Multi-surface problem in refraction at spherical surfaces.

[qweeerty]

Homework Statement

A thick-walled spherical shell made of transparent glass(n=1.50) has internal radius R and external radius 2R. The central cavity and region surrounding the shell are filled with
air (n=1.00). The center of symmetry of the object is located at the origin of an xy coordinate system. Paraxial light rays traveling parallel to the x-axis enter the sphere from the left as shown. These rays will refract four times before they passing entirely through the
spherical shell and emerge into the air beyond x = 2R. The final image is located at one of the two focal points of this complex optical system.

(a) Calculate the x-coordinates of the three intermediate image points and the final focal point, and identify each image as real or virtual. Note: the image due to each successive refraction acts as the object for the next refraction. You will find that the real image due to the first surface acts as a virtual object at the second surface, with negative object distance.

(b) Calculate the position of the focal point for parallel light entering from the left as before, but now with the interior region filled with glass (n=1.50) to make a homogenous solid glass sphere.

Homework Equations

n_a/s+n_b/s'=n_b-n_a/R
i/s+i/s'=i/f
m=-s'(n_a)/s(n_b)
n_asin$$\alpha$$=n_bsin$$\beta$$

The Attempt at a Solution

To be honest, I don't even know where to start with this. I just don't understand how you're supposed to get an image distance if you don't know s. I'm not looking for an answer, I just want to know how to get started.

 

[mark726]

To get started, you can begin by identifying what information is given in the problem and what information is being asked for. In this case, the problem provides the refractive indices (n) and radii (R) of the spherical shell and the surrounding medium, as well as the location of the light source (x=0). It also states that the final image is located at one of the two focal points, but does not specify which one.

The problem is asking for the x-coordinates of the three intermediate image points and the final focal point, as well as whether each image is real or virtual. It also asks for the position of the focal point for parallel light entering from the left when the interior region is filled with glass.

Next, you can think about what equations or principles may be relevant for solving this problem. Some possible equations to consider are Snell's law, the lens equation, and the magnification equation. You may also want to review the concept of refraction and how it affects the path of light rays passing through different mediums.

To solve the problem, you can start by considering the path of the light rays as they pass through the spherical shell and the surrounding medium. Use Snell's law to determine the angles of refraction at each interface, and use the lens equation to relate the object distance (s) and image distance (s') at each surface. Keep in mind that the image from the previous refraction acts as the object for the next refraction.

Once you have determined the image distances at each surface, you can use the magnification equation to determine the position of the intermediate image points and the final focal point. Remember to consider whether each image is real or virtual, and use the sign conventions for object and image distances.

For part (b), you can use the same approach but with the new refractive index for the interior region. This will change the angles of refraction and therefore the path of the light rays through the spherical shell.

It may also be helpful to draw a diagram of the situation to visualize the path of the light rays and keep track of the different distances and angles involved.