A bright object is placed on one side of a converging lens...

A bright object is placed on one side of a converging lens of focal length $f,$ and a white screen for viewing the image is on the opposite side. The distance $d_{\mathrm{T}}=d_{\mathrm{i}}+d_{\mathrm{o}}$ between the object and the screen is kept fixed, but the lens can be moved. ( $a )$ Show that if $d_{\mathrm{T}}>4 f,$ there will be two positions where the lens can be placed and a sharp image is formed. $(c)$ Determine a formula for the distance between the two lens positions in part $(a),$ and the ratio of the image sizes.

Key Concepts

Magnification and Image Size Ratio

Magnification, defined as the ratio of the image size to the object size, is given by the negative ratio of the image distance to the object distance. In this context, the different solutions for the lens position not only yield distinct object and image distances but also different magnifications, leading to a specific ratio of the image sizes. This concept is essential in understanding how the positioning of the lens affects the scale of the resulting image.

Discriminant Condition for Multiple Solutions

The quadratic equation derived from combining the thin lens equation with the fixed total distance constraint has a discriminant that determines the nature and number of its solutions. For the case of a converging lens with a fixed object?screen separation, a positive discriminant (d_T > 4f) guarantees two distinct lens positions that both produce a sharp image on the screen.

Conjugate Points

Conjugate points refer to the paired positions of the object and its corresponding sharp image as defined by the thin lens equation. When an object is placed at a certain distance from a lens, there exists a unique image location on the other side, and these positions are interchangeable in the lens formula, emphasizing the symmetry inherent in image formation.

Fixed Total Distance Constraint

In setups where the object and the screen are fixed in position, the sum of the object distance and the image distance is held constant. This fixed total distance introduces a constraint that, when combined with the thin lens equation, leads to a quadratic relation which can yield two valid solutions for the lens position, provided the constraint is suitably satisfied relative to the lens’s focal length.

Thin Lens Equation

This fundamental relation in geometric optics relates the object distance (d_o), the image distance (d_i), and the focal length (f) of a converging lens by the formula 1/f = 1/d_o + 1/d_i. It is the starting point for analyzing how lenses form images and is used to derive further conditions for image formation under various constraints.

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