Show that placing a 1 -diopter lens in front of a 2 -diopter lens gives the equivalent of a single 3 -diopter lens (i.e., the powers of closely spaced lenses add).

Optics, a branch of physics, is the study of light and its behavior. Understanding optics involves exploring how light interacts with various media, like lenses, mirrors, and human eyes. The fundamental concept behind optics is refraction, where light bends as it passes through materials of different densities. Refraction is the principle that allows lenses to focus light, an essential function in devices such as cameras, glasses, and microscopes. Lenses come in different shapes: convex lenses, which curve outward and converge light, and concave lenses, which curve inward and diverge light.

When a beam of light passes through a lens, its path changes to meet at a focal point; the distance from this point to the lens is known as the focal length. By altering the focal length, one can adjust the degree to which light converges or diverges, which is key in improving vision or capturing a sharp photograph.

The lens power formula is instrumental in optometry and optical physics. It quantifies the degree to which a lens can bend or converge light rays. Defined in diopters (D), the power (P) of a lens is the reciprocal of its focal length (f) in meters. Mathematically, it is denoted as:
\[ P = \frac{1}{f} \]
In practice, this means a lens with a shorter focal length (indicating a strong bending force) has a higher power. Conversely, a lens with a longer focal length has a lower power. Diopters provide a convenient way to describe the refractive strength of lenses, especially when comparing or combining multiple lenses, as in prescription eyeglasses or complex optical systems.

The concept of equivalent focal length is particularly useful when dealing with multiple lenses in close proximity, as is common in optical devices like telescopes or binoculars. It refers to the single focal length that characterizes the combined effect of two or more lenses. To calculate the equivalent focal length (Feq) of two closely spaced lenses, the following formula is used:
\[ \frac{1}{F_{eq}} = \frac{1}{F_1} + \frac{1}{F_2} \]
where \(F_1\) and \(F_2\) are the focal lengths of the individual lenses. By combining their optical powers, you can obtain an \effective\ focal length that defines how the multi-lens system focuses light. This is key to designing lenses that work together to achieve a desired magnification or corrective vision property.

Applying this concept helps explain complex optical systems in simpler terms, making it easier to design and understand the behavior of lenses when they're used in combination.