The minimum sum of the distances of a real object and a real image from a concave mirror of radius of curvature 20cm is:

Options

A. a.10 cm
B. b.40 cm
C. c.20 cm
D. d.none of these

Answer

To solve the problem of finding the minimum sum of the distances of a real object and a real image from a concave mirror with a radius of curvature of 20 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between radius of curvature and focal length**:
The focal length (f) of a concave mirror is given by the formula:
\[ f = \frac{R}{2} \]
where R is the radius of curvature. Here, \( R = 20 \, \text{cm} \), so:
\[ f = \frac{20}{2} = 10 \, \text{cm} \]

2. **Define object distance (u) and image distance (v)**:
For a concave mirror, the object distance (u) is taken as negative when the object is in front of the mirror. The image distance (v) is also negative for real images. We need to find the minimum value of \( |u| + |v| \).

3. **Use the mirror formula**:
The mirror formula relates the object distance (u), image distance (v), and focal length (f):
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

4. **Express u in terms of v**:
Rearranging the mirror formula gives:
\[ \frac{1}{u} = \frac{1}{f} - \frac{1}{v} \]
Therefore,
\[ u = \frac{fv}{v - f} \]

5. **Find the sum of distances**:
The sum of the distances can be expressed as:
\[ S = |u| + |v| = -u - v \]
Substituting \( u \) from the previous step:
\[ S = -\left(\frac{fv}{v - f}\right) - v \]

6. **Minimize the sum S**:
To find the minimum sum, we can differentiate \( S \) with respect to \( v \) and set the derivative to zero. However, a simpler approach is to recognize that the minimum occurs when the object and image distances are equal in magnitude, i.e., \( |u| = |v| \).

7. **Set u = -v**:
If we set \( |u| = |v| \), then:
\[ u = -v \]
Substituting into the mirror formula gives:
\[ \frac{1}{f} = \frac{1}{-v} + \frac{1}{v} \]
This leads to:
\[ \frac{1}{f} = 0 \quad \text{(which is not possible)} \]
Instead, we can find the minimum sum by recognizing that the minimum occurs when both distances are equal to twice the focal length.

8. **Calculate the minimum sum**:
Since \( f = 10 \, \text{cm} \), the minimum sum \( |u| + |v| \) occurs at:
\[ |u| + |v| = 2f + 2f = 4f = 4 \times 10 = 40 \, \text{cm} \]

### Final Answer:
The minimum sum of the distances of the real object and the real image from the concave mirror is:
\[ \boxed{40 \, \text{cm}} \]