Thin Lenses

This section explores the behavior of thin converging and diverging lenses when a parallel beam of light passes through them. We will cover the principles of refraction and how lenses manipulate light to focus or spread it.

Converging Lenses

Action on a Parallel Beam of Light

A converging lens is thicker in the middle than at the edges. When a parallel beam of light strikes a converging lens, the rays are refracted (bent) towards a single point on the other side of the lens. This point is called the focal point (F). The distance from the lens to the focal point is the focal length (f).

The path of light rays passing through a converging lens can be visualized as follows:

1. Rays parallel to the principal axis are refracted towards the principal axis, passing through the focal point.
2. Rays passing through the center of the lens are not refracted.
3. Rays passing through the edge of the lens are refracted outwards.

Focal Length and Object Distance

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

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

Where:

• f is the focal length of the lens (in meters).
• u is the distance of the object from the lens (in meters).
• v is the distance of the image from the lens (in meters).

The sign convention for distances is important:

• Object distances (u) are typically taken as negative if the object is on the same side of the lens as the incoming light.
• Image distances (v) are positive if the image is on the opposite side of the lens from the object; otherwise, they are negative.

Image Formation with Converging Lenses

Converging lenses can form real or virtual images depending on the object's position relative to the focal point.

• Object beyond 2f: Real, inverted, and diminished image.
• Object at 2f: Real, inverted, and the same size as the object.
• Object between f and 2f: Real, inverted, and magnified image.
• Object at f: No image is formed.
• Object between the lens and f: Virtual, upright, and magnified image.

Diverging Lenses

Action on a Parallel Beam of Light

A diverging lens is thinner in the middle than at the edges. When a parallel beam of light strikes a diverging lens, the rays are refracted outwards, as if they originated from a virtual focal point on the same side of the lens as the incoming light.

The path of light rays passing through a diverging lens can be visualized as follows:

1. Rays parallel to the principal axis are refracted outwards, appearing to diverge from a point behind the lens.
2. Rays passing through the center of the lens are not refracted.
3. Rays passing through the edge of the lens are refracted inwards.

Focal Length and Object Distance

Diverging lenses also have a focal length (f), which is negative. The relationship between the focal length (f), the object distance (u), and the image distance (v) for a thin lens is given by the lens formula:

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

The sign convention for distances remains the same as for converging lenses.

Image Formation with Diverging Lenses

Diverging lenses always form virtual, upright, and diminished images, regardless of the object's position.

Summary

Understanding how thin lenses affect parallel beams of light is fundamental to optics. Converging lenses focus light to form real or virtual images, while diverging lenses spread light, always forming virtual, upright, and diminished images. The lens formula provides a quantitative relationship between the object, image, and focal distances.