SAT Physics Geometric Optics - Thin Lenses

SAT Physics Geometric Optics - Thin Lenses

THIN LENSES
Lenses use refraction to change the way an object appears. When light traveling in air enters the denser lens, it slows and bends due to refraction. When it exits the lens and moves back into air, the light speeds up and experiences a second refraction. The type and degree of curvature of the two lens surfaces dictate where the image will appear and how it will be magnified.
    Beginning physics introduces the geometric optics of extremely simplified lenses. The surfaces of these lenses are spherical. As a result, they are not perfect. The images formed by spherical lenses have a slight aberration, causing the images to lack sharpness. The spheres forming these lenses are the same size, creating symmetrical lenses. In addition, the lenses are assumed to be extremely thin even though they may not appear very thin in diagrams. These assumptions greatly simplify lens mathematics.
    Figure 16.7 demonstrates the symmetrical and spherical nature of a simple Convex lens. Although the diagram appears to be two circles, the lens is actually formed by the intersection of the two spheres. The optical axis is a horizontal line running through the center of the lens. All vertical measurements are made from the optical axis. A vertical line (usually not drawn) also passes through the center of the lens. All horizontal measurements are made from the center of the lens. Two key points, f and 2f are shown on both sides of the lens. The symbols f and 2f also represent distances measured from the center of the lens to these key points. The point 2f is at the center of curvature of each sphere. The distance 2f is, therefore, equal to the radius, R. As a result, the focal length, f can be calculated as follows.

This means that the focal point, f is located midway between the center of curvature and the surface of the lens. The lenses are considered so thin that they have negligible thickness (despite the obvious thickness shown in lens diagrams). As a result, the focal distance is measured from the center of the lens rather than the lens surface. Although only a convex lens is shown in Figure 16.7, the focal length formula holds true for all spherical lens and mirrors discussed in this chapter.

Figure 16.7. A simple convex lens

Converging Lenses, Object Outside f
Converging lenses bring light together. The convex lens is a converging lens. The location and size of the image formed by a lens or mirror can be plotted using ray tracing. In ray tracing, an object is represented by a vertical arrow extending upward from the optical axis, as shown in Figure 16.8. Two different light rays, both starting at the top of the object (tip of the vertical arrow), are drawn moving through the lens. Where they intersect is the location where the image will be focused. Look at the figure and trace the following key rays.

• Light parallel to the optical axis converges on the far focal point.
• Light that passes through the center of the lens moves in a straight line.
• Light passing through the focus exits parallel to the optical axis.

Figure 16.8 shows these two key rays of light originating at the object and intersecting at the image formed by this converging convex lens.

Figure 16.8. Convex lens

The rules for positive and negative signs for d₀, d_i, h₀, and h_t are the same as those for the pinhole camera, discussed earlier. There is one additional sign to consider. Converging optical instruments, such as the convex lens in Figure 16.8, have a positive focal distance (+f). When the object is positioned outside of the focal point, f, the image will be inverted and real. For example, movies are projected on a screen. Therefore, they are real images and inverted. In order to show a movie upright in the theater, the film must be fed into the projector inverted.
    There is a key geometric relationship between the focal length, f, of the lens, the object distance, d₀,and the image distance, d_i:

All of these variables are in the denominator and require a value to be inverted during the solution process. For example, when solving for f, the answer for 1/f will be one of the available choices. After solving for 1/f, remember to invert to find f. The magnification formula and the sign conventions discussed in the pinhole camera section also apply to lenses and mirrors.

Table 16.2 summarizes the signs for the variables in Figure 16.8 and in the equations shown above.

Table 16.2 A Converging Convex Lens with an Object Outside of f

Problems on the exam may simply address trends as the object is moved either toward the focal point or away from the focal point. For the converging convex lens, remember the following points:

1. Object outside of 2f : Small image (M< 1) and inside 2f on the far side.
2. Object at 2f : Image and object are the same size (M = 1) and at 2f on the far side.
3. Object between 2f and f : Large image (M> 1) and outside 2f on the far side.
4. As objects move toward f, the image distance and the image size increase.

Converging Lenses, Object at f
When the object is placed at the focal point of a converging lens, the ray traces are parallel to each other and never intersect. No image is formed. However, if this scenario is reversed and the object is positioned infinitely far away, then all the light rays arriving at the lens will be parallel to the optical axis. Every ray of light will converge at the focal point, creating an image at f. Wavelengths of light are so incredibly small that in comparison, an object 100 meters away might as well be at infinity. You should know that the image of a distant object will be located at the focal point on the opposite side of a converging convex lens.

Converging Lenses, Object Inside f
An interesting phenomenon occurs when the object is moved inside the focal point of a converging convex lens. In Figure 16.9, the ray traces used in Figure 16.8 do not intersect on the far side of the lens. However, if they are traced backward (dashed lines), then an intersection is found on the near side of the lens. This creates an upright image, which is a virtual image that cannot be projected onto a screen. The formulas used previously remain the same. However, the signs on the upright image height (+ h_i) and image distance (- d_i) reverse when the object moves inside the focal point. Even though the ray traces seem to diverge, this is still a converging convex lens with a positive focal point (+f). The actual light rays are still being converged on, and pass through, the focal point. The image will have the greatest magnification when the object is closest to the focal point. As the object is moved from the focal point, f, toward the lens, the image decreases in size and moves toward the lens. An example of this is the magnifying lens. An object placed between the focal point and magnifying lens will be upright, virtual, and magnified.

Figure 16.9 Magnifying convex lens

Table 16.3 summarizes the signs for the variables in Figure 16.9 and in the equations shown on page 325.

Table 16.3 A Converging Convex Lens with an Object Inside of f

Convex Lenses

The object viewed by a convex lens is positioned outside of the focus, as shown in the diagram above. Which of the following correctly describes the image?
(A) No image is formed
(B) Real and upright
(C) Real and inverted
(D) Virtual and upright
(E) Virtual and inverted
 
WHAT'S THE TRICK?

Convex lenses produce three possible outcomes. When the object is inside of f, the image is virtual and upright. When the object is at f, the image cannot be formed. When the object is outside of f, the image is real and inverted, which matches the scenario given in the problem. The answer is C.

Diverging Lenses
The concave lens is a diverging lens. Diverging optical instruments spread rays of light. Instead of the light converging on the far focus, it spreads out from the near focus, as shown in Figure 16.10 and summarized as follows.

• Light parallel to the optic axis diverges in a line from the near focal point.
• Light that passes through the center of the lens moves in a straight line.
• The resulting rays diverge and their back traces intersect. (Note that the back trace of the straight ray passing through the center of the lens coincides with the ray itself.)
                                        
                                                                              Figure 16.10. concave lens

The resulting image is upright and virtual. Again, all the equations and rules for variable signs are the same throughout this chapter. They are summarized in Table 16.4. Unlike the converging lens, the diverging concave lens is capable of producing only a small (M< 1) image that is upright and virtual. Whether the object is outside f at f or inside f has no bearing. As the object moves toward the lens, the image also moves toward the lens and becomes larger.

Table 16.4 A Diverging Concave Lens

Concave Lens

Which of the following is NOT true for a concave lens?
(A) Concave lens are divergent.
(B) The image is virtual.
(C) The image is upright.
(D) The image is larger than the object.
(E) The image forms on the near side of the lens.
 
WHAT'S THE TRICK?

In this case, knowing the characteristics of concave lenses is a must. Concave lenses are divergent. They can create only virtual images that are upright and appear on the near side of the lens. These lenses can create only images that are always smaller than the object regardless of the location of the object. Therefore, the correct answer is D.

SPHERICAL MIRRORS
Spherical mirrors are simply a small section of a single sphere that have a reflective surface. For concave mirrors, the reflective surface is the inside surface. For convex mirrors, it is the outside surface. Mirror optics problems are nearly identical to lens problems. The equations are the same. The rules for the variable signs are the same. The trends of image location are the same. However, there are a few key differences.

1. Although lenses consist of two intersecting spheres with two focal points, mirrors consist of a single spherical surface with only one focal point.
2. Although converging lenses are convex and diverging lenses are concave, mirrors are the opposite. Converging mirrors are concave, and diverging mirrors are convex.
3. Although light passes through a lens and creates real images on the far side and virtual images on the near side, mirrors reflect light back to the near side. For mirrors real images form on the near side and virtual images form on the far side.

Converging Mirrors, Object Outside of f
The concave mirror converges parallel rays of light through the focal point. Ray tracing for mirrors requires you to use a different strategy than that for a lens.

• Light parallel to the optical axis is reflected through the focus.
• Light through the focus reflects parallel to the optical axis.

The ray trace in Figure 16.11 demonstrates how the image can be found when the object is located outside the focus.

Figure 16.11. Concave mirror

The mathematics, signs, and trends, shown in Table 16.5, are identical to those of the converging lens with an object outside its focus.

Table 16.5 Converging Mirrors with the Object Outside of f

Converging Mirrors, Object at f
This is the same as for a converging lens. When the object is positioned at the focus, no image is formed since the light rays are parallel and cannot intersect. However, if the object is positioned far away, the light rays arriving at the mirror will be essentially parallel. Parallel rays striking a converging optical instrument are refracted and focused at the focal point, f. A practical example of this is the collection of light from a distant star through a concave, reflecting telescope.

Converging Mirrors, Object Inside of f
Just as with the converging lens, when the object is moved inside the focus, an upright, virtual image is created. The ray trace rules are a bit more complicated for this scenario and are depicted in Figure 16.12. The ray parallel to the optical axis reflects through the focus as before. However, a ray drawn from the tip of the object through the focus will not strike the mirror. A new ray must be drawn. The mirror is spherical and 2f is at the center of the sphere. Any ray of light starting at the center of a sphere, 2 f, will be reflected straight back to the center, 2f. A ray of light starting at 2f and passing through the very tip of the object will reflect right back toward 2f as shown in Figure 16.12. This ray diverges from the ray passing through f When back ray traces are drawn they intersect to form an upright image on the far side of the mirror. No light passes through the mirror. Although this image can be seen with the eye, it cannot be projected onto a screen. It is a virtual image.

Figure 16.12. Virtual Image in a concave mirror

All the equations, variable signs, and image trends are identical to those seen in the converging lens when the object is located inside of f. The main difference is that virtual images form on the far side of mirrors. Table 16.6 summarizes this information.

Table 16.6. Converging Mirrors with the Object Inside of f

Diverging Mirrors
The diverging mirror is very similar to the diverging lens. It is convex rather than concave. The ray trace in Figure 16.13 appears very different, but it uses the same logic as seen earlier.

1. Light parallel to the optical axis reflects in a line drawn from the focus.
2. Light aimed at the focus reflects parallel to the optical axis.
3. The reflected rays diverge so the image is formed by their back traces.

Figure 16.13. Diverging convex mirror

The resulting image is upright and virtual. It is very similar to the image seen in the diverging lens. The image is always small (M < 1), upright, and virtual. As the object moves toward the mirror, the image becomes larger and moves toward the mirror. The main difference is that this virtual image is on the far side of the mirror. Table 16.7 lists the characteristics of a diverging mirror.

Table 16.7. Characteristics of a Diverging Mirror

Convergent optical instruments, whether they are lenses or mirrors, share many characteristics. This is also true for divergent optical instruments. Table 16.8 summarizes the key characteristics for the lenses and mirrors tested in beginning physics.

Table 16.8. Key Characteristics of Lenses and Mirrors