Understanding the Problem: Lens Properties and Image Formation

The question asks us to determine the focal length and the nature (converging or diverging) of a lens given specific information about an object and its image. We are told the object is placed 10 cm in front of the lens and the image formed is real, inverted, and of the same size as the object.

Analyzing the Image Characteristics

The properties of the image (real, inverted, and same size) give us crucial clues about the lens and the object/image positions:

• Real Image: A real image is formed when light rays actually converge at a point. Real images can be projected onto a screen. For a lens, real images are typically formed on the opposite side of the lens from the object.
• Inverted Image: An inverted image means the image is upside down relative to the object. Lenses that form real images always form inverted images.
• Same Size as Object: This means the magnification (ratio of image height to object height) is 1. However, since the image is inverted, the linear magnification \(m\) is \(-1\).

Relationship Between Object Distance, Image Distance, and Focal Length for Same Size Image

For a lens to form a real, inverted image of the same size as the object, the object must be placed at a specific distance from the lens. This occurs when the object is placed at a distance equal to twice the focal length (\(2F\)) from a converging lens. In this special case, the real and inverted image is also formed at a distance of \(2F\) on the opposite side of the lens.

Given that the object is placed 10 cm in front of the lens, and the image is of the same size, this 10 cm distance must correspond to \(2F\).

So, Object distance \(u = 10 \text{ cm}\).

Since the image is real and of the same size as the object, the image distance \(v\) must also be 10 cm.

For lens calculations using the standard sign convention:

• Object distance, \(u\), is taken as negative when placed in front of the lens (left side, typically): \(u = -10 \text{ cm}\).
• Image distance, \(v\), is positive for real images formed on the opposite side (right side, typically): \(v = +10 \text{ cm}\).

Using the Lens Formula to Find Focal Length

The relationship between object distance (\(u\)), image distance (\(v\)), and focal length (\(f\)) for a lens is given by the lens formula:

\(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\)

Now, substitute the values of \(u\) and \(v\) with their appropriate signs:

\(\frac{1}{f} = \frac{1}{10 \text{ cm}} - \frac{1}{-10 \text{ cm}}\)

\(\frac{1}{f} = \frac{1}{10 \text{ cm}} + \frac{1}{10 \text{ cm}}\)

\(\frac{1}{f} = \frac{1 + 1}{10 \text{ cm}}\)

\(\frac{1}{f} = \frac{2}{10 \text{ cm}}\)

\(\frac{1}{f} = \frac{1}{5 \text{ cm}}\)

Therefore, the focal length is:

\(f = 5 \text{ cm}\)

Determining the Nature of the Lens

The sign of the focal length tells us the nature of the lens:

• A positive focal length indicates a converging lens (convex lens).
• A negative focal length indicates a diverging lens (concave lens).

Since we calculated \(f = +5 \text{ cm}\), the lens is a converging lens.

Conclusion

Based on the calculations using the lens formula and the analysis of the image properties, the focal length of the lens is 5 cm and the lens is converging in nature.

Alternatively, recognizing the special case where the image is real, inverted, and the same size as the object, we know that the object must be placed at \(2F\) for a converging lens. Given the object distance is 10 cm, we have:

\(u = 2F\)

\(10 \text{ cm} = 2F\)

\(F = \frac{10 \text{ cm}}{2}\)

\(F = 5 \text{ cm}\)

This confirms the focal length is 5 cm. The formation of a real image implies it is a converging lens.

Revision Table: Lens Properties for Different Object Positions (Converging Lens)

Additional Information: Types of Lenses and Their Uses

Lenses are optical devices used to refract light and form images. They are primarily classified into two types based on their shape and how they affect parallel light rays:

1. Converging Lens (Convex Lens):

• Thicker in the middle and thinner at the edges.
• Converges parallel light rays to a point called the principal focus (F).
• Has a positive focal length.
• Can form both real and virtual images depending on the object's position.
• Used in magnifying glasses, cameras, telescopes, microscopes, and to correct hypermetropia (farsightedness).

2. Diverging Lens (Concave Lens):

• Thinner in the middle and thicker at the edges.
• Diverges parallel light rays; they appear to come from a point called the principal focus (F) on the same side as the object.
• Has a negative focal length.
• Always forms virtual, erect, and diminished images, regardless of the object's position.
• Used in peepholes in doors, flashlights, and to correct myopia (nearsightedness).