lens that causes light rays parallel to central axis to converge Thin Lenses Lens Transparent object with two refracting surfaces whose central axes coincide Light refracts from air into lens, crosses through lens, refracts back into air Converging lens lens that causes light rays parallel to central axis to converge Diverging lens lens that causes light rays parallel to central axis to diverge

Demonstration: Lens board Converging Diverging Parallel ray refracts through the focal point. A ray through the focal point refracts parallel to the central axis. A ray through the center of curvature refracts straight.

Lens Equation ( < 0 ) True for thin lens and paraxial rays. magnification m = h’/h = - q/p

Thin Lenses f > 0 Convergent lens A lens is a piece of transparent material with two refracting surfaces whose central axes coincide. A lens is thin if its thickness is small compared to all other lengths (s, s’, radii of curvature). r is positive if the center of curvature is on the same side of the surface as the refracted light f > 0 Convergent lens r1>0 r2<0 Net convex – thicker in the middle Parallel rays converge to real focus. f > 0 Divergent lens f < 0 r1<0 r2>0 Net concave – thinner in the middle Parallel rays diverge from virtual focus. f < 0

Signs in the Lens Equation for Thin Lenses p is positive for real object q is positive for real image q is negative for virtual image m is positive if image is upright m is negative if image is inverted f is positive if converging lens f is negative if diverging lens p is negative for virtual object

Properties of Images - Summary For converging lenses ( f > 0): If the object is inside the focal point, the image is virtual (q < 0), enlarged, has the same orientation, and farther from the lens. If the object is outside the focal point, the image is real (q > 0), reduced or enlarged (depending on the object distance), inverted, and farther from or closer to the lens on the other side. If the object is at the focal point, no image is formed. For diverging lenses ( f < 0): The image is always virtual (q < 0), reduced, has the same orientation, and closer to the lens. Power of a lens = 1/f (m-1) (diopters or D)

Quiz Question 1 An object 1 cm tall is 20 cm in front (i.e., left) of a lens of focal length -20 cm and has an image 10cm also in front (left) of the lens. Which of the following is a correct description of the lens and its image? Diverging lens. The image size is magnified by a factor of 2. Image has the same orientation as the object . b). Diverging lens. The image size is magnified by a factor of 2 and inverted c). Diverging lens. The image size is reduced by a factor of 2. Image has the same orientation as the object . d). Diverging lens. The image size is reduced by a factor of 2 and inverted

Example: Simple Magnifying Lens You would like to read the fine print of a legal document. In order to enlarge it, you use a converging lens (|f| = 6 cm). If you place the lens 4 cm from the paper, what is the magnification of the image? p = + 4 cm f = + 6 cm q = ? m = ?

1/p + 1/q = 1/f q = pf/(p – f) q = (4cm)(6cm)/(4 cm – 6 cm) q = -12 cm Example: Solution 1/p + 1/q = 1/f q = pf/(p – f) q = (4cm)(6cm)/(4 cm – 6 cm) q = -12 cm m = - q/p m = - (-12cm)/(+4cm) m = 3

Lenses in Combination First lens: Second lens: ( < f2 here) What if p2> f2 ? Total transverse magnification In this example, p1> 0, q1> 0, p2> 0, q2< 0

An object 2 cm tall is 15 cm in front (i.e., left) of a Quiz Question 2 An object 2 cm tall is 15 cm in front (i.e., left) of a diverging lens of focal length −10 cm. Which of the following is a correct description of its image? a) The image is enlarged. b) The image is real. c) The image is upright. d) The image is in back (right) of the lens. e) None of the above is correct.

a) The image is diminished. b) The image is inverted. Quiz Question 3 An object 2 cm tall is 15 cm in front (i.e., left) of a converging lens of focal length 10 cm. Which of the following is a correct description of its image? a) The image is diminished. b) The image is inverted. c) The image is virtual. d) The image is in front (left) of the lens. e) None of the above is correct.

b) The image is inverted. c) The image is enlarged. Quiz Question 4 An object 2 cm tall is 10 cm in front (i.e., left) of a converging lens of focal length 15 cm. Which of the following is a correct description of its image? a) The image is real. b) The image is inverted. c) The image is enlarged. d) The image is in back (right) of the lens. e) None of the above is correct.

Aberrations nblue > nred Chromatic aberration correct Cameras, … correct nblue > nred Spherical aberration Parabolic mirrror

p ↓ => f↓ to keep q at ≈ 2.5 cm The Eye ≈ 2.5 cm f depends on p p ↓ => f↓ to keep q at ≈ 2.5 cm

Corrective Lenses Farsighted eye focuses rays from nearby object behind the retina Nearsighted eye focuses rays from distant object in front of the retina A converging lens corrects this defect A diverging lens corrects this defect