An object is placed in front of a 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 parallax between the images formed by the two mirrors. What is the radius of curvature of the convex mirror?

Options

A. 25cm
B. 7cm
C. 18cm
D. 27cm

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the setup
We have an object placed in front of a convex mirror at a distance of 50 cm. A plane mirror is introduced that covers the lower half of the convex mirror. The distance between the object and the plane mirror is 30 cm.

### Step 2: Determine the position of the plane mirror
Since the object is 50 cm away from the convex mirror and the plane mirror is 30 cm away from the object, we can find the distance from the convex mirror to the plane mirror:
- Distance from object to convex mirror = 50 cm
- Distance from object to plane mirror = 30 cm
- Therefore, distance from convex mirror to plane mirror = 50 cm - 30 cm = 20 cm.

### Step 3: Calculate the image position for the plane mirror
For a plane mirror, the image is formed at the same distance behind the mirror as the object is in front of it. Thus, if the object is 30 cm away from the plane mirror, the image formed by the plane mirror will also be 30 cm behind it:
- Distance of image from the plane mirror = 30 cm
- Therefore, the image distance from the convex mirror = 20 cm (to the mirror) + 30 cm (behind the mirror) = 50 cm.

### Step 4: Use the mirror formula for the convex mirror
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) is the focal length,
- \( v \) is the image distance,
- \( u \) is the object distance.

For the convex mirror:
- The image distance \( v = -50 \) cm (since the image formed by the convex mirror is virtual and behind the mirror).
- The object distance \( u = -50 \) cm (the object is in front of the mirror).

Substituting these values into the mirror formula:
\[ \frac{1}{f} = \frac{1}{-50} + \frac{1}{-50} \]
\[ \frac{1}{f} = -\frac{1}{50} - \frac{1}{50} = -\frac{2}{50} = -\frac{1}{25} \]
Thus, the focal length \( f = -25 \) cm (the negative sign indicates it is a convex mirror).

### Step 5: Calculate the radius of curvature
The radius of curvature \( R \) is related to the focal length by the equation:
\[ R = 2f \]
Substituting the value of \( f \):
\[ R = 2 \times (-25) = -50 \text{ cm} \]
Since the radius of curvature is a positive quantity, we take the absolute value:
\[ R = 50 \text{ cm} \]

### Final Answer
The radius of curvature of the convex mirror is **50 cm**.

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