Physics Worksheet 12B: Light, Refraction, and Thin Converging Lenses

Focal Length Determination via Autocollimation

In the study of optics, specifically concerning thin converging lenses, a common experimental setup involves a light source, a convex lens, and a plane mirror. When a light source ( $S$ ) is positioned in front of a convex lens and its image is observed to coincide exactly with the position of the source itself, several physical conditions are met. This method is often referred to as the autocollimation method for finding the focal length.

According to the provided diagram in Worksheet 12B, the light source ( $S$ ) is placed at a specific distance from a convex lens, with a plane mirror positioned behind the lens. The diagram indicates that the distance from the source ( $S$ ) to the convex lens is $20\,cm$ , and the total distance from the source ( $S$ ) to the plane mirror is $35\,cm$ . It follows that the distance between the convex lens and the plane mirror is $35\,cm - 20\,cm = 15\,cm$ .

For the image to coincide with the object ( $S$ ), the light rays originating from $S$ must pass through the lens and emerge as a beam of parallel rays. When these parallel rays strike the plane mirror at a normal incidence (an angle of $90^\circ$ to the mirror surface), they are reflected directly back along their original paths. Upon passing back through the convex lens in the reverse direction, these parallel rays converge at the principal focus of the lens. Since the rays were made parallel by the lens initially, the object must have been placed at the focal point. Therefore, the focal length ( $f$ ) of the convex lens is exactly equal to the distance between the source and the lens, which is $f = 20\,cm$ .

Image Displacement and Point Source Dynamics

The behavior of images formed by a thin converging lens ( $L$ ) depends explicitly on the position of the point light source relative to the lens's optical center and principal axis. In the provided scenario, a point light source ( $P$ ) is placed in front of a lens, forming a real image at point ( $Q$ ).

When the light source is moved from position $P$ to a new position $P'$ (where $P'$ is further away from the lens than $P$ ), the physical location of the image must shift in accordance with the thin lens equation: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ . Here, $u$ represents the object distance, $v$ represents the image distance, and $f$ is the focal length. If the object distance ( $u$ ) increases (moving $P$ to $P'$ ), the value of $\frac{1}{u}$ decreases. To maintain the equality with the constant $\frac{1}{f}$ , the value of $\frac{1}{v}$ must increase, which means the image distance ( $v$ ) must decrease. Thus, the image moves closer to the lens.

Furthermore, for a converging lens, the image of a point source located above the principal axis will be formed below the principal axis due to the inversion property of real images. If $P'$ is moved further from the lens and potentially shifted vertically, the image will move towards the lens and in the opposite vertical direction relative to the principal axis. In the multiple-choice selection provided, moving the source to $P'$ results in the image moving to point $C$ , consistent with the geometric optics of real image formation beyond the focal length.

Characteristics of Images Formed by Distant Objects

Convex lenses are used to focus light rays from various distances, and the nature of the resulting image depends on the object's proximity to the lens relative to its focal length ( $f$ ) and twice the focal length ( $2f$ ).

When a convex lens is positioned to focus light rays from a distant object (often considered to be at infinity), the incident rays are essentially parallel to each other. These rays are refracted by the lens and converge at the principal focus. The resulting image exhibits three primary characteristics:

It is a real image because the light rays actually converge at a point and can be captured on a screen.
It is an inverted image, which is a standard property of real images produced by a single thin converging lens when the object is located beyond the focal length.
It is diminished in size. The statement that the image is the "same size as the object" only occurs when the object is placed exactly at $2f$ . For a distant object, the image is much smaller than the original object.

Based on these principles, specifically for a distant object, only the descriptions that the image is inverted and real are correct. This corresponds to the selection comprising points 2 and 3 only.