HW07'sol - Classical Electrodynamics II PHY 506 Spring 2011...

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1 C:\User\Teaching\505-506\S11\HW07'sol.doc PHY 506 Classical Electrodynamics II Spring 2011 Homework 7 with solutions Problem 7.1 (10 points). Prove that the following quantities: (i) E 2 c 2 B 2 , and (ii) E B are Lorentz- invariant. Solution : Both facts can be proved by writing these expressions in Cartesian components,     z z y y x x z y x z y x B E B E B E B B B c E E E B c E B E , 2 2 2 2 2 2 2 2 2 2 , and then using Eqs. (9.134) of the lecture notes for the transformation of each component. A shorter way to reach the same goal is to notice that according to Eqs. (9.125) and (9.131) of the lecture notes, 1 , 2 2 2 2 2  F F c B c E , 4 G F c B E so that both combinations are double scalar products, and hence Lorentz-invariant. Problem 7.2 (10 points). Find the trajectory of a relativistic particle in a uniform electrostatic field E for the case of arbitrary initial velocity u (0). Hint : You are encouraged to explore alternative ways of integration the equation of motion, different from the one used in class for case u (0) E . Solutions : (i) An elegant alternative way to solve this problem is to integrate the 4-vector equation (9.145), u qF d dp , directly, considering the proper time of the particle as an argument. For the nonvanishing components of 4-velocity 2 we get equations c d u d d u d u d c d z x z ) ( ) ( , 0 ) ( ) ( , ) ( ) ( , where qE / cm is a constant parameter with the reciprocal time (s -1 ) dimensionality. The middle equation is elementary, and yields  C u u c cu u z x x x 2 / 1 2 2 2 ) 0 ( ) 0 ( ) 0 ( const . The remaining two equations may be combined (by the additional differentiation of any of them over and substitution of the remaining equation) to give similar second-order differential equations 1 Actually, the first of this expressions has been discussed in class – see Eq. (9.217). 2 I am using the same coordinate system choice as discussed in Sec. 9.6 of the lecture notes, with axis z along the electric field, and axis x in the plane of motion, so that u y = 0 for any .
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2 C:\User\Teaching\505-506\S11\HW07'sol.doc ), ( ) ( ), ( ) ( 2 2 2 2 2 2 z z u u d d c c d d with similar solutions ) . ( sinh ), ( cosh 0 0 A u A c z ( * ) Constants
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This note was uploaded on 09/10/2011 for the course PHY 506 taught by Professor Stephens,p during the Spring '08 term at SUNY Stony Brook.

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HW07'sol - Classical Electrodynamics II PHY 506 Spring 2011...

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