1. Lenses
© 2007

2. Refraction in Prisms
Two prisms base to base
If we apply the laws of refraction to two prisms, the rays bend toward the base, converging light.
Parallel rays, however, do not converge to a focus, leaving images distorted and unclear.

3. Refraction in Prisms (Cont.)
Two prisms apex to apex
Similarly, inverted prisms cause parallel light rays to bend toward the base (away from the center).
Again there is no clear virtual focus, and once again, images are distorted and unclear.

4. Converging and Diverging Lens
If a smooth surface replaces the prisms, a well-defined focus produces clear images.
Converging Lens
Diverging Lens
Real focus
Virtual focus
Double-convex
Double-concave

5. The Focal Length of Lenses
Converging Lens
Diverging Lens
Focal length f F f - f +
The focal length f is positive for a real focus (converging) and negative for a virtual focus.

6. The Principal Focus
Since light can pass through a lens in either direction, there are two focal points for each lens.
Left to right
The principal focal point F is shown here. Yellow F is the other one. F
Right to left
Now suppose light moves from right to left instead . . . F F

7. Types of Converging Lenses
In order for a lens to converge light it must be thicker near the midpoint to allow more bending.
Plano-convex lens
Double-convex lens
Converging meniscus lens

8. Types of Diverging Lenses
In order for a lens to diverge light, it must be thinner near the midpoint to allow more bending.
Double-concave lens
Plano-concave lens
Diverging meniscus lens

9. Terms for Image Construction
The near focal point is the focus F on the same side of the lens as the incident light.
The far focal point is the focus F on the opposite side to the incident light.
Converging Lens
Diverging Lens F
Far focus F
Far focus F
Near focus F
Near focus

10. Image Construction:
Ray 1: A ray parallel to the lens axis passes through the far focus of a converging lens or appears to come from the near focus of a diverging lens.
Converging Lens
Diverging Lens
Ray 1
Ray 1 F F

11. Image Construction:
Ray 2: A ray passing through the near focal point of a converging lens or proceeding toward the far focal point of a diverging lens is refracted parallel to the lens axis.
Converging Lens
Diverging Lens F
Ray 1
Ray 2
Ray 2

12. Image Construction:
Ray 3: A ray passing through the center of any lens continues in a straight line. The refraction at the first surface is balanced by the refraction at the second surface.
Converging Lens
Diverging Lens F
Ray 1
Ray 2
Ray 3
Ray 3

13. Images’ Tracing Points
Draw an arrow to represent the location of an object, then draw any two of the rays from the tip of the arrow. The image is where lines cross.
1. Is the image erect or inverted?
2. Is the image real or virtual?
Real images are always on the opposite side of the lens. Virtual images are on the same side.
3. Is it enlarged, diminished, or same size?

14. Object Outside 2F F 2F Real; inverted; diminished
1. The image is inverted; i.e., opposite to the object orientation.
2. The image is real; i.e., formed by actual light rays in front of mirror.
3. The image is diminished in size; i.e., smaller than the object.
Image is located between F and 2F

15. Object at 2F F 2F Real; inverted; same size
1. The image is inverted; i.e., opposite to the object orientation.
2. The image is real; i.e., formed by actual light rays in front of the mirror.
3. The image is the same size as the object.
Image is located at 2F on other side

16. Object Between 2F and F F 2F Real; inverted; enlarged
1. The image is inverted; i.e., opposite to the object orientation.
2. The image is real; formed by actual light rays on the opposite side
3. The image is enlarged in size; i.e., larger than the object.
Image is located beyond 2F

17. Object at Focal Length F
Parallel rays; no image formed
When the object is located at the focal length, the rays of light are parallel. The lines never cross, and no image is formed.

18. Object Inside F F 2F Virtual; erect; enlarged
1. The image is erect; i.e., same orientation as the object.
2. The image is virtual; i.e., formed where light does NOT go.
3. The image is enlarged in size; i.e., larger than the object.
Image is located on near side of lens

19. Review of Image Formations
Parallel rays; no image formed F 2F
Real; inverted; enlarged F 2F
Real; inverted; same size
Object Outside 2F Region F 2F
Real; inverted; diminished F 2F
Virtual; erect; enlarged

20. Diverging Lens Imaging
All images formed by diverging lenses are erect, virtual, and diminished. Images get larger as object approaches.
Diverging Lens F
Diverging Lens F

21. Analytical Approach to ImagingF 2F so f si ho
-hi
Lens Equation:
Magnification:

22. Same Sign Convention as For Mirrors
1. Object so and image si distances are positive for real and images negative for virtual images.
2. Image height hi and magnifi-cation M are positive for erect negative for inverted images.
3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.

23. Example 1. A magnifying glass consists of a converging lens of focal length 25 cm. A bug is 8 mm long and placed 15 cm from the lens. What are the nature, size, and location of the image? F
so = 15 cm; f= 25 cm
si = cm
The fact that si is negative means that the image is virtual (on same side as object).

24. Example 1 Cont.) A magnifying glass consists of a converging lens of focal length 25 cm. A bug is 8 mm long and placed 15 cm from the lens. What is the size of the image? F
so = 15 cm; si = cm ho hi
hi = +20 mm
The fact that hi is positive means that the image is erect. It is also larger than object.

25. Example 2: What is the magnification of a diverging lens (f = -20 cm) if the object is located 35 cm from the center of the lens? F
First we find si then M
si = cm
M =

26. Example 3: Derive an expression for calculating the magnification of a lens when the object distance and focal length are given.
From last equation: si = -soM
Substituting for si in second equation gives . . .
Thus, . . .
Use this expression to verify answer in Example 4.

27. The principal focus is denoted by the red F.
Summary
A converging lens is one that refracts and converges parallel light to a real focus beyond the lens. It is thicker near the middle. F
The principal focus is denoted by the red F. F
A diverging lens is one that refracts and diverges parallel light which appears to come from a virtual focus in front of the lens.

28. Summary of Math Approachso f si ho
-hi
Lens Equation:
Magnification:

29. Summary of Sign Convention
1. Object so and image si distances are positive for real and images negative for virtual images.
2. Image height hi and magnifi-cation M are positive for erect negative for inverted images.
3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.