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 step by step, we will follow the information given in the question and the principles of optics.

### 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 placed such that it covers the lower half of the convex mirror, and the distance from the object to the plane mirror is 30 cm.

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

### Step 3: Find the image formed by the plane mirror
A plane mirror forms an image at the same distance behind the mirror as the object is in front of it. Therefore, the image formed by the plane mirror will be:
- Image distance (V_plane) = - Distance from the plane mirror to the object = -30 cm (the negative sign indicates that the image is virtual and located behind the mirror).

### Step 4: Find the image formed by the convex mirror
The image formed by the convex mirror can be calculated using the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
where:
- \( u = -50 \) cm (object distance for the convex mirror, negative as per sign convention),
- \( v \) is the image distance for the convex mirror (which we need to find),
- \( f \) is the focal length of the convex mirror.

### Step 5: Establish the condition of no parallax
Since there is no parallax between the images formed by the two mirrors, the image distance from the convex mirror must equal the distance of the image formed by the plane mirror:
- Therefore, \( v = -10 \) cm (the image distance from the convex mirror).

### Step 6: Substitute values into the mirror formula
Now we can substitute the known values into the mirror formula:
\[ \frac{1}{f} = \frac{1}{-10} + \frac{1}{-50} \]
Calculating the right-hand side:
\[ \frac{1}{f} = -\frac{1}{10} - \frac{1}{50} \]
Finding a common denominator (which is 50):
\[ \frac{1}{f} = -\frac{5}{50} - \frac{1}{50} = -\frac{6}{50} = -\frac{3}{25} \]
Thus,
\[ f = -\frac{25}{3} \text{ cm} \approx -8.33 \text{ cm} \]

### Step 7: Calculate the radius of curvature
The radius of curvature \( R \) is related to the focal length \( f \) by the formula:
\[ R = 2f \]
Substituting the value of \( f \):
\[ R = 2 \times -\frac{25}{3} = -\frac{50}{3} \text{ cm} \approx -16.67 \text{ cm} \]
Since the radius of curvature is taken as a positive quantity, we can state:
\[ R = \frac{50}{3} \text{ cm} \approx 16.67 \text{ cm} \]

### Final Answer:
The radius of curvature of the convex mirror is approximately \( 16.67 \) cm.