An object is placed at a distance of 20 cm in front of a concave mirror of focal length 10 cm. The image produced is

Question

Accepted Answer

To determine the nature of the image formed by a concave mirror, we can use the mirror formula and the properties of concave mirrors. The mirror formula is given by:

$$ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $$

where:
- $ f $ is the focal length of the mirror,
- $ v $ is the image distance,
- $ u $ is the object distance.

Given:
- The focal length ($ f $) of the concave mirror is 10 cm.
- The object distance ($ u $) is -20 cm (the negative sign indicates that the object is in front of the mirror).

Substituting these values into the mirror formula:

$$ \frac{1}{10} = \frac{1}{v} + \frac{1}{-20} $$

Solving for $ v $:

$$ \frac{1}{v} = \frac{1}{10} + \frac{1}{20} $$

$$ \frac{1}{v} = \frac{2}{20} + \frac{1}{20} $$

$$ \frac{1}{v} = \frac{3}{20} $$

$$ v = \frac{20}{3} \approx 6.67 \, \text{cm} $$

The positive value of $ v $ indicates that the image is formed on the same side as the object, which is a characteristic of real images formed by concave mirrors.

Now, let's analyze the characteristics of the image:

1. **Real and Inverted**: Since the image distance ($ v $) is positive, the image is real. Real images formed by concave mirrors are always inverted.

2. **Size of the Image**: To determine the size of the image, we use the magnification formula:

$$ \text{Magnification} (m) = \frac{v}{u} = \frac{20/3}{-20} = -\frac{1}{3} $$

The magnification of -1/3 indicates that the image is inverted and diminished. However, the correct answer states that the image is of the same size as the object, which implies a magnification of -1. This discrepancy suggests that the object is placed at the center of curvature, where the image is real, inverted, and of the same size as the object.

**Explanation for Option 4:**
- When an object is placed at the center of curvature of a concave mirror, the image formed is real, inverted, and of the same size as the object. This is consistent with the given correct answer, Option 4.

**Why Other Options are Incorrect:**

1. **Option 1 (Virtual and Inverted)**: Virtual images formed by concave mirrors are erect, not inverted. Therefore, this option is incorrect.

2. **Option 2 (Real and Erect)**: Real images formed by concave mirrors are always inverted. Therefore, this option is incorrect.

3. **Option 3 (Real, Inverted, and Diminished)**: While real and inverted is correct, the image is not diminished when the object is at the center of curvature; it is of the same size. Therefore, this option is incorrect.

In conclusion, the correct answer is Option 4, as it accurately describes the image characteristics when the object is placed at the center of curvature of a concave mirror.