Radius of curvature of concave mirror is `40cm` and the size of image is twice as that of object, then the object distance is

Options

A. `60cm`
B. `20cm`
C. `40cm`
D. `30cm`

Explanation

To find the object distance for a concave mirror given the radius of curvature and the magnification, we can follow these steps:

### Step 1: Determine the Focal Length
The focal length (f) of a concave mirror is related to its radius of curvature (R) by the formula:
\[ f = -\frac{R}{2} \]
Given that the radius of curvature \( R = 40 \, \text{cm} \):
\[ f = -\frac{40}{2} = -20 \, \text{cm} \]

### Step 2: Use the Magnification Formula
The magnification (m) is given as the ratio of the height of the image (h') to the height of the object (h):
\[ m = \frac{h'}{h} \]
It is also related to the object distance (u) and the image distance (v) by the formula:
\[ m = -\frac{v}{u} \]
Given that the image size is twice that of the object, we have:
\[ m = 2 \]
Thus, we can write:
\[ 2 = -\frac{v}{u} \]
This implies:
\[ v = -2u \]

### Step 3: Use the Mirror Formula
The mirror formula relates the object distance (u), the image distance (v), and the focal length (f):
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Substituting \( v = -2u \) into the mirror formula gives:
\[ \frac{1}{-20} = \frac{1}{u} + \frac{1}{-2u} \]

### Step 4: Simplify the Equation
To simplify the equation:
\[ \frac{1}{-20} = \frac{1}{u} - \frac{1}{2u} \]
Finding a common denominator for the right side:
\[ \frac{1}{-20} = \frac{2 - 1}{2u} = \frac{1}{2u} \]
Thus, we have:
\[ \frac{1}{-20} = \frac{1}{2u} \]

### Step 5: Solve for Object Distance (u)
Cross-multiplying gives:
\[ 2u = -20 \]
\[ u = -10 \, \text{cm} \]

### Conclusion
The object distance is \( u = -10 \, \text{cm} \). The negative sign indicates that the object is placed in front of the mirror, which is consistent with the convention for concave mirrors.

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