Parallel rays from a faraway source strike a converging lens of focal length \(20 \mathrm{cm}\) at an angle of 15 degrees with the horizontal direction. Find the vertical position of the real image observed on a screen in the focal plane.

Optics is a branch of physics that deals with the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. It encompasses the study of visible, ultraviolet, and infrared light. When studying optics, we encounter concepts such as reflection, refraction, diffraction, and interference.

In problems involving optics, especially with lenses, we often examine how light rays change direction when passing through different mediums—like air to glass—and how this affects image formation. For our problem with the converging lens, the fundamental aspect of optics we are dealing with is refraction, which is the bending of light as it enters a medium where its speed is different.

The formation of real images by lenses is a core concept in optics. A real image is produced when light rays converge at a point after passing through the lens. In contrast to a virtual image, a real image can be projected onto a screen and is formed on the same side of the lens as the object.

A converging lens, also known as a convex lens, is thicker at the center than at the edges, causing incoming parallel rays to converge at a point known as the focal point. The distance between the lens and this focal point is the focal length. The real image is formed at the focal plane, which is perpendicular to the principal axis of the lens and passes through the focal point.

Lenses have specific properties that determine how they influence light passing through them. The focal length, which is the distance from the lens to the focal point, is one such property. It is a measure of how strongly the lens converges or diverges light.

A lens with a shorter focal length will have a stronger bending effect on incoming light, resulting in a shorter distance for light rays to converge and form an image. In the exercise provided, the converging lens has a focal length of 20 cm, illustrating that it draws parallel rays from a faraway source to meet at the focal point 20 cm from the lens on the principal axis. Furthermore, the lens' thickness, curvature, and the material from which it is made influence its behavior and effectively its focal length.

Trigonometry is an essential tool in physics, particularly in optics for analyzing light paths and image formation. It allows us to connect angles with side lengths of triangles—an invaluable relationship when determining dimensions that are not immediately apparent.

By applying trigonometric functions such as sine, cosine, and tangent, we can solve for unknown variables in optical systems. For example, in the given problem, we use the sine function due to the relationship it holds with the opposite side and hypotenuse of a right triangle. This relationship gives us a method to calculate the vertical position of an image formed by the lens when light enters at an angle. Here, the hypotenuse represents the focal length, and the angle given is the incident angle of the light ray with the horizontal, allowing us to determine the exact point at which the real image is formed on the screen, vertically offset from the principal axis.