:

• 1/f = 1/v + 1/u

Here, f is the focal length, v is the image distance, and u is the object distance. For a convex mirror, the focal length is positive, and since the object is placed at 50 cm, we have:

• u = -50 cm (the object distance is taken as negative in mirror conventions)

Finding the Focal Length

We need to express the focal length in terms of the radius of curvature (R) of the convex mirror. The relationship is:

• f = R/2

Now, substituting this into the mirror formula:

• 1/(R/2) = 1/v - 1/50

Image Formation by the Plane Mirror

Next, we consider the plane mirror. The distance between the object and the plane mirror is 30 cm, meaning the image formed by the plane mirror will be at a distance of 30 cm behind the mirror. Since the plane mirror reflects the image directly, the image distance for the plane mirror is:

• v_plane = +30 cm

Equating the Image Distances

Since there is no gap between the images formed by the two mirrors, the image distance from the convex mirror must equal the image distance from the plane mirror. Therefore, we can set:

• v_convex = v_plane

Substituting the values, we have:

• v = 30 cm

Solving for the Radius of Curvature

Now we can substitute this value back into the mirror formula:

• 1/(R/2) = 1/30 + 1/50

Finding a common denominator (which is 150) gives:

• 1/(R/2) = (5 + 3)/150 = 8/150

Thus, we can simplify this to:

• 1/(R/2) = 4/75

Taking the reciprocal, we find:

• R/2 = 75/4

Multiplying both sides by 2 gives:

• R = 75/2 = 37.5 cm

Final Result

The radius of curvature of the convex mirror is therefore 37.5 cm. This value indicates how curved the mirror is, which affects the way it reflects light and forms images.