PAP111_Lecture18 - PAP111 Lecture18 Equations N

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PAP 111  Mechanics and Relativity Lecture 18 Relativistic Mechanics The Lorentz Transformation Equation
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Galilean Transformation  Equations and Electromagnetism The application of Galilean Relativity to mechanics has  led to the set of Galilean Transformation and Velocity  Equations. Newton’s Laws are invariant with respect to these set of  Equations. What about Maxwell’s theory of Electromagnetism? Are  they invariant with respect to these set of Equations? The answer to these questions can be obtained by  applying the Galilean transformation equation to a major  result of Maxwell’s theory – the EM wave equations.
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Galilean Transformation and  the EM Wave Equation 0 ) , ( 1 ) , ( 2 2 2 2 2 = - t t x E c x t x E 0 ' ) ' , ' ( 1 ' ' ) ' , ' ( 2 ' ) ' , ' ( 1 2 2 2 2 2 2 2 2 2 = - + - t t x E c t x t x E c v x t x E c v vt x x - = ' t t = ' The wave equation for EM waves as deduced from Maxwell’s  equations is given by By performing a Galilean co-ordinate transformation along the  x   direction:                 ;         , we obtain This shows that the physical laws of electromagnetism do not  maintain the same form in all inertial frames under the Galilean  transformation.
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Lorentz Transformation  Equations In the following, we will derive a set of transformation equations.  Maxwell’s equations will be invariant with respect to these equations.  We shall determine these equations through the postulates of special  relativity. Let us observe an event from two inertial frames S and S’. The event  is recorded as x, y, z, t in the S frame, and x’, y’, z’, t’ in the S’ frame.  The x and x’ axis are common for the two frames and the  corresponding frames are parallel. The relative velocity of the two  frames is v along the x-x’ axis. Also, at the instant the origin O and O’  of the two frames coincide, the respective clocks are set to t = 0 and  t’  = 0. (Refer to Lecture 4 for a Figure of these S and S’ frames.) We now seek the equations of transformation which relates one  observer’s space-time coordinates of an event with the other  observer’s coordinates of the same event.
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Lorentz Transformation  Equations In addition to the postulates of relativity, we will also assume that  space and time are homogeneous (i.e., all points in space and time  are equivalent), and space is isotropic (i.e., space is equivalent with  respect to direction. E.g. ‘up’ space is the same as ‘down’ space).
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