Final Review. A Puzzle... Special Relativity. Direction of the Force. Moving at the Speed of Light
|
|
- Hilary Wiggins
- 5 years ago
- Views:
Transcription
1 Final Review A Puzzle... Diretion of the Fore A point harge q is loated a fixed height h above an infinite horizontal onduting plane. Another point harge q is loated a height z (with z > h) above the plane. The two harges lies on the same vertial line. If z is only slightly larger than h, then the fore on the top harge is learly upward. But for larger values of z, is the fore still always upward? Hint: Try to solve this without doing any alulations. Think dipole. There are two negative image harges on the other side of the plane, at the mirror-image loations. For very large z values of the top harge q, the lower q and its image harge -q look like a dipole from afar, whih has a repulsive (upward) field that falls off like z 3. But the attrative (downward) field from the other image harge -q behaves like (2 z) 2. This has a smaller power of z in the denominator, so it dominates for large z. The fore on the top harge q is therefore downward for large z. So the answer to the stated question is "No." Moving at the Speed of Light One of the interesting quirks about the veloity addition formula is that if you start off moving at in one frame, then you move in in another frame. This begs some interesting questions, suh as what happens if you aelerate a ar to the speed of light, and you turn on your headlights. Would the light move at speed relative to you, would it all pool inside of the headlight, or would something altogether different happen? Mihael Stevens has an amazing YouTube video analyzing this very question. Speial Relativity The following problems all deal with the setup shown below. Two trains, A and B, eah have proper length L and move in the same diretion. A s speed is 4, and B s speed is 3. A starts behind B. Printed by Wolfram Mathematia Student Edition
2 2 Leture 8 - Final Review.nb We define the following two events, Event E : "The front of A passing the bak ofb" Event E 2 : "The bak of A passing the front ofb" () Length Contration/Time Dilation What is the differene in time Δt C and spae Δx C between E and E 2, as viewed by person C on the ground? Relative to C on the ground, the γ fators assoiated with A and B are Therefore, their lengths in the ground frame are γ A (C frame) = γ B (C frame) = /2 = 3 (2) /2 = 4 (3) L (C L frame) A = γ(c frame) = 3 A L (4) (C L frame) L B = γ(c frame) = 4 B L () While overtaking B, A must travel farther than B, by an exess distane equal to the sum of the lengths of the (C trains, whih is L frame) (C A + L frame) B = 3 L + 4 L = 7 L. The relative speed of the two trains (as viewed by C on the ground) is the differene of the speeds, whih is. The total time is therefore Δt C = 7 L = 7 L (6) The distane between both events equals the distane that A travels minus the length of A (sine E happens at the front of A and E 2 happens at the rear of A). Therefore for the two events. Veloity Addition Δx C = Δt C 4-3 L = L (7). Find Δt B, Δx B between E and E 2, as viewed by a stationary observer in B s frame? Repeat for Δt A, Δx A for a stationary observer in A s frame. 2. Person D walks at onstant speed from the bak of train B to its front, suh that he oinides with both events E and E 2. Compute Δt D, Δx D.. In B s referene frame, A moves at a slower speed found by the veloity addition formula Printed by Wolfram Mathematia Student Edition
3 Leture 8 - Final Review.nb 3 with an assoiated γ fator whih implies that A has a length while B has its rest length u A (B frame) = 4-3 γ A (B frame) = L A (B frame) = Therefore the time between events E and E 2 equals The distane between the two events is simply - 4 Δt B = L A (B frame) +L B (B frame) 3 = /2 = 3 (8) 2 (9) L γ(b frame) = 2 A 3 L (0) L B (B frame) = L () u(b frame) = 2 3 L = L A 3 (2) Δx B = L (3) It is straightforward to repeat all of these alulations in A s frame. Everything will be symmetri (exept with everything now moving in the opposite diretion), so that Δt A = L (4) Δx A = -L () 2. In B s frame, D must walk with speed Δx B = in order to oinides with both events. In D s referene frame, Δt B A will have veloity and B will have veloity u A (D frame) = u B (D frame) = = (6) = - (7) (D In hindsight, it makes sense that u frame) (D A = -u frame) B beause of the symmetry of the problem (i.e. that D oinides with both events and that A and B both have proper length L). The γ fator and length of A will be (D γ frame) A = = - 2 /2 2 6 (8) L A (D frame) = and, by symmetry, this must equal the length of B, Therefore, Lastly, sine D oinides with both events, Δt D = L A (D frame) +L B (D frame) L γ(d frame) = 2 6 L A (9) L B (D frame) = 2 6 L (20) u(d frame) A -u(d frame) = 2 6 L = 2 L 6 B (2) Δx B = 0 (22) Note: There are several double heks we an perform. For example, the speed of D with respet to the ground frame (i.e. C frame) an be obtained by relativistially adding 3 and or subtrating from 4. These both give the same answer, namely, as they must. The γ fator between D and the ground is therefore 7 (C γ frame) D = = /2 2 6 (23) We an now use time dilation to say that someone on the ground sees the overtaking take a time of Printed by Wolfram Mathematia Student Edition
4 4 Leture 8 - Final Review.nb Δt C = Δt D γ D (C frame) = 2 L 6 Likewise, the γ fator between D and either train equals say ground overtaking 7 = 7 L, in agreement with equation (6) from the previous problem. 2 6 γ D (A frame) = γ D (B frame) = = - 2 /2 2 6 (24) Therefore, the time between events as viewed in A or B equals Δt A = Δt B = Δt D γ D (A frame) = 2 L 6 agreement with equations (2) and (4) above. = L in 2 6 Note that we annot use simple time dilation to relate the ground to A or B, beause the two events don t happen at the same plae in the train frames! But sine both events happen at the same plae in D s frame, namely right at D, we an indeed use time dilation to go from D s frame to any other frame. Lorentz Transformations Verify that the values of Δx and Δt found in the above problems satisfy the Lorentz transformations between the six pairs of frames, namely AB, AC, AD, BC, BD, CD. Gathering together all of the values found above, Δt A B C D L L 7 L 2 L 6 Δx -L L L 0 We also list the various veloities and γ fators from the above examples between eah pair of frames The Lorentz transformations are AB AC AD BC BD CD v 3 3 γ Δx = γ (Δx' + v Δt) (2) Δt = γ Δt' + v Δx' 2 (26) For eah of the six pairs, we ll transform from the faster frame to the slower frame. This means that the oordinates of the faster frame will be on the right-hand side of the Lorentz transformations. The sign on the right-hand side of the Lorentz transformations will therefore always be a +. In the AB ase, for example, we will write, Frames B and A, in that order, to signify that the B oordinates are on the left-hand side, and the A oordinates are on Printed by Wolfram Mathematia Student Edition
5 Leture 8 - Final Review.nb the right-hand side. We ll simply list the Lorentz transformations for the six ases, and you an hek that they do indeed all work out. Frames B and A Frames C and A Frames D and A Frames C and B Frames B and D Frames C and D Eletrostatis L = 3 2 -L + 3 L L = 3 2 L + 3 (-L) 2 L = 3 -L + 4 L 7 L = 3 L + 4 (-L) 2 0 = 2 6 -L + L 2 L 6 = L ( ) (-L) 2 L = 4 L + 3 L 7 L = 4 L + 3 (L) 2 L = L 6 L = L 6 + ( ) (0) 2 L = L 6 7 L = 7 2 L (0) 2 (27) (28) (29) (30) (3) (32) Sphere and Cones (a) Consider a fixed hollow spherial shell with radius R and surfae harge density σ. A partile with mass m and harge -q that is initially at rest falls in from infinity. What is its speed when it reahes the enter of the shell? (Assume that a tiny hole has been ut in the shell, to let the harge through.) (b) Consider two fixed hollow onial shells (that is, ie ream ones without the ie ream) with base radius R, slant height L, and surfae harge density σ, arranged as shown in figure (b) below. A partile with mass m and harge -q that is initially at rest falls in from infinity, along the perpendiular bisetor line, as shown. What is its speed when it reahes the tip of the ones? (The answers to both parts of this problem should relate very niely!) Printed by Wolfram Mathematia Student Edition
6 6 Leture 8 - Final Review.nb s (a) This problem is terrifially simple if we use the eletri potential. Reall that for a spherial shell, the potential equals ϕ[r] = k Q where Q = 4 π R 2 σ for r R. Inside the sphere, the eletri field is zero so that the potential must r be a onstant; by ontinuity, the eletri potential inside the shell equals ϕ[r] = k Q for r R. R Therefore the work required to bring a partile from to the enter of the spherial shell equals -q ϕ[0], so that the partiles energy due to free fall would be 2 m v2 = q ϕ[0] = k q Q or equivalently R v = 2 k q Q /2 = 2 q R σ R ϵ 0 /2 (33) We ould also do this (more painfully) by integrating the eletri field and solving the orresponding differential equation. The eletri field for r R equals E = k Q r so that the fore aelerating the partile radially inward r 2 satisfies m v dv dr = m r = - k Q q r (34) 2 where we have used the relation d2 r = r = dr dr dt 2 dr dv = dv v. Separating the variables and integrating t differential dr equation above, v 0 m v dv = r - k Q q dr r (3) 2 2 m v2 = k Q q r (36) This formula is valid from r = until r = R, at whih point the eletri field beomes zero and the veloity remains onstant until the partile hits the enter, moving at speed 2 m v2 = k Q q R (37) The solution proeeds as above. (b) Sine ones may seem like foreign objets, let start by first integrating over the surfae area of the one to make sure that we orretly find its surfae area (the surfae area of a one is π R L, as per Wikipedia). We will integrate in rings from z = 0 to z = H = L 2 - R 2, as shown below. Printed by Wolfram Mathematia Student Edition
7 Leture 8 - Final Review.nb 7 The surfae area of a ring between z and z + d z is given by (surfae area of ring between z and z + dz) = 2 π (radius of ring) (width of ring) (38) Noting the similar triangles, the radius of a ring at height z equals z R. The vertial height of the ring is dz, but H we want to know the slant width (i.e. the diagonal length) of the ribbon. Sine the ring goes vertially up by d z and outward by R H dz, the slant width is + R H 2 /2 d z. Thus (surfae area of ring between z and z + dz) = 2 π z R H + R H 2 /2 dz (39) and the surfae area of the entire one equals surfae area = 0 H 2 π z R H + R H 2 /2 dz = π R (H 2 + R 2 ) /2 = π R L as desired. Having onfirmed this neat fat, let us return to the problem. The area element of a one is given by Equation (40), sine we just showed that this integral overs the full surfae of the one and yields the orret surfae area formula. The potential at the base of the two ones, ϕ[0], equals twie the ontribution from a single one, written in the familiar form k dq, equals r H ϕ[0] = 2 k 2 π z R H + R /2 H 2 dz σ 0 z + R H 2 /2 = 4 π k σ R H 0 H dz = σ R ϵ 0 Sine this has the exat same form as the potential ϕ[0] = k Q R = σ R ϵ 0 we found in Part (a), the solution proeeds as above, and we one again find the same veloity v = 2 q R σ /2. What a remarkable oinidene! ϵ 0 Condutors (40) (4) Image Charges for Two Planes This setion investigates the following problem very deeply! A point harge q is loated between two parallel infinite onduting planes, a distane b from one and l - b from the other. Where should image harges be loated so that the eletri field is everywhere perpendiular to the Printed by Wolfram Mathematia Student Edition
8 8 Leture 8 - Final Review.nb planes? image harges everywhere perpendiular Set one plate at z = 0 and one at z = l, and let the positron be at z = b. The problem is learly one dimensional, so we fix x = y = 0 for all points we disuss. If we just onsider the bottom plate, we an plae a negative harge at -b, but then we need to take are of both of these harges with images beyond the top plate with a negative harge at z = 2 l - b and a positive harge at z = 2 l + b. In other words, there will be a asade of image harges (of alternating signs!) In the figure below, the two given planes are indiated by the bold lines, and the given real harge is labeled R. It turns out that we will need an infinite number of image harges, as shown. Solid dots are positive, hollow dots are negative (assuming the given real harge is positive). The following Manipulate shows the real harge q (in blak) together with the positive image harges (blue) and negative image harges (orange). The real onduting planes are in the middle and are darker than the other planes at z = -l, ±2 l, ±3 l... b Show E Out[3]= Real harge Q Image harge Q Image harge -Q The pattern learly emerges: Printed by Wolfram Mathematia Student Edition
9 Leture 8 - Final Review.nb 9 Positive Charges Negative Charges l + b 4 l - b 2 l + b 2 l - b 0 l + b 0 l - b -2 l + b -2 l - b -4 l + b -4 l - b If you want, you an group the harges into two sets the odds and evens, as indiated by the onneting lines in the figure above. Eah odd harge orrets the effet of the previous odd harge, with respet to alternating planes. Likewise for the evens. In the speial ase where the given real harge is loated midway between the two planes, all the image harges are similarly loated midway between the (imaginary) planes in the figure above. So the net fore on the given harge is zero, as it should be. The Limit b l In the limit b l when the harge q is very lose to one of the plates, find an approximate expression for the fore on the harge. To zeroth order, the fore on the harge q omes solely from the image harge nearby at z = -b, whih is a fore F k q2 (2 b) 2 (43) downwards. But we an do muh better than that! All the remaining pairs of harges an be approximated as dipoles; two of these dipoles are at a distane 2 l, another two are at 4 l, and so on. Reall the formula 2 k q d for the r 3 eletri field of a dipole along its axis (where d is the distane between the two harges and r is the distane from the dipole s enter). Note that every dipole pushes the harge q in the +z-diretion, the total downwards fore on the dipole equals F = k q2-2 {2 k q 2 (2 b)} (2 b) 2 = k q2 4 b 2 - k q2 b l (2 l) 3 (4 l) 3 (6 l) 3 + where the first fator of 2 in the seond term omes from the fat that there are two pairs of dipoles at distane 2 l, 4 l... (on the left and right side of the harge). The fator in parenthesis equals Zeta[3].202 (44) Sum, {n,, } 3 n Zeta[3] You an also alulate the total fore by looking at the fores from the positive and negative image harges separately. From the figure above, the fore on the real harge q from the other positive harges will always be 0 by symmetry, but the downwards fore from the negative harges equals F = k q2 - k q 2 (2 b) 2 n= = k q2 - k q2 4 b 2 4 l 2 n= n 2 - (2 n l-2 b) 2 (2 n l+2 b) b n l 2 + b n l 2 k q2 - k q2 4 b 2 4 l 2 n= + 2 b n 2 n l b = k q2 - k q2 b 4 b 2 l 3 n= n 3 n l (4) Printed by Wolfram Mathematia Student Edition
10 0 Leture 8 - Final Review.nb where in the third step we used the Taylor series - 2 ϵ + O[ϵ] 2. This result mathes equation (44), as (+ϵ) 2 expeted. Ciruits Two Light Bulbs Certain light bulbs an be treated as resistors, with the brightness of the bulb proportional to the power dissipated in the bulb s resistor. (a) Two light bulbs are onneted in parallel, and then onneted to a battery, as shown in figure (a). You observe that bulb is twie as bright as bulb 2. Whih bulb s resistor is larger, and by what fator? (b) The bulbs are now onneted in series, as shown in figure (b). Whih bulb is brighter, and by what fator? How bright is eah bulb ompared with bulb in Part (a)? (a) The power dissipated takes the form V2. Both bulbs have the same voltage drop V, so if bulb is twie as bring R as bulb 2, it must have half the resistane, R 2 = 2 R. In parallel, the larger resistor is dimmer. (b) The power dissipated also takes the form I 2 R. Both bulbs now have the same urrent I, so if Bulb 2 has twie the resistane, as we found in Part (a), then it is twie as bright - the opposite of the ase in Part (a). In series, the larger resistor is brighter. We an also ompare the total power dissipated in eah ase. If the resistanes are R and 2 R, then in Part (a) the total power dissipated is V2 R + V2 2 R = 3 V2 2 R. In Part (b) the total power is I2 R + I 2 (2 R) = 3 I 2 R, where I = V. So the 3 R power is V2 3 R. This is 2 V2 of the power in Part (a). In units of 9 R, the power in Part (a) are and, while in Part (b) 2 they are 9 and 2 9. Attenuator Chain (a) Find the equivalent resistane between terminals A and B in the infinite ladder of resistors shown below. Hint: Call the input resistane R, and note that it will not be hanged by adding a new set of resistors to the front end of the hain to make it one unit longer. (b) Show that, if voltage V 0 is applied at the input to suh a hain, the voltage at suessive nodes dereases in a geometri series. What should the ratio of the resistors be so that the ladder is an attenuator that halves the voltage Printed by Wolfram Mathematia Student Edition
11 Leture 8 - Final Review.nb geometri voltage at every step? () Obviously a truly infinite ladder would not be pratial. Can you suggest a way to terminate it after a few setions without introduing any error in its attenuation? (a) If R is the effetive resistane of the infinite hain, then the hain is equivalent to the iruit shown below. Thus, R = R + R2 R R2 +R (46) whih implies R= R + R2 +4 R R2 2 /2 (47) where we have hosen the positive root. (b) To demonstrate the stated geometri series result, onsider four points A, A', B, B' that form a square somewhere within the iruit. Given the voltage V ' between A' and B', what is the voltage V between A and B? Let the urrent flowing towards A ' be I, so that (using the effetive resistane R of the infinite hain) the urrent splits into I = I2 = R2 R+R2 R R+R2 I (48) I (49) What will be the urrent between A and B? By symmetry, it must be V V' = I I = R2 R2 +R I I2, I so that (0) (As a quik hek, you an make sure that the voltage around the square loop going from A ' to A to B to B' is zero, as it must be.) This result is independent of where along the hain we pik the adjaent nodes, so the voltages Printed by Wolfram Mathematia Student Edition
12 2 Leture 8 - Final Review.nb be.) independent along pik adjaent voltages aross suessive nodes derease in a geometri series. If we want V = V' 2, then we must have R = R 2. Equation (47) then yields 2 R 2 - R = (R R R 2 ) /2 () (2 R 2 - R ) 2 = R R R 2 (2) R 2 = 2 R (3) If we instead wanted V (that is, the voltage hardly dereases), then we need R R V' 2, whih implies R R 2. On the other hand, if we want V (that is, the voltage dereases quikly), then we need R R V' 2, whih implies R R 2. These results make intuitive sense. () To terminate the ladder after any setion, without hanging its resistane from that of the infinite hain, we an simply onnet a single resistor R given by Equation (47) in parallel with the last R 2, beause this R mimis the rest of the infinite hain. Mathematia Initialization Printed by Wolfram Mathematia Student Edition
Lecture 3 - Lorentz Transformations
Leture - Lorentz Transformations A Puzzle... Example A ruler is positioned perpendiular to a wall. A stik of length L flies by at speed v. It travels in front of the ruler, so that it obsures part of the
More informationChapter 26 Lecture Notes
Chapter 26 Leture Notes Physis 2424 - Strauss Formulas: t = t0 1 v L = L0 1 v m = m0 1 v E = m 0 2 + KE = m 2 KE = m 2 -m 0 2 mv 0 p= mv = 1 v E 2 = p 2 2 + m 2 0 4 v + u u = 2 1 + vu There were two revolutions
More informationThe homopolar generator: an analytical example
The homopolar generator: an analytial example Hendrik van Hees August 7, 214 1 Introdution It is surprising that the homopolar generator, invented in one of Faraday s ingenious experiments in 1831, still
More informationPHYSICS 212 FINAL EXAM 21 March 2003
PHYSIS INAL EXAM Marh 00 Eam is losed book, losed notes. Use only the provided formula sheet. Write all work and answers in eam booklets. The baks of pages will not be graded unless you so ruest on the
More informationMillennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion
Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six
More informationSpecial Relativity. Relativity
10/17/01 Speial Relativity Leture 17 Relativity There is no absolute motion. Everything is relative. Suppose two people are alone in spae and traveling towards one another As measured by the Doppler shift!
More informationGeneration of EM waves
Generation of EM waves Susan Lea Spring 015 1 The Green s funtion In Lorentz gauge, we obtained the wave equation: A 4π J 1 The orresponding Green s funtion for the problem satisfies the simpler differential
More informationName Solutions to Test 1 September 23, 2016
Name Solutions to Test 1 September 3, 016 This test onsists of three parts. Please note that in parts II and III, you an skip one question of those offered. Possibly useful formulas: F qequb x xvt E Evpx
More informationEinstein s theory of special relativity
Einstein s theory of speial relatiity Announements: First homework assignment is online. You will need to read about time dilation (1.8) to answer problem #3 and for the definition of γ for problem #4.
More informationChapter 35. Special Theory of Relativity (1905)
Chapter 35 Speial Theory of Relatiity (1905) 1. Postulates of the Speial Theory of Relatiity: A. The laws of physis are the same in all oordinate systems either at rest or moing at onstant eloity with
More informationTHEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE?
THEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE? The stars are spheres of hot gas. Most of them shine beause they are fusing hydrogen into helium in their entral parts. In this problem we use onepts of
More informationRelativity in Classical Physics
Relativity in Classial Physis Main Points Introdution Galilean (Newtonian) Relativity Relativity & Eletromagnetism Mihelson-Morley Experiment Introdution The theory of relativity deals with the study of
More informationRelativity fundamentals explained well (I hope) Walter F. Smith, Haverford College
Relativity fundamentals explained well (I hope) Walter F. Smith, Haverford College 3-14-06 1 Propagation of waves through a medium As you ll reall from last semester, when the speed of sound is measured
More informationTHE TWIN PARADOX A RELATIVISTIC DOMAIN RESOLUTION
THE TWIN PARADOX A RELATIVISTIC DOMAIN RESOLUTION Peter G.Bass P.G.Bass www.relativitydomains.om January 0 ABSTRACT This short paper shows that the so alled "Twin Paradox" of Speial Relativity, is in fat
More informationEinstein s Three Mistakes in Special Relativity Revealed. Copyright Joseph A. Rybczyk
Einstein s Three Mistakes in Speial Relativity Revealed Copyright Joseph A. Rybzyk Abstrat When the evidene supported priniples of eletromagneti propagation are properly applied, the derived theory is
More informationGreen s function for the wave equation
Green s funtion for the wave equation Non-relativisti ase January 2019 1 The wave equations In the Lorentz Gauge, the wave equations for the potentials are (Notes 1 eqns 43 and 44): 1 2 A 2 2 2 A = µ 0
More informationSpecial and General Relativity
9/16/009 Speial and General Relativity Inertial referene frame: a referene frame in whih an aeleration is the result of a fore. Examples of Inertial Referene Frames 1. This room. Experiment: Drop a ball.
More informationModes are solutions, of Maxwell s equation applied to a specific device.
Mirowave Integrated Ciruits Prof. Jayanta Mukherjee Department of Eletrial Engineering Indian Institute of Tehnology, Bombay Mod 01, Le 06 Mirowave omponents Welome to another module of this NPTEL mok
More informationThe Laws of Acceleration
The Laws of Aeleration The Relationships between Time, Veloity, and Rate of Aeleration Copyright 2001 Joseph A. Rybzyk Abstrat Presented is a theory in fundamental theoretial physis that establishes the
More informationRelativistic Addition of Velocities *
OpenStax-CNX module: m42540 1 Relativisti Addition of Veloities * OpenStax This work is produed by OpenStax-CNX and liensed under the Creative Commons Attribution Liense 3.0 Abstrat Calulate relativisti
More informationarxiv: v1 [physics.gen-ph] 5 Jan 2018
The Real Quaternion Relativity Viktor Ariel arxiv:1801.03393v1 [physis.gen-ph] 5 Jan 2018 In this work, we use real quaternions and the basi onept of the final speed of light in an attempt to enhane the
More informationCHAPTER 26 The Special Theory of Relativity
CHAPTER 6 The Speial Theory of Relativity Units Galilean-Newtonian Relativity Postulates of the Speial Theory of Relativity Simultaneity Time Dilation and the Twin Paradox Length Contration Four-Dimensional
More informationPhysics 2D Lecture Slides Lecture 7: Jan 14th 2004
Quiz is This Friday Quiz will over Setions.-.6 (inlusive) Remaining material will be arried over to Quiz Bring Blue Book, hek alulator battery Write all answers in indelible ink else no grade! Write answers
More informationRelativistic Dynamics
Chapter 7 Relativisti Dynamis 7.1 General Priniples of Dynamis 7.2 Relativisti Ation As stated in Setion A.2, all of dynamis is derived from the priniple of least ation. Thus it is our hore to find a suitable
More informationCritical Reflections on the Hafele and Keating Experiment
Critial Refletions on the Hafele and Keating Experiment W.Nawrot In 1971 Hafele and Keating performed their famous experiment whih onfirmed the time dilation predited by SRT by use of marosopi loks. As
More informationSimple Considerations on the Cosmological Redshift
Apeiron, Vol. 5, No. 3, July 8 35 Simple Considerations on the Cosmologial Redshift José Franiso Garía Juliá C/ Dr. Maro Mereniano, 65, 5. 465 Valenia (Spain) E-mail: jose.garia@dival.es Generally, the
More informationPh1c Analytic Quiz 2 Solution
Ph1 Analyti Quiz 2 olution Chefung Chan, pring 2007 Problem 1 (6 points total) A small loop of width w and height h falls with veloity v, under the influene of gravity, into a uniform magneti field B between
More informationPhysicsAndMathsTutor.com 1
PhysisAndMathsTutor.om. (a (i beam splitter [or semi-silvered mirror] (ii a ompensator [or a glass blok] allows for the thikness of the (semi-silvered mirror to obtain equal optial path lengths in the
More information8.022 (E&M) Lecture 11
8.0 (E&M) Leture Topis: Introdution to Speial Relatiit Length ontration and Time dilation Lorentz transformations Veloit transformation Speial relatiit Read for the hallenge? Speial relatiit seems eas
More information12.1 Events at the same proper distance from some event
Chapter 1 Uniform Aeleration 1.1 Events at the same proper distane from some event Consider the set of events that are at a fixed proper distane from some event. Loating the origin of spae-time at this
More informationRelativistic effects in earth-orbiting Doppler lidar return signals
3530 J. Opt. So. Am. A/ Vol. 4, No. 11/ November 007 Neil Ashby Relativisti effets in earth-orbiting Doppler lidar return signals Neil Ashby 1,, * 1 Department of Physis, University of Colorado, Boulder,
More informationDeveloping Excel Macros for Solving Heat Diffusion Problems
Session 50 Developing Exel Maros for Solving Heat Diffusion Problems N. N. Sarker and M. A. Ketkar Department of Engineering Tehnology Prairie View A&M University Prairie View, TX 77446 Abstrat This paper
More informationThe Gravitational Potential for a Moving Observer, Mercury s Perihelion, Photon Deflection and Time Delay of a Solar Grazing Photon
Albuquerque, NM 0 POCEEDINGS of the NPA 457 The Gravitational Potential for a Moving Observer, Merury s Perihelion, Photon Defletion and Time Delay of a Solar Grazing Photon Curtis E. enshaw Tele-Consultants,
More informationIf velocity of A relative to ground = velocity of B relative to ground = the velocity of A relative to B =
L Physis MC nswers Year:1989 Question Number: 3,0,,4,6,9,30,31,36,40,4 1989MC (3) If eloity of relatie to ground = and eloity of relatie to ground =, then the eloity of relatie to = X X Y Y Suppose that
More informationChapter Outline The Relativity of Time and Time Dilation The Relativistic Addition of Velocities Relativistic Energy and E= mc 2
Chapter 9 Relativeity Chapter Outline 9-1 The Postulate t of Speial Relativity it 9- The Relativity of Time and Time Dilation 9-3 The Relativity of Length and Length Contration 9-4 The Relativisti Addition
More informationF = F x x + F y. y + F z
ECTION 6: etor Calulus MATH20411 You met vetors in the first year. etor alulus is essentially alulus on vetors. We will need to differentiate vetors and perform integrals involving vetors. In partiular,
More informationWhere as discussed previously we interpret solutions to this partial differential equation in the weak sense: b
Consider the pure initial value problem for a homogeneous system of onservation laws with no soure terms in one spae dimension: Where as disussed previously we interpret solutions to this partial differential
More informationThe gravitational phenomena without the curved spacetime
The gravitational phenomena without the urved spaetime Mirosław J. Kubiak Abstrat: In this paper was presented a desription of the gravitational phenomena in the new medium, different than the urved spaetime,
More informationarxiv:gr-qc/ v7 14 Dec 2003
Propagation of light in non-inertial referene frames Vesselin Petkov Siene College, Conordia University 1455 De Maisonneuve Boulevard West Montreal, Quebe, Canada H3G 1M8 vpetkov@alor.onordia.a arxiv:gr-q/9909081v7
More informationGreen s function for the wave equation
Green s funtion for the wave equation Non relativisti ase 1 The wave equations In the Lorentz Gauge, the wave equations for the potentials in Lorentz Gauge Gaussian units are: r 2 A 1 2 A 2 t = 4π 2 j
More informationarxiv:gr-qc/ v2 6 Feb 2004
Hubble Red Shift and the Anomalous Aeleration of Pioneer 0 and arxiv:gr-q/0402024v2 6 Feb 2004 Kostadin Trenčevski Faulty of Natural Sienes and Mathematis, P.O.Box 62, 000 Skopje, Maedonia Abstrat It this
More information( x vt) m (0.80)(3 10 m/s)( s) 1200 m m/s m/s m s 330 s c. 3.
Solutions to HW 10 Problems and Exerises: 37.. Visualize: At t t t 0 s, the origins of the S, S, and S referene frames oinide. Solve: We have 1 ( v/ ) 1 (0.0) 1.667. (a) Using the Lorentz transformations,
More informationMass Transfer 2. Diffusion in Dilute Solutions
Mass Transfer. iffusion in ilute Solutions. iffusion aross thin films and membranes. iffusion into a semi-infinite slab (strength of weld, tooth deay).3 Eamples.4 ilute diffusion and onvetion Graham (85)
More informationDO PHYSICS ONLINE. SPECIAL RELATIVITY Frames of Reference
DO PHYSICS ONLINE SPACE SPECIAL RELATIVITY Frames of Referene Spae travel Apollo 11 spaeraft: Earth Moon v ~ 40x10 3 km.h -1 Voyager spaeraft: v ~ 60x10 3 km.h -1 (no sling shot effet) Ulysses spaeraft:
More informationModule 5: Red Recedes, Blue Approaches. UNC-TFA H.S. Astronomy Collaboration, Copyright 2012
Objetives/Key Points Module 5: Red Reedes, Blue Approahes UNC-TFA H.S. Astronomy Collaboration, Copyright 2012 Students will be able to: 1. math the diretion of motion of a soure (approahing or reeding)
More informationIllustrating the relativity of simultaneity Bernhard Rothenstein 1), Stefan Popescu 2) and George J. Spix 3)
Illustrating the relativity of simultaneity ernhard Rothenstein 1), Stefan Popesu ) and George J. Spix 3) 1) Politehnia University of Timisoara, Physis Department, Timisoara, Romania, bernhard_rothenstein@yahoo.om
More informationBeams on Elastic Foundation
Professor Terje Haukaas University of British Columbia, Vanouver www.inrisk.ub.a Beams on Elasti Foundation Beams on elasti foundation, suh as that in Figure 1, appear in building foundations, floating
More informationPhysics 218, Spring February 2004
Physis 8 Spring 004 0 February 004 Today in Physis 8: dispersion in onduting dia Semilassial theory of ondutivity Condutivity and dispersion in tals and in very dilute ondutors : group veloity plasma frequeny
More informationPhysics 6C. Special Relativity. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Physis 6C Speial Relatiity Two Main Ideas The Postulates of Speial Relatiity Light traels at the same speed in all inertial referene frames. Laws of physis yield idential results in all inertial referene
More informationThe Special Theory of Relativity
The Speial Theory of Relatiity Galilean Newtonian Relatiity Galileo Galilei Isaa Newton Definition of an inertial referene frame: One in whih Newton s first law is alid. onstant if F0 Earth is rotating
More informationSimultaneity. CHAPTER 2 Special Theory of Relativity 2. Gedanken (Thought) experiments. The complete Lorentz Transformation. Re-evaluation of Time!
CHAPTER Speial Theory of Relatiity. The Need for Aether. The Mihelson-Morley Eperiment.3 Einstein s Postulates.4 The Lorentz Transformation.5 Time Dilation and Length Contration.6 Addition of Veloities.7
More information22.01 Fall 2015, Problem Set 6 (Normal Version Solutions)
.0 Fall 05, Problem Set 6 (Normal Version Solutions) Due: November, :59PM on Stellar November 4, 05 Complete all the assigned problems, and do make sure to show your intermediate work. Please upload your
More informationDIGITAL DISTANCE RELAYING SCHEME FOR PARALLEL TRANSMISSION LINES DURING INTER-CIRCUIT FAULTS
CHAPTER 4 DIGITAL DISTANCE RELAYING SCHEME FOR PARALLEL TRANSMISSION LINES DURING INTER-CIRCUIT FAULTS 4.1 INTRODUCTION Around the world, environmental and ost onsiousness are foring utilities to install
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E')
22.54 Neutron Interations and Appliations (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E') Referenes -- J. R. Lamarsh, Introdution to Nulear Reator Theory (Addison-Wesley, Reading, 1966),
More informationTowards an Absolute Cosmic Distance Gauge by using Redshift Spectra from Light Fatigue.
Towards an Absolute Cosmi Distane Gauge by using Redshift Spetra from Light Fatigue. Desribed by using the Maxwell Analogy for Gravitation. T. De Mees - thierrydemees @ pandora.be Abstrat Light is an eletromagneti
More informationClass XII - Physics Electromagnetic Waves Chapter-wise Problems
Class XII - Physis Eletromagneti Waves Chapter-wise Problems Multiple Choie Question :- 8 One requires ev of energy to dissoiate a arbon monoxide moleule into arbon and oxygen atoms The minimum frequeny
More informationQ2. [40 points] Bishop-Hill Model: Calculation of Taylor Factors for Multiple Slip
27-750, A.D. Rollett Due: 20 th Ot., 2011. Homework 5, Volume Frations, Single and Multiple Slip Crystal Plastiity Note the 2 extra redit questions (at the end). Q1. [40 points] Single Slip: Calulating
More informationCasimir self-energy of a free electron
Casimir self-energy of a free eletron Allan Rosenwaig* Arist Instruments, In. Fremont, CA 94538 Abstrat We derive the eletromagneti self-energy and the radiative orretion to the gyromagneti ratio of a
More informationAddition of velocities. Taking differentials of the Lorentz transformation, relative velocities may be calculated:
Addition of veloities Taking differentials of the Lorentz transformation, relative veloities may be allated: So that defining veloities as: x dx/dt, y dy/dt, x dx /dt, et. it is easily shown that: With
More informationMeasuring & Inducing Neural Activity Using Extracellular Fields I: Inverse systems approach
Measuring & Induing Neural Ativity Using Extraellular Fields I: Inverse systems approah Keith Dillon Department of Eletrial and Computer Engineering University of California San Diego 9500 Gilman Dr. La
More informationExperiment 03: Work and Energy
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physis Department Physis 8.01 Purpose of the Experiment: Experiment 03: Work and Energy In this experiment you allow a art to roll down an inlined ramp and run into
More informationTutorial 8: Solutions
Tutorial 8: Solutions 1. * (a) Light from the Sun arrives at the Earth, an average of 1.5 10 11 m away, at the rate 1.4 10 3 Watts/m of area perpendiular to the diretion of the light. Assume that sunlight
More informationAharonov-Bohm effect. Dan Solomon.
Aharonov-Bohm effet. Dan Solomon. In the figure the magneti field is onfined to a solenoid of radius r 0 and is direted in the z- diretion, out of the paper. The solenoid is surrounded by a barrier that
More information1 sin 2 r = 1 n 2 sin 2 i
Physis 505 Fall 005 Homework Assignment #11 Solutions Textbook problems: Ch. 7: 7.3, 7.5, 7.8, 7.16 7.3 Two plane semi-infinite slabs of the same uniform, isotropi, nonpermeable, lossless dieletri with
More informationINTRO VIDEOS. LESSON 9.5: The Doppler Effect
DEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS INTRO VIDEOS Big Bang Theory of the Doppler Effet Doppler Effet LESSON 9.5: The Doppler Effet 1. Essential Idea: The Doppler Effet desribes the phenomenon
More informationVelocity Addition in Space/Time David Barwacz 4/23/
Veloity Addition in Spae/Time 003 David arwaz 4/3/003 daveb@triton.net http://members.triton.net/daveb Abstrat Using the spae/time geometry developed in the previous paper ( Non-orthogonal Spae- Time geometry,
More informationElectromagnetic radiation of the travelling spin wave propagating in an antiferromagnetic plate. Exact solution.
arxiv:physis/99536v1 [physis.lass-ph] 15 May 1999 Eletromagneti radiation of the travelling spin wave propagating in an antiferromagneti plate. Exat solution. A.A.Zhmudsky November 19, 16 Abstrat The exat
More informationLightning electromagnetic environment in the presence of a tall grounded strike object
JOURNAL OF GEOPHYSICAL RESEARCH, VOL.,, doi:.9/4jd555, 5 Lightning eletromagneti environment in the presene of a tall grounded strike objet Yoshihiro Baba Department of Eletrial Engineering, Doshisha University,
More information4. (12) Write out an equation for Poynting s theorem in differential form. Explain in words what each term means physically.
Eletrodynamis I Exam 3 - Part A - Closed Book KSU 205/2/8 Name Eletrodynami Sore = 24 / 24 points Instrutions: Use SI units. Where appropriate, define all variables or symbols you use, in words. Try to
More informationCALCULATION OF NONLINEAR TUNE SHIFT USING BEAM POSITION MEASUREMENT RESULTS
International Journal of Modern Physis A Vol. 24, No. 5 (2009) 974 986 World Sientifi Publishing Company CALCULATION OF NONLINEAR TUNE SHIFT USING BEAM POSITION MEASUREMENT RESULTS PAVEL SNOPOK, MARTIN
More informationThe Hanging Chain. John McCuan. January 19, 2006
The Hanging Chain John MCuan January 19, 2006 1 Introdution We onsider a hain of length L attahed to two points (a, u a and (b, u b in the plane. It is assumed that the hain hangs in the plane under a
More informationEvaluation of effect of blade internal modes on sensitivity of Advanced LIGO
Evaluation of effet of blade internal modes on sensitivity of Advaned LIGO T0074-00-R Norna A Robertson 5 th Otober 00. Introdution The urrent model used to estimate the isolation ahieved by the quadruple
More informationDirectional Coupler. 4-port Network
Diretional Coupler 4-port Network 3 4 A diretional oupler is a 4-port network exhibiting: All ports mathed on the referene load (i.e. S =S =S 33 =S 44 =0) Two pair of ports unoupled (i.e. the orresponding
More information10.2 The Occurrence of Critical Flow; Controls
10. The Ourrene of Critial Flow; Controls In addition to the type of problem in whih both q and E are initially presribed; there is a problem whih is of pratial interest: Given a value of q, what fators
More informationPHYSICS 432/532: Cosmology Midterm Exam Solution Key (2018) 1. [40 points] Short answer (8 points each)
PHYSICS 432/532: Cosmology Midterm Exam Solution Key (2018) 1. [40 points] Short answer (8 points eah) (a) A galaxy is observed with a redshift of 0.02. How far away is the galaxy, and what is its lookbak
More informationTENSOR FORM OF SPECIAL RELATIVITY
TENSOR FORM OF SPECIAL RELATIVITY We begin by realling that the fundamental priniple of Speial Relativity is that all physial laws must look the same to all inertial observers. This is easiest done by
More information+Ze. n = N/V = 6.02 x x (Z Z c ) m /A, (1.1) Avogadro s number
In 1897, J. J. Thomson disovered eletrons. In 1905, Einstein interpreted the photoeletri effet In 1911 - Rutherford proved that atoms are omposed of a point-like positively harged, massive nuleus surrounded
More informationarxiv:physics/ v4 [physics.gen-ph] 9 Oct 2006
The simplest derivation of the Lorentz transformation J.-M. Lévy Laboratoire de Physique Nuléaire et de Hautes Energies, CNRS - IN2P3 - Universités Paris VI et Paris VII, Paris. Email: jmlevy@in2p3.fr
More informationThe Concept of Mass as Interfering Photons, and the Originating Mechanism of Gravitation D.T. Froedge
The Conept of Mass as Interfering Photons, and the Originating Mehanism of Gravitation D.T. Froedge V04 Formerly Auburn University Phys-dtfroedge@glasgow-ky.om Abstrat For most purposes in physis the onept
More informationGeneral Equilibrium. What happens to cause a reaction to come to equilibrium?
General Equilibrium Chemial Equilibrium Most hemial reations that are enountered are reversible. In other words, they go fairly easily in either the forward or reverse diretions. The thing to remember
More informationmax min z i i=1 x j k s.t. j=1 x j j:i T j
AM 221: Advaned Optimization Spring 2016 Prof. Yaron Singer Leture 22 April 18th 1 Overview In this leture, we will study the pipage rounding tehnique whih is a deterministi rounding proedure that an be
More informationPhysics (Theory) There are 30 questions in total. Question Nos. 1 to 8 are very short answer type questions and carry one mark each.
Physis (Theory) Tie allowed: 3 hours] [Maxiu arks:7 General Instrutions: (i) ll uestions are opulsory. (ii) (iii) (iii) (iv) (v) There are 3 uestions in total. Question Nos. to 8 are very short answer
More informationWRAP-AROUND GUSSET PLATES
WRAP-AROUND GUSSET PLATES Where a horizontal brae is loated at a beam-to-olumn intersetion, the gusset plate must be ut out around the olumn as shown in Figure. These are alled wrap-around gusset plates.
More informationFour-dimensional equation of motion for viscous compressible substance with regard to the acceleration field, pressure field and dissipation field
Four-dimensional equation of motion for visous ompressible substane with regard to the aeleration field, pressure field and dissipation field Sergey G. Fedosin PO box 6488, Sviazeva str. -79, Perm, Russia
More informationCanimals. borrowed, with thanks, from Malaspina University College/Kwantlen University College
Canimals borrowed, with thanks, from Malaspina University College/Kwantlen University College http://ommons.wikimedia.org/wiki/file:ursus_maritimus_steve_amstrup.jpg Purpose Investigate the rate of heat
More informationQUANTUM MECHANICS II PHYS 517. Solutions to Problem Set # 1
QUANTUM MECHANICS II PHYS 57 Solutions to Problem Set #. The hamiltonian for a lassial harmoni osillator an be written in many different forms, suh as use ω = k/m H = p m + kx H = P + Q hω a. Find a anonial
More informationAstr 5465 Mar. 29, 2018 Galactic Dynamics I: Disks
Galati Dynamis Overview Astr 5465 Mar. 29, 2018 Subjet is omplex but we will hit the highlights Our goal is to develop an appreiation of the subjet whih we an use to interpret observational data See Binney
More informationLecture 7 - Momentum. A Puzzle... Momentum. Basics (1)
Lecture 7 - omentum A Puzzle... An Experiment on Energy The shortest configuration of string joining three given points is the one where all three angles at the point of intersection equal 120. How could
More informationV. Interacting Particles
V. Interating Partiles V.A The Cumulant Expansion The examples studied in the previous setion involve non-interating partiles. It is preisely the lak of interations that renders these problems exatly solvable.
More information23.1 Tuning controllers, in the large view Quoting from Section 16.7:
Lesson 23. Tuning a real ontroller - modeling, proess identifiation, fine tuning 23.0 Context We have learned to view proesses as dynami systems, taking are to identify their input, intermediate, and output
More informationPhysics for Scientists & Engineers 2
Review Maxwell s Equations Physis for Sientists & Engineers 2 Spring Semester 2005 Leture 32 Name Equation Desription Gauss Law for Eletri E d A = q en Fields " 0 Gauss Law for Magneti Fields Faraday s
More informationF = c where ^ı is a unit vector along the ray. The normal component is. Iν cos 2 θ. d dadt. dp normal (θ,φ) = dpcos θ = df ν
INTRODUCTION So far, the only information we have been able to get about the universe beyond the solar system is from the eletromagneti radiation that reahes us (and a few osmi rays). So doing Astrophysis
More informationMetric of Universe The Causes of Red Shift.
Metri of Universe The Causes of Red Shift. ELKIN IGOR. ielkin@yande.ru Annotation Poinare and Einstein supposed that it is pratially impossible to determine one-way speed of light, that s why speed of
More informationa) What is the duration of the trip according to Ginette? b) What is the duration of the trip according to Tony?
Ginette stays on Earth while Tony travels towards a star loated 4.6 lightyears away from Earth. The speed of Tony s ship is 80% of the speed of light. www.how-to-draw-artoons-online.om/artoon-earth.html
More informationSOME FUNDAMENTAL ASPECTS OF COMPRESSIBLE FLOW
SOE FUNDAENAL ASECS OF CORESSIBLE FLOW ah number gas veloity mah number, speed of sound a a R < : subsoni : transoni > : supersoni >> : hypersoni art three : ah Number 7 Isentropi flow in a streamtube
More informationOn the Absolute Meaning of Motion
On the Absolute Meaning of Motion H. Edwards Publiation link: https://doi.org/10.1016/j.rinp.2017.09.053 Keywords: Kinematis; Gravity; Atomi Cloks; Cosmi Mirowave Bakground Abstrat The present manusript
More informationDynamics of the Electromagnetic Fields
Chapter 3 Dynamis of the Eletromagneti Fields 3.1 Maxwell Displaement Current In the early 1860s (during the Amerian ivil war!) eletriity inluding indution was well established experimentally. A big row
More informationMaximum Entropy and Exponential Families
Maximum Entropy and Exponential Families April 9, 209 Abstrat The goal of this note is to derive the exponential form of probability distribution from more basi onsiderations, in partiular Entropy. It
More information19 Lecture 19: Cosmic Microwave Background Radiation
PHYS 652: Astrophysis 97 19 Leture 19: Cosmi Mirowave Bakground Radiation Observe the void its emptiness emits a pure light. Chuang-tzu The Big Piture: Today we are disussing the osmi mirowave bakground
More informationExperiment 3: Basic Electronic Circuits II (tbc 1/7/2007)
Experiment 3: Basi Eletroni iruits II (tb /7/007) Objetive: a) To study the first-order dynamis of a apaitive iruits with the appliation of Kirhoff s law, Ohm s law and apaitane formula. b) To learn how
More information