Lenses
Converging and divergent lens
Converging lenses are thicker in the center and thinner at the edges, with a shape resembling a convex surface. These lenses have the ability to make parallel light rays refract and converge to a single point known as the focus. Depending on the distance between the object and the lens, a converging lens can form real and inverted images (when the object is beyond the focal point) or virtual and upright images (when the object is between the lens and the focal point).
Diverging lenses are thinner in the center and thicker at the edges, with a shape resembling a concave surface. These lenses cause parallel light rays to spread out as if they were emanating from a virtual focal point located on the same side of the lens as the light source. Diverging lenses always form virtual, upright, and smaller images than the object, as the rays do not converge but diverge.
Both lenses rely on the law of refraction of light, which describes how light rays change direction when passing from one medium to another (e.g., from air to glass or plastic). The curvature and refractive index of the lens determine how light rays are bent. The position of the images formed by these lenses can be calculated using optical equations such as the lens maker's equation:
$$\frac{1}{f} = \frac{1}{s} + \frac{1}{s^{\prime}}$$
where \(f\) is the focal length, \(s\) is the object distance, and \(s^{\prime}\) is the image distance.
Instructions
Adjust the gray sliders to modify the lens properties. The \(C\) parameter represents the 'convexity factor', which influences the shape of the lens: if it is positive, the lens becomes convergent; if negative, divergent. The values of \(n_{1}\) and \(n_{2}\) correspond to the refractive index of the external medium and the lens, respectively. Manipulate the blue dot of the curved slider to alter the direction of the incident light rays. Likewise, move the blue dot located on the lens to the left or right to adjust its thickness."
Two converging lens system
Instructions
Move the points on the arrow representing the object, the focal lengths marked along the horizontal axis, or the crosses at the center of the lenses to adjust the characteristics of the optical system. Distances indicated as \(o_{n}\)refer to object positions, \(i_{n}\)to image positions, and \(f_{n}\) to focal lengths.
To determine the distance of the second image, we use the following equations:
$$\begin{split} \frac{1}{o_{1}} + \frac{1}{i_{1}} &= \frac{1}{f_{1}}\\ \frac{1}{o_{2}} + \frac{1}{i_{2}} &= \frac{1}{f_{2}} \end{split}$$
and we add them:
$$\begin{split} \frac{1}{o_{1}} &+ \frac{1}{i_{1}} + \frac{1}{o_{2}} + \frac{1}{i_{2}} = \frac{1}{f_{1}} + \frac{1}{f_{2}},\\ \frac{1}{i_{2}} &= \frac{1}{f_{1}} + \frac{1}{f_{2}} - \left( \frac{1}{o_{1}} + \frac{1}{o_{2}} + \frac{1}{i_{1}} \right). \end{split}$$