The radius of curvature of a convex mirror is 60cm. When an object is A, its image is formed at B. If the size of image is half that of the object, then the distance between A and B is

Options

A. 30cm
B. 60cm
C. 45 cm
D. 90 cm

Explanation

To solve the problem step by step, we will use the properties of mirrors, particularly convex mirrors, and the formulas related to focal length, magnification, and the mirror equation.

### Step 1: Determine the Focal Length
The radius of curvature (R) of the convex mirror is given as 60 cm. The focal length (f) of a mirror is related to its radius of curvature by the formula:
\[ f = \frac{R}{2} \]
Substituting the value:
\[ f = \frac{60 \, \text{cm}}{2} = 30 \, \text{cm} \]

### Step 2: Understand the Magnification
The problem states that the size of the image is half that of the object. Magnification (m) is defined as:
\[ m = \frac{\text{Height of Image}}{\text{Height of Object}} \]
Since the image is half the size of the object, we have:
\[ m = \frac{1}{2} \]
For mirrors, magnification is also given by:
\[ m = -\frac{v}{u} \]
where \( v \) is the image distance and \( u \) is the object distance. Therefore, we can write:
\[ -\frac{v}{u} = \frac{1}{2} \]
This implies:
\[ v = -\frac{1}{2} u \]

### Step 3: Use the Mirror Equation
The mirror equation is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Substituting the focal length and the expression for \( v \):
\[ \frac{1}{30} = \frac{1}{u} + \frac{1}{-\frac{1}{2}u} \]
This simplifies to:
\[ \frac{1}{30} = \frac{1}{u} - \frac{2}{u} \]
\[ \frac{1}{30} = -\frac{1}{u} \]
Thus, we can find \( u \):
\[ u = -30 \, \text{cm} \]

### Step 4: Calculate the Image Distance
Using the relationship \( v = -\frac{1}{2} u \):
\[ v = -\frac{1}{2} \times (-30) = 15 \, \text{cm} \]
This means the image is formed 15 cm behind the mirror.

### Step 5: Calculate the Distance Between A and B
The distance between points A (object) and B (image) is given by the absolute values of the object distance and the image distance:
\[ \text{Distance} = |u| + |v| \]
Substituting the values:
\[ \text{Distance} = |-30| + |15| = 30 + 15 = 45 \, \text{cm} \]

### Final Answer
The distance between A and B is **45 cm**.

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