Physics 1202: Lecture 20 Today’s Agenda Announcements: Team problems today Team 14: Gregory Desautels, Benjamin Hallisey, Kyle Mcginnis Team 15: Austin Dion, Nicholas Gandza, Paul Macgillis-Falcon Homework #9: due Friday NO HOMEWORK next week. Midterm 2: Tuesday April 10: covers Ch. 24-27. Midterm sample + To-Know sheet on web already Review session: chapters 24-27 1

24-AC Current L C ~ e R w

Suppose: Phasors for L,C,R i i wt w ß i i wt w i i wt w

w - dependence in AC Circuits The maximum current & voltage are related via the impedence Z Currents AC-circuits as a function of frequency:

24-5 RLC Circuits Phasor diagram Follow the loop Total V useful to analyze an RLC circuit Follow the loop VR = R Imax (in phase) VL= XL Imax (leads by 90o) VC= XC Imax (lags by 90o) Total V V= VR + VL+ VC = Z Imax

Phasors:LCR f Imax R Imax (XL-XC) Vmax = Imax Z Þ

24-5 RLC Circuits The phase angle for an RLC circuit is: If XL = XC, the phase angle is zero, and the voltage and current are in phase. The power factor:

Resonance For fixed R,C,L the current im will be a maximum at the resonant frequency w0 which makes the impedance Z purely resistive. ie: reaches a maximum when: XL=XC the frequency at which this condition is obtained is given from: Þ Note that this resonant frequency is identical to the natural frequency of the LC circuit by itself! At this frequency, the current and the driving voltage are in phase!

24-6 Resonance in Electrical Circuits In an RLC circuit with an ac power source, the impedance is a minimum at the resonant frequency: XL=XC

Resonance The current in an LCR circuit depends on the values of the elements and on the driving frequency through the relation “ Impedance Triangle” Z |  R | XL-XC | 1 2 x im 2wo w R=Ro m / R0 Suppose you plot the current versus w, the source voltage frequency, you would get: R=2Ro

Power in LCR Circuit The power can be expressed in term of i max: Þ This result is often rewritten in terms of rms values: Þ Power delivered depends on the phase, f, the “power factor” phase depends on the values of L, C, R, and w

Chap. 25 f ( x f ( x ) x z y x

E & B in Electromagnetic Wave Plane Harmonic Wave: where: y x z Note: the direction of propagation is given by the cross product where are the unit vectors in the (E,B) directions. Nothing special about (Ey,Bz); eg could have (Ey,-Bx) Note cyclical relation:

Lecture 14, ACT 1 (a) + z direction (b) -z direction (c) +y direction Suppose the electric field in an e-m wave is given by: In what direction is this wave traveling ? (a) + z direction (b) -z direction (c) +y direction (d) -y direction To determine the direction, set phase = 0: Therefore wave moves in + z direction! Another way:

E & B in Electromagnetic Wave Plane Harmonic Wave: where: y x z From general properties of waves : Þ

25-4 Energy and Momentum in Electromagnetic Waves The energy a wave delivers to a unit area in a unit time is called the intensity.

25-4 Energy and Momentum in Electromagnetic Waves Substituting for the energy density: An electromagnetic wave also carries momentum:

25-5 Polarization Polarized light has its electric fields all in the same direction. Unpolarized light has its electric fields in random directions. The polarization of an EM wave refers to the direction of its electric field

Polarizers I = I0 cos2q Made of long molecules (polymers) Block electric field along their length Electric field perpendicular passes through E E. H. Land (1909 – 1991): Polaroid E So Eafter=E cosq Recall that I ~ E2 I = I0 cos2q

25-5 Polarization For unpolarized light passing through a polarizer the transmitted intensity is half the initial intensity A polarizer and an analyzer can be combined

26 R q h h’ o-R R-i o i

26-1 The Reflection of Light The law of reflection states that the angle of incidence equals the angle of reflection: © 2017 Pearson Education, Inc.

26-3 Spherical Mirrors: definitions Spherical mirrors have a central axis (a radius of the sphere) and a center of curvature (the center of the sphere) Concave mirror Convex mirror © 2017 Pearson Education, Inc.

26-3 Concave Spherical Mirrors Consider parallel rays They hit a spherical mirror They come together at the focal point This is a ray diagram for finding the focal point of a concave mirror.

26-3 Convex Spherical Mirrors Consider parallel rays They hit a spherical mirror They appear to have come from the focal point, if the mirror is convex © 2017 Pearson Education, Inc.

Diagram: Concave Mirror, o > R Active Figure 36.15 Ray diagrams for spherical mirrors, along with corresponding photographs of the images of candles. (a) When the object is located so that the center of curvature lies between the object and a concave mirror surface, the image is real, inverted, and reduced in size. The object is outside the center of curvature of the mirror The image is real The image is inverted The image is smaller than the object

Diagram: Concave Mirror, o < f Active Figure 36.15 Ray diagrams for spherical mirrors, along with corresponding photographs of the images of candles. (a) When the object is located so that the center of curvature lies between the object and a concave mirror surface, the image is real, inverted, and reduced in size. The object is between the mirror and the focal point The image is virtual The image is upright The image is larger than the object

Diagram: Convex Mirror Active Figure 36.15 Ray diagrams for spherical mirrors, along with corresponding photographs of the images of candles. (a) When the object is located so that the center of curvature lies between the object and a concave mirror surface, the image is real, inverted, and reduced in size. The object is in front of a convex mirror The image is virtual The image is upright The image is smaller than the object

Mirror – Lens Definitions Some important terminology we introduced last class, o = distance from object to mirror (or lens) i = distance from mirror to image o positive, i positive if on same side of mirror as o. R = radius of curvature of spherical mirror f = focal length, = R/2 for spherical mirrors. Concave, Convex, and Spherical mirrors. M = magnification, (size of image) / (size of object) negative means inverted image R g q a object b h image o i

o i f h’ h 26.5

EM wave at an interface What happens when light hits a surface of a material? Three Possibilities Reflected Refracted (transmitted) Absorbed incident ray reflected ray MATERIAL 1 MATERIAL 2 refracted ray

25-6: Index of Refraction The wave incident on an interface can not only reflect, but it can also propagate into the second material. Claim the speed of an electromagnetic wave is different in matter than it is in vacuum. Recall, from Maxwell’s eqns in vacuum: How are Maxwell’s eqns in matter different? e0 ® e , m0 ® m (both increase) Therefore, the speed of light in matter is smaller and related to the speed of light in vacuum by: where n = index of refraction of the material: The index of refraction is frequency dependent: For example nblue > nred

The two triangles above each have hypotenuse L Snell’s Law l1 l2 L n1 n2 q1 q2 From the last slide: q1 q2 The two triangles above each have hypotenuse L ⇒ \ ⇒ But,

26-5 Refraction: Basic properties Light may refract into a material where its speed is lower angle of refraction is less than the angle of incidence The ray bends toward the normal Light may refract into a material where its speed is higher angle of refraction is more than the angle of incidence The ray bends away from the normal If n1 = n2 Þ no effect If light enters normal Þ no effect

Total Internal Reflection Consider light moving from glass (n1=1.5) to air (n2=1.0) incident ray reflected refracted q2 q1 qr GLASS AIR n2 n1 ie light is bent away from the normal. as q1 gets bigger, q2 gets bigger, but q2 can never get bigger than 90° !! In general, if sin q1  sin qC  (n2 / n1), we have NO refracted ray; we have TOTAL INTERNAL REFLECTION. For example, light in water which is incident on an air surface with angle q1 > qc = sin-1(1.0/1.5) = 41.8° will be totally reflected. This property is the basis for the optical fibers used in communication.

Converging Lens Principal Rays F Image P.A. Object F 1) Rays parallel to principal axis pass through focal point. 2) Rays through center of lens are not refracted. 3) Rays through F emerge parallel to principal axis. Image is: real, inverted and enlarged (in this case). Assumptions: • monochromatic light incident on a thin lens. • rays are all “near” the principal axis.

Diverging Lens Principal Rays F P.A. Image Object F 1) Rays parallel to principal axis pass through focal point. 2) Rays through center of lens are not refracted. 3) Rays toward F emerge parallel to principal axis. Image is virtual, upright and reduced

The Mirror/Lens Equation We have derived, in the paraxial (and thin lens) approximation, the same equations for mirrors and lenses: when the following sign conventions are used: Variable f > 0 f < 0 o > 0 o < 0 i > 0 i < 0 Mirror concave convex real (front) virtual (back) Lens converging diverging real (back) virtual (front)

3 Cases for Converging Lenses Object Image Past 2F Inverted Reduced Real This could be used in a camera. Big object on small film Image Object Between F & 2F Inverted Enlarged Real This could be used as a projector. Small slide on big screen Image Object Inside F Upright Enlarged Virtual This is a magnifying glass

Lecture 18, ACT 5 (a) left half of image disappears object A lens is used to image an object on a screen. The right half of the lens is covered. What is the nature of the image on the screen? lens (a) left half of image disappears (b) right half of image disappears screen (c) entire image reduced in intensity All rays from the object are brought to a focus at the screen by the lens. The covering simply blocks half of the rays. Therefore the intensity is reduced but the image is of the entire object!

26-8 Dispersion and the Rainbow The index of refraction n varies slightly with the frequency f of light (or wavelength l) of light in general, the higher f, the higher the index of refraction n This means that refracted light is “spread out” in a rainbow of colors This is known as dispersion

Prisms f The index of refraction for a material usually decreases with increasing wavelength Violet light refracts more than red light when passing from air into a material

27 Optical Instruments

The EYE I2 I1 ~fo ~fe L objective eyepiece

27-1 The Human Eye The ciliary muscles The near point The far point adjust the shape of the lens to accommodate near and far vision. The near point the closest point to the eye that the lens is able to focus normal vision ~ 25 cm from the eye it increases with age as the lens becomes less flexible The far point farthest point at which the eye can focus it is infinitely far away, if vision is normal

The Lens Equation Convergent Lens: h i o f h’

27-2 Corrective Optics & Human Eye A nearsighted person: far point at a finite distance objects farther away will appear blurry: lens focus too strong so the image is formed in front of the retina. Use diverging lens f chosen for a distant object to form image at the far point Strength of corrective lenses: © 2017 Pearson Education, Inc.

27-2 Corrective Optics & Human Eye A farsighted person: see distant objects clearly but cannot focus on close objects—the near point is too far away lens not strong enough: image focus is behind the retina. Use a converging lens Augment the converging power of the eye The final image is past the near point © 2017 Pearson Education, Inc.

27-3 The Magnifying Glass A simple convex lens makes objects appear bigger by making them appear closer Similar to a corrective lens for farsightedness it brings the near point closer to the eye Angular size of an object angle it subtends on the retina, and depends both on the size of the object and its distance from the eye assuming it is small

Recap of Today’s Topic : Announcements: Team problems today Team 14: Gregory Desautels, Benjamin Hallisey, Kyle Mcginnis Team 15: Austin Dion, Nicholas Gandza, Paul Macgillis-Falcon Homework #9: due Friday NO HOMEWORK next week. Midterm 2: Tuesday April 10: covers Ch. 24-27. Midterm sample + To-Know sheet on web already Review session: chapters 24-27