Lenses form images through the refraction of light, which is the bending of light rays as they pass through materials with different optical densities. Unlike mirrors that reflect light, lenses allow light to pass through and change direction according to the law of refraction. This fundamental difference is crucial in understanding how optical devices like telescopes, cameras, and microscopes create images.

Convex lenses, also known as converging lenses, bend parallel incoming light rays toward a focal point on the opposite side of the lens. To accurately determine the image formed by a convex lens, ray diagrams are used, which involve drawing at least two of three principal rays. The first ray, called the P-ray, travels parallel to the principal axis and then refracts through the far focal point. The second ray, the F-ray, passes through the near focal point on the same side as the object and then refracts parallel to the principal axis. The third ray, the M-ray, passes straight through the midpoint of the lens without bending significantly.

Where these rays intersect indicates the location of the image. When the object is placed beyond the focal length ($d_o > f$), the rays converge on the opposite side of the lens, producing a real and inverted image. The size of this image can vary—it may be reduced or enlarged depending on the object's distance from the lens. This relationship is governed by the lens formula:

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$

where $f$ is the focal length, $d_o$ is the object distance, and $d_i$ is the image distance. The magnification $M$ is given by:

$$M = -\frac{d_i}{d_o}$$

The negative sign indicates image inversion when the image is real.

When the object is placed closer to the lens than the focal length ($d_o < f$), the rays diverge on the opposite side, and the image cannot be formed by actual ray convergence. Instead, by tracing the refracted rays backward, they appear to originate from a point behind the lens. This creates a virtual, upright, and enlarged image. Virtual images cannot be projected onto a screen because the light rays do not actually meet but only appear to do so from the observer's perspective.

Sign conventions for convex lenses are similar to those used for mirrors, with the focal length $f$ being positive ($f > 0$) for convex lenses and negative ($f < 0$) for concave lenses. Object distances are positive when the object is on the incoming light side, and image distances are positive when the image forms on the opposite side of the lens.

Understanding these principles allows for the analysis and prediction of image formation in various optical systems, enhancing comprehension of how lenses manipulate light to produce real and virtual images with different orientations and sizes.

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