Physics 1202: Lecture 20 Today’s Agenda Announcements: –Lectures posted on: –HW assignments, etc. Homework #6:Homework #6: –Due next Friday

Einstein’s relativity (1905) Einstein incorporated this result in his 2 postulates 2- Constant speed of light: The speed of light c is the same in all inertial reference frames, regardless of the relative velocity of the source and receiver of the light. 1- Principle of Relativity: The laws of Physics are the same on all inertial reference frame. These “simple” postulates have big implications …

Relativistic Factor  No material particle can reach c –  ≥ 1 for 0 ≤ v < c Length contraction: Time dilation: Velocity addition:

Relativistic Doppler Effect Doppler effect for light Application: redshift of astrophysical objects Note: for v<<c, –get back “normal” doppler effect (e.g. for sound)

Relativistic momentum Momentum increases with v In the limit of non-relativistic speeds ( v << c,  ≈ 1 ): which agrees with the classic formula!

Relativistic Kinetic Energy It scales with   For small v, it has to agree with classic expression

Relativistic Energy Kinetic energy Rest Energy Total Energy Total relativistic energy Rest energy Energy-momentum relation:

Energy-Momentum relation Square the total energy  where  since

Waves, Wavefronts, and Rays Consider a light wave (not necessarily visible) whose E field is described by This wave travels in the +x direction and has no dependence on y or z, i.e. it is a plane wave. 3-D Representation Wave Fronts RAYS

Waves, Wavefronts, and Rays For a point source: This wave travels in the +r direction and has no dependence on angle, i.e. it is a spherical wave. 3-D Representation

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 refracted ray MATERIAL 1 MATERIAL 2

Geometric Optics What happens to EM waves (usually light) in different materials? –index of refraction, n. Restriction: waves whose wavelength is much shorter than the objects with which it interacts. Assume that light propagates in straight lines, called rays. Our primary focus will be on the REFLECTION and REFRACTION of these rays at the interface of two materials. incident ray reflected ray refracted ray MATERIAL 1 MATERIAL 2

Reflection The angle of incidence equals the angle of reflection  i =  r, where both angles are measured from the normal: ·Note also, that all rays lie in the “plane of incidence” ii rr ii rr x EiEi ii ErEr rr ·Why? »This law is quite general; we supply a limited justification when surface is a good conductor, Electric field lines are perpendicular to the conducting surface. The components of E parallel to the surface of the incident and reflected wave must cancel!!

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?    ,    ·Therefore, the speed of light in matter is related to the speed of light in vacuum by: ·The index of refraction is frequency dependent: For example n blue > n red where n = index of refraction of the material:

Refraction How is the angle of refraction related to the angle of incidence? –Unlike reflection,  1 cannot equal  2 !! »n 1  n 2  v 1  v 2 but, the frequencies (f 1,f 2 ) must be the same  the wavelengths must be different! Therefore,  2 must be different from  1 !! 11 22 n1n1 n2n2 1 2

Snell’s Law From the last slide: 11 11 1 2 L n1n1 n2n2 11 22 22 22 22 The two triangles above each have hypotenuse L  But,

Lecture 20, ACT 1 Which of the following ray diagrams could represent the passage of light from air through glass and back to air? (a) (b)(c)

Lecture 20, ACT 2 Which of the following ray diagrams could represent the passage of light from air through glass and back to air? (a) (b) (c) (d)

Prisms Index of refraction frequency ultraviolet absorption bands white light prism A prism does two things, 1.Bends light the same way at both entrance and exit interfaces. 2.Splits colours due to dispersion.

Prisms 11 22 Entering For air/glass interface, we use n(air)=1, n(glass)=n 33 44 Exiting

Prisms 11 22 33 44 Overall Deflection At both deflections the amount of downward deflection depends on n (and the prism apex angle,  ). The overall downward deflection goes like,  ~ A(  ) + B n Different colors will bend different amounts ! 

Lecture 20, ACT 3 White light is passed through a prism as shown. Since n(blue) > n(red), which color will end up higher on the screen ? ? ? A) BLUE B) RED

LIKE SO!In second rainbow pattern is reversed

Total Internal Reflection –Consider light moving from glass (n 1 =1.5) to air (n 2 =1.0) ie light is bent away from the normal. as  1 gets bigger,  2 gets bigger, but  2 can never get bigger than 90  !! In general, if sin  1  sin  C  (n 2 / n 1 ), we have NO refracted ray; we have TOTAL INTERNAL REFLECTION. For example, light in water which is incident on an air surface with angle  1 >  c = sin -1 (1.0/1.5) = 41.8  will be totally reflected. This property is the basis for the optical fibers used in communication. incident ray reflected ray refracted ray 22 11 rr GLASS AIR n2n2 n1n1 2

ACT 4: Critical Angle... air n =1.00 glass n =1.5 air n =1.00 cc water n =1.33 glass n =1.5 water n =1.33 cc Case I Case II An optical fiber is cladded by another dielectric. In case I this is water, with an index of refraction of 1.33, while in case II this is air with an index of refraction of Compare the critical angles for total internal reflection in these two cases a)  cI >  cII b)  cI =  cII c)  cI <  cII

ACT 5: Fiber Optics air n =1.00 glass n =1.5 air n =1.00 cc water n =1.33 glass n =1.5 water n =1.33 cc Case I Case II The same two fibers are used to transmit light from a laser in one room to an experiment in another. Which makes a better fiber, the one in water ( I ) or the one in air ( II ) ? a)  Water b)  Air

Problem You have a prism that from the side forms a triangle of sides 2cm x 2cm x 2  2cm, and has an index of refraction of 1.5. It is arranged (in air) so that one 2cm side is parallel to the ground, and the other to the left. You direct a laser beam into the prism from the left. At the first interaction with the prism surface, all of the ray is transmitted into the prism. a)Draw a diagram indicating what happens to the ray at the second and third interaction with the prism surface. Include all reflected and transmitted rays. Indicate the relevant angles. b)Repeat the problem for a prism that is arranged identically but submerged in water.

Solution At the first interface  =0 o, no deflection of initial light direction. At 2nd interface  =45 o, from glas to air ? Critical angle: sin(  c )=1.0/1.5 =>  c = 41.8 o < 45 o Thus, at 2nd interface light undergoes total internal reflection At 3rd interface  =0 o, again no deflection of the light beam A) Prism in air B) Prism in water (n=1.33) At the first interface  =0 o, the same situation. At 2nd interface now the critical angle: sin(  c )=1.33/1.5 =>  c = 62 o > 45 o Now at 2nd interface some light is refracted out the prism n 1 sin(  1 ) = n 2 sin(  2 ) => at  2 = 52.9 o Some light is still reflected, as in A) ! At 3rd interface  =0 o, the same as A)