April 11 Physics 54 LectureProfessor Henry Greenside

Key Points from Previous Lecture

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Today’s TopicsConclusion of Chapter 33: “Reflection and Refraction”• Total internal reflection.

Chapter 34: “Lenses and Optical Instruments”• Converging versus diverging lenses.• The lensmaker’s equation for thin lenses.• The two foci of a thin lens.• The three-principal rays for predicting the image formed by a

thin lens.

Demo: Vanishing Beaker

Snell’s law makes an elementary but interesting prediction: you need to have a mismatch in the indices of refraction in order for direction of light rays to change. Similarly, the Maxwell equations predict that light will reflect at an interface only if the indices of refraction differ.

So what happens if an object is placed in a fluid of identical index of refraction? See movie at site: http://groups.physics.umn.edu/demo/optics/movies/6A4030.mov

Read: The Invisible Man by H. G. Wells.

Total Internal Reflection

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Underwater Internal Reflection

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Reflection From Water Surface

Applications of Internal Reflection

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Demo: Light Pipe Dipped in Mineral Oil

Fiber Optics Imply A Radio-Silent Earth

Arecibo Radio Antenna

Chapter 34: Lenses and Optical Instruments

Lenses With Spherical Surfaces Easiest to Grind

Easiest lenses to make have two different spherical sides, one from sphere of radius R1, other with radius of R2. These radii can be positive or negative, in which case one has respectively convex or concave lenses.

Converging and Diverging Kinds of Lenses

PRS: Air-Filled Bag as Underwater Lens

Which acts like a converging lens: (a) or (b)?

PRS: Air-Filled Bag as Underwater LensSolution

Which acts like a converging lens: Answer: (b)

Demo: Thin Lens Has Two Symmetric Foci

A lens has two foci (plural of focus) and they are symmetrically centered about lens provided that the lens is sufficiently thin.

Note: As shown, rays of different color refract slightly differently (“dispersion”) so do not quite come to focus at same point, even within the thin lens approximation. Such a distortion is called “chromatic aberration” and is important to avoid since the aberration causes a blurring of the optical image.

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Sign conventions: f > 0 : lens is converging f < 0 : lens is diverging convex: Ri > 0 concave: Ri < 0 flat: Ri = infinity

Equation is analogous to mirror result f=R/2.

Note symmetry of formula: a thin lens has the same focal length on both sides even if lens is

not symmetric (R1, R2 different).

How f of Thin Lens Determined? Lensmaker’s Equation

Thin lens, small angle approximations!

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Example of Lensmaker’s Equation: Convex Meniscus

Lens made from convex spherical surface with R1 = 22.4cm, concave spherical surface with R2 = -46.2. Deduce f ~ 87 cm. Since f >0, this is a converging lens.

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Watch out for minus sign if concave surface!

How to Make a Powerful Lens?1. Can create a powerful lens (short focal length f ) by making index of refraction n as big

as possible (think diamond!) or radii Ri of curvature of lens surfaces as small as possible, which means making the lens small and fat. This is how Leeuwenhoek achieved large magnifications with a single lens, also how modern microscopes achieve large magnification with their objective lenses. Obvious limitations: small image, image is dim (not much light gets through), diffraction causes blurring, hard to make.

2. Second approach: recognize that refraction occurs only at interface between two media and not inside given medium, so try making many successive thin layers with different indices of refraction. This is how the octopus eye achieves a short focal length with a big thin lens, an approach now being used by some digital camera makers.

Antoni van Leeuwenhoek Octopus Eye (1632-1723)

Ray Diagrams for Lenses

As for mirrors, you should be able to draw three of the infinitely many rays emerging from the tip of the object to identify the image: the ray parallel to the principal axis (P-ray); ray through center of lens (C-ray), and ray going through the nearest focus (F-ray). For sufficiently thin lens (small angles of refraction), these three rays intersect at about the same place, the tip of the image I.

Fig. 33-22 on p. 823 explains why C-ray of thin lens continues on about the same straight path for a sufficiently thin lens.

Java Applet for Lenses

http://www.phy.ntnu.edu.tw/java/Lens/lens_e.html

Please take time to play with this applet and study how varying the object size and position affects the image size, image location, and whether the image is virtual or real.

In this applet, you can change the lens from converging to diverging by clicking on the plus or minus sign for the focal length f (which reverses the sign of the focal length).

Make sure that you can confirm your qualitative observations via the thin-lens formula, 1/do + 1/di = 1/f. For example, if the focal length f is a positive constant (the case for a glass converging lens of fixed geometry), decreasing the object distance do causes the reciprocal 1/do to become larger, which means that 1/di must become smaller, which means di must become larger, i.e., moving an object closer to the focus of a converging lens causes the image to move further away from the focus and closer to the lens. (But what if do < f ?)

How to Find Focal Length of Thin Converging Lens