4/13/2015. Outlines CHAPTER 12 ELECTRODYNAMICS & RELATIVITY. 1. The special theory of relativity. 2. Relativistic Mechanics

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1 CHAPTER 12 ELECTRODYNAMICS & RELATIVITY Lee Chow Department of Physics University of Central Florida Orlando, FL Outlines 1. The special theory of relativity 2. Relativistic Mechanics 3. Relativistic Electrodynamics 2 Einstein s postulates What is an inertial reference frame? The laws of Physics apply in all inertial reference frames The speed of light in vacuum is a constant regardless of the motion of the source or observer. Consequences 1. Length contraction,, 2. Time dilation,, 3. Simultaneity is not absolute. 3 Review Train-Platform problem A Platform reference frame Two lightnings strike the ends of the train simultaneously. For observer in the platform frame, he can show that these two lightnings happen simultaneously. Will these two events happen at the same time for observer in the train??? The answer is NO. The observer in the middle of the train will receive the signal from B first. B 4 Train reference frame Example 12.3 A B The barn and ladder paradox A B 5 6 1

2 The Lorentz Transformation In classical theory, (before Einstein) for different inertial reference frames, we have Galilean transformation: If we take into account of length contraction and time dilation, we have Lorentz transformation: The Structure of Spacetime If we let,,,, Then Lorentz transformation takes on a very simple form in the matrix notation Four-vectors 8 Covariant and Contravariant vectors Invariance of interval between two events Covariant vectors Contravariant vectors,,,,,, Position, velocity, and acceleration are all contravariant vectors, while derivative of a function such as the gradient of a scalar function is a covariant vector. Define interval between two events as the inner product (dot product) of two 4-vectors For I < Time-like I > Space-like I = Light-like,,,,,, 9 10 If a vector is transformed under the rule: Space-time diagram Examples:, (,,,,. Covariant vector If a vector is transformed under the rule: Contravariant vector Examples: position vector,, (,,,, (V/c,,,, etc

3 12.2 Relativistic Mechanics Proper time & Proper velocity Let s recall the definition of Proper time and Proper length: Proper time--- Time interval measured with one clock. Proper length Length measured when the object is at rest. Assume you are in the train reference frame. The train is moving with a velocity of relative the platform frame. (1) Time measured by you using your watch 13 dt Time measured by clocks on the ground The moving cloak ticks slower. 14 The relative velocity of the train is (2) where and dt are distance and time measured in platform reference frame.. Define a proper velocity as follow: Proper velocity This is a hybrid quantity, where is measured in the ground-frame, while is in the moving frame. 15 l This proper velocity has some advantages over the ordinary velocity, since it transforms easily under Lorentz transformation. Here we define as follow: and is a 4-vector, transforms under Lorentz transformation just like a coordinate. 16 For the ordinary velocity, the Lorentz transformation is given by: / (7) Relativistic Energy and Momentum We define Momentum as mass times proper velocity Relativistic momentum And then extend the concept to 4-vector notation: with We further define relativistic energy as / /

4 The energy-momentum 4-vector can be expressed as From eq. (6) we can see that if,,,,, Note that the rest mass is, when. The energy-moment 4-vector also satisfy the Lorentz transformation. If, The inner product of the energy-momentum 4-vector is given by For massless particles (12) (13) 21 Inner products of 4-Vectors Inner product of and Assume there are different frames, such that So the inner product of two vectors is Invariant under coordinate transformations. Skip and Relativistic Electrodynamics Unlike Newtonian mechanics, classical electrodynamics is consistent with the special relativity. Namely, Maxwell equations and Lorentz force law can be applied to any initial reference frame. Classical mechanics on the other hand needs to be modified in order to satisfy the special relativity. We know change of magnetic flux induces emf & E field, while change of the electric flux generates magnetic field. Now we want to demonstrate that field and field are just two sides of a coin. 23 Interaction between a current and a charge q S-frame Assume we have a string of positive line charge moving to the right and a string of negative line charge moving to the right. The total current is given by Meanwhile there is a charge q, a distance s away, is moving to the right at. ( u < v)there is no electric force, because the net charge of the wire is zero. But q experiences the Lorentz force. 24 4

5 Now from the -frame, the charge is stationary, but the two strings of line charges move at different speed. From velocity transformation, we have Because >, there is more length contraction for the negatively charged string, hence higher line charge density. To calculate, we first assume that charge is invariant under Lorentz transformation, so all the effect of the relative velocity is to cause the length contraction, so we calculation of charge density is affected by the volume measurement. where is the line charge density when the charges are not moving and This will generate a net negative charge on the wire, so there is a pure Coulomb force between q and the wire. 25 Now substitute eq. (16) in eq. (19), we obtain 26 In the S frame, the stationary charge q senses a net line charge density As a result of length contraction, the initially neutral wire will be charged in a different reference. 27 The field in the -frame due to is given by and Re-arrange the above, 28 From equation 12.68, we can see that the attractive force on q is perpendicular to the relative velocity, so the force in the S-frame is given by Since The expression in the parenthesis is the magnetic field at a distance S away from the wire due to a current in the wire How the Fields Transform We will use examples to show how fields transform. First we will review the notation introduced in this chapter. Space-Time 4-vector,,, Charge density 4-vector Energy-Momentum 4 vector,,, Vector potential 4-vector,,,,,, Note: Charge is invariant under Lorentz transformation. 29 Will Field transform like 4-vector? 30 5

6 A parallel plate capacitor at rest in frame However -Frame -Frame S-frame is in relative motion with respect to the Frame. The fields in the two reference frames are: ) The field in the y-direction (parallel to the velocity), is independent of the relative velocity. Example A point charge q is at rest in frame what is the field of q in the S frame? The length in x-direction will be contracted (l l) ) In the frame, the electric field is given by If we written it for each component, then Now if we use the expressions (31) and (32), the field in the S-frame can be written below: We need to convert the above expressions into variables used in the S-frame. 33 Substitute (34) into (35) b At t=0, q passes through the origin 34 We end up with / / Define as a vector from q to P, as shown in Figure on previous page, we can express eq. (36) as a vector, / 35 Eq. (37) is the same as eq. (12.93) in the book. It is also the same as eq. (10.75) in the book. For example, if we let and / / / / / /. Next we will study a case where and are non-zero. 36 6

7 After some math, we arrive at the following equations and Now if we go back to the parallel-plate capacitor problem, in the S-frame, there are and fields and Now in the -frame, (moving at a velocity relative to S-frame, and relative to -frame) In general, the and fields transform as follow: and 37 This is not the same as Lorentz transformation, and components of or are not 4-vectors. 38 Two special cases of the transformations are: Electrodynamics in tensor form and the field tensor Continueity equation in 3D 1. Pure electric field. If in the S-frame 2. Pure Magnetic Field If in the S-frame Define,,, and (39) Lorentz conditions The field Tensor Define,,, Lorentz condition reduced eq. (41) to