Understanding Concave Mirror Image Formation

This question asks about the characteristics of the image formed by a concave spherical mirror when an object is placed at a specific distance from the mirror's pole. We are given the radius of curvature and the object distance. To find the image position, size, and nature, we need to use the mirror formula and the magnification formula.

Calculating Focal Length of the Concave Mirror

The relationship between the focal length (\(f\)) and the radius of curvature (\(R\)) for a spherical mirror is given by \(f = R/2\). For a concave mirror, the focal length is considered negative according to the sign convention when the object is real.

Given:

• Radius of curvature, \(R = 30 \text{ cm}\)

So, the focal length is:

\[ f = \frac{-R}{2} = \frac{-30 \text{ cm}}{2} = -15 \text{ cm} \]

The focal length of the concave mirror is -15 cm.

Applying the Mirror Formula

The mirror formula relates the object distance (\(u\)), the image distance (\(v\)), and the focal length (\(f\)) of a spherical mirror:

\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]

We are given that the object is placed 15 cm away from the pole in front of the mirror. According to the sign convention, the object distance (\(u\)) for a real object placed in front of the mirror is negative.

Given:

• Object distance, \(u = -15 \text{ cm}\)
• Focal length, \(f = -15 \text{ cm}\) (calculated above)

Substitute these values into the mirror formula:

\[ \frac{1}{v} + \frac{1}{-15 \text{ cm}} = \frac{1}{-15 \text{ cm}} \]

\[ \frac{1}{v} - \frac{1}{15} = -\frac{1}{15} \]

To find \(1/v\), rearrange the equation:

\[ \frac{1}{v} = -\frac{1}{15} + \frac{1}{15} \]

\[ \frac{1}{v} = 0 \]

This result means that \(v\) approaches infinity (\(v \rightarrow \infty\)).

Determining Image Position and Nature

The image distance being at infinity (\(v \rightarrow \infty\)) indicates that the image is formed very far away from the mirror. For a real object placed at the focal point of a concave mirror, the rays of light after reflection become parallel, forming an image at infinity. When an image is formed at infinity by a concave mirror with a real object, the image is real and inverted.

Determining Image Size (Magnification)

The linear magnification (\(m\)) of the image is given by the formula:

\[ m = -\frac{v}{u} \]

In this case, \(u = -15 \text{ cm}\) and \(v \rightarrow \infty\).

\[ m = -\frac{\infty}{-15} = \infty \]

A magnification of infinity (\(m \rightarrow \infty\)) means that the image is highly enlarged.

Summary of Image Characteristics

Based on our calculations for the concave mirror:

• Position: At infinity
• Nature: Real and Inverted
• Size: Highly Enlarged

Analyzing the Options

Let's compare our findings with the given options for the position, size, and nature of the image formed by the concave mirror:

Our calculated characteristics (Position: At infinity, Size: Highly enlarged, Nature: Real and inverted) match Option 2 exactly.

Conclusion

When an object is placed at the focal point of a concave spherical mirror, the image is formed at infinity. This image is highly enlarged, real, and inverted. The given object distance (15 cm) is equal to the focal length (half of the 30 cm radius of curvature), confirming the object is at the focal point.

Revision Table: Concave Mirror Image Formation

Additional Information on Concave Mirrors

Concave mirrors are converging mirrors, meaning they tend to converge parallel rays of light to a single point (the principal focus). They are used in various applications, including:

• Shaving mirrors: Used to get an enlarged, virtual image when the face is placed close to the mirror (between P and F).
• Headlights of cars/torches: A bulb is placed at the focus so that the light rays after reflection become parallel, forming a strong beam.
• Reflecting telescopes: Large concave mirrors are used to gather light from distant stars.
• Solar furnaces: Used to concentrate sunlight at the focus to produce heat.

Understanding the position of the object relative to the focal point and the center of curvature is key to predicting the characteristics of the image formed by a concave mirror.