An object is located in water \(30 \mathrm{cm}\) from the vertex of a convex surface made of Plexiglas with a radius of curvature of \(80 \mathrm{cm} .\) Where does the image form by refraction and what is its magnification? \(n_{\text {water }}=4 / 3\) and \(n_{\text {Plexiglas }}=1.65\)

The lens maker's formula is a critical equation in optics, especially when it comes to understanding refraction at spherical surfaces. It relates the radii of curvature of the lens surfaces, the refractive indices of the lens material and the surrounding medium, and the focal length of the lens. The formula is generally expressed as

Understanding the Formula

The typical form for a single surface lens in terms of refractive indices (n) and radius of curvature (R) is as follows:
\[\frac{n_2}{s_2} - \frac{n_1}{s_1} = \frac{n_2 - n_1}{R}\]
Here, \(n_1\) is the refractive index of the medium where the object is placed, \(n_2\) is the refractive index of the lens material, \(s_1\) is the object distance, \(s_2\) is the image distance, and R is the radius of curvature of the lens surface.

Application in Problem-solving

To use this formula in practical scenarios like in the given exercise, one must identify the known values and strategically resolve for the unknowns, such as the image distance. This formula forms the basis for designing lenses with precise focal points to create the desired image size and position.

Image distance in the context of optical physics is the distance between the image and a reference point on the optical device, often the center of the lens or the vertex of a curved mirror. It's a crucial factor in our understanding of how lenses and mirrors form images.

Calculating Image Distance

For refractive surfaces, the image distance \(s_2\) can be found using the lens maker's formula mentioned earlier. We determine how far the image lies from the refractive surface and hence where the observer would see the virtual or real image formed by the lens. This concept not only aids in academic problems but also plays a significant role in practical applications like photography, vision correction, and in designing optical instruments.

Significance in Optical Systems

Accurately calculating the image distance aids in producing clear images, focusing cameras, and ensuring precision in complex optical systems used in telescopes, microscopes, and other imaging equipment. The exercise example illustrates clearly how knowing the object distance, the refractive indices, and the curvature allows for precise determination of where an image will form.

Magnification is an optical measure indicating how much larger or smaller an image appears, compared to the object itself. It's a dimensionless figure which gives a sense of the scaling change due to an optical system.

Finding Magnification In Refraction

In the context of refraction at spherical surfaces, as demonstrated in the given exercise, magnification can be determined using the formula:
\[M = \frac{n_1 s_2}{n_2 s_1}\]
where \(n_1\) and \(n_2\) are the refractive indices, and \(s_1\) and \(s_2\) are the object and image distances respectively. When \(M\) is greater than 1, the image is magnified; when it is less than 1, the image is minified. A negative value indicates that the image is formed inverted.

Role in Optics

Magnification is key in various applications from reading small text with a magnifying glass to exploring the cosmos with a telescope. Each optical system is designed to achieve a specific magnification to suit its purpose. In the case of our exercise, the magnification tells us that the image of the object as seen in the Plexiglas will be somewhat smaller than the object itself, a crucial piece of information for understanding optical phenomena.