Magnetic Forces from Special Relativity

Introduction

It is possible to transform the speeds of both the fixed positive charges in the wire as well as the moving negative charges in the wire to a frame where the test particle is stationary.  Then the magnetic forces on the test particle disappear and we find that we are left with a net electric charge on the wire segments that are parallel to the direction of the test particle motion and the direction of the current on that wire segment.

Figure 1: Current loop and test charge as viewed in laboratory frame.  The velocity of the test charge is vt so its beta is vt/c.  The velocity of the negative charge current is vI.

Velocity Transformations

Looking at Figure 1 we see that prior to transformation:

v t =| v t | x ^ v Itop =| v I | x ^ v Ibottom =| v I | x ^ v Ileft =| v I | y ^ v Iright =| v I | y ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaCODam aaBaaaleaacaWG0baabeaakiabg2da9iabgkHiTiaacYhacaWG2bWa aSbaaSqaaiaadshaaeqaaOGaaiiFaiqacIhagaqcaaqaaiaahAhada WgaaWcbaGaamysaiaaykW7caWG0bGaam4BaiaadchaaeqaaOGaeyyp a0JaeyOeI0IaaiiFaiaadAhadaWgaaWcbaGaamysaaqabaGccaGG8b GabiiEayaajaaabaGaaCODamaaBaaaleaacaWGjbGaaGPaVlaadkga caWGVbGaamiDaiaadshacaWGVbGaamyBaaqabaGccqGH9aqpcaGG8b GaamODamaaBaaaleaacaWGjbaabeaakiaacYhaceGG4bGbaKaaaeaa caWG2bWaaSbaaSqaaiaadMeacaaMc8UaamiBaiaadwgacaWGMbGaam iDaaqabaGccqGH9aqpcqGHsislcaGG8bGaamODamaaBaaaleaacaWG jbaabeaakiaacYhaceWH5bGbaKaaaeaacaWG2bWaaSbaaSqaaiaadM eacaaMc8UaamOCaiaadMgacaWGNbGaamiAaiaadshaaeqaaOGaeyyp a0JaaiiFaiaadAhadaWgaaWcbaGaamysaaqabaGccaGG8bGabCyEay aajaaaaaa@78B1@  

(1.1)

 

After the transformation to set the test particle speed, vt, to zero we have:

v t =| v t | x ^ v Itop = ( | v t || v I | ) 1 | v I v t | c 2 x ^ v Ibottom = ( | v t |+| v I | ) 1+ | v I v t | c 2 x ^ v Ileft =| v I | y ^ +| v t | x ^ v Ileft =| v I | y ^ +| v t | x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaCODam aaBaaaleaacaWG0baabeaakiabg2da9iaacYhacaWG2bWaaSbaaSqa aiaadshaaeqaaOGaaiiFaiqacIhagaqcaaqaaiaahAhadaWgaaWcba GaamysaiaaykW7caWG0bGaam4BaiaadchaaeqaaOGaeyypa0ZaaSaa aeaadaqadaqaaiaacYhacaWG2bWaaSbaaSqaaiaadshaaeqaaOGaai iFaiabgkHiTiaacYhacaWG2bWaaSbaaSqaaiaadMeaaeqaaOGaaiiF aaGaayjkaiaawMcaaaqaaiaaigdacqGHsisldaWcaaqaaiaacYhaca WG2bWaaSbaaSqaaiaadMeaaeqaaOGaamODamaaBaaaleaacaWG0baa beaakiaacYhaaeaacaWGJbWaaWbaaSqabeaacaaIYaaaaaaaaaGcce GG4bGbaKaaaeaacaWH2bWaaSbaaSqaaiaadMeacaaMc8UaamOyaiaa d+gacaWG0bGaamiDaiaad+gacaWGTbaabeaakiabg2da9maalaaaba WaaeWaaeaacaGG8bGaamODamaaBaaaleaacaWG0baabeaakiaacYha cqGHRaWkcaGG8bGaamODamaaBaaaleaacaWGjbaabeaakiaacYhaai aawIcacaGLPaaaaeaacaaIXaGaey4kaSYaaSaaaeaacaGG8bGaamOD amaaBaaaleaacaWGjbaabeaakiaadAhadaWgaaWcbaGaamiDaaqaba GccaGG8baabaGaam4yamaaCaaaleqabaGaaGOmaaaaaaaaaOGabiiE ayaajaaabaGaaCODamaaBaaaleaacaWGjbGaaGPaVlaadYgacaWGLb GaamOzaiaadshaaeqaaOGaeyypa0JaeyOeI0IaaiiFaiaadAhadaWg aaWcbaGaamysaaqabaGccaGG8bGabCyEayaajaGaey4kaSIaaiiFai aadAhadaWgaaWcbaGaamiDaaqabaGccaGG8bGabiiEayaajaaabaGa aCODamaaBaaaleaacaWGjbGaaGPaVlaadYgacaWGLbGaamOzaiaads haaeqaaOGaeyypa0JaaiiFaiaadAhadaWgaaWcbaGaamysaaqabaGc caGG8bGabCyEayaajaGaey4kaSIaaiiFaiaadAhadaWgaaWcbaGaam iDaaqabaGccaGG8bGabiiEayaajaaaaaa@A2D9@  

(1.2)

Let us define some value of  the various Lorentz factors, γ.

γ t = 1 1 v t 2 c 2 1 1 β t 2 γ I = 1 1 v I 2 c 2 1 1 β I 2 γ Ileft = 1 1 v Ileft 2 c 2 1 1 β Ileft 2 γ Iright = 1 1 v Iright 2 c 2 1 1 β Iright 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeq4SdC 2aaSbaaSqaaiaadshaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaWa aOaaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWG2bWaa0baaSqaaiaads haaeaacaaIYaaaaaGcbaGaam4yamaaCaaaleqabaGaaGOmaaaaaaaa beaaaaGccqGHHjIUdaWcaaqaaiaaigdaaeaadaGcaaqaaiaaigdacq GHsislcqaHYoGydaqhaaWcbaGaamiDaaqaaiaaikdaaaaabeaaaaaa keaacqaHZoWzdaWgaaWcbaGaamysaaqabaGccqGH9aqpdaWcaaqaai aaigdaaeaadaGcaaqaaiaaigdacqGHsisldaWcaaqaaiaadAhadaqh aaWcbaGaamysaaqaaiaaikdaaaaakeaacaWGJbWaaWbaaSqabeaaca aIYaaaaaaaaeqaaaaakiabggMi6oaalaaabaGaaGymaaqaamaakaaa baGaaGymaiabgkHiTiabek7aInaaDaaaleaacaWGjbaabaGaaGOmaa aaaeqaaaaaaOqaaiabeo7aNnaaBaaaleaacaWGjbGaaGPaVlaadYga caWGLbGaamOzaiaadshaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaaba WaaOaaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWG2bWaa0baaSqaaiaa dMeacaaMc8UaamiBaiaadwgacaWGMbGaamiDaaqaaiaaikdaaaaake aacaWGJbWaaWbaaSqabeaacaaIYaaaaaaaaeqaaaaakiabggMi6oaa laaabaGaaGymaaqaamaakaaabaGaaGymaiabgkHiTiabek7aInaaDa aaleaacaWGjbGaaGPaVlaadYgacaWGLbGaamOzaiaadshaaeaacaaI YaaaaaqabaaaaaGcbaGaeq4SdC2aaSbaaSqaaiaadMeacaaMc8Uaam OCaiaadMgacaWGNbGaamiAaiaadshaaeqaaOGaeyypa0ZaaSaaaeaa caaIXaaabaWaaOaaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWG2bWaa0 baaSqaaiaadMeacaaMc8UaamOCaiaadMgacaWGNbGaamiAaiaadsha aeaacaaIYaaaaaGcbaGaam4yamaaCaaaleqabaGaaGOmaaaaaaaabe aaaaGccqGHHjIUdaWcaaqaaiaaigdaaeaadaGcaaqaaiaaigdacqGH sislcqaHYoGydaqhaaWcbaGaamysaiaaykW7caWGYbGaamyAaiaadE gacaWGObGaamiDaaqaaiaaikdaaaaabeaaaaaaaaa@A333@  

(1.3)

β Itop = β t β I 1+ β I β t β Ibottom = β t β I 1+ β I β t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqOSdi 2aaSbaaSqaaiaadMeacaaMc8UaamiDaiaad+gacaWGWbaabeaakiab g2da9maalaaabaGaeqOSdi2aaSbaaSqaaiaadshaaeqaaOGaeyOeI0 IaeqOSdi2aaSbaaSqaaiaadMeaaeqaaaGcbaGaaGymaiabgUcaRiab ek7aInaaBaaaleaacaWGjbaabeaakiabek7aInaaBaaaleaacaWG0b aabeaaaaaakeaacqaHYoGydaWgaaWcbaGaamysaiaaykW7caWGIbGa am4BaiaadshacaWG0bGaam4Baiaad2gaaeqaaOGaeyypa0ZaaSaaae aacqaHYoGydaWgaaWcbaGaamiDaaqabaGccqGHsislcqaHYoGydaWg aaWcbaGaamysaaqabaaakeaacaaIXaGaey4kaSIaeqOSdi2aaSbaaS qaaiaadMeaaeqaaOGaeqOSdi2aaSbaaSqaaiaadshaaeqaaaaaaaaa @63F1@  

(1.4)

To compute the charge density for these betas we must obtain their Lorentz boost γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo7aNbaa@3793@ .

1 ( γ Itop ' ) 2 =1 ( β t β I 1 β I β t ) 2 1 ( γ Ibottom ' ) 2 =1 ( β t + β I 1+ β I β t ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaWaaSaaae aacaaIXaaabaWaaeWaaeaacqaHZoWzdaqhaaWcbaGaamysaiaaykW7 caWG0bGaam4BaiaadchaaeaaaaGccaGGNaaacaGLOaGaayzkaaWaaW baaSqabeaacaaIYaaaaaaakiabg2da9iaaigdacqGHsisldaqadaqa amaalaaabaGaeqOSdi2aaSbaaSqaaiaadshaaeqaaOGaeyOeI0Iaeq OSdi2aaSbaaSqaaiaadMeaaeqaaaGcbaGaaGymaiabgkHiTiabek7a InaaBaaaleaacaWGjbaabeaakiabek7aInaaBaaaleaacaWG0baabe aaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaakeaadaWc aaqaaiaaigdaaeaadaqadaqaaiabeo7aNnaaBaaaleaacaWGjbGaaG PaVlaadkgacaWGVbGaamiDaiaadshacaWGVbGaamyBaaqabaGccaGG NaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaakiabg2da9i aaigdacqGHsisldaqadaqaamaalaaabaGaeqOSdi2aaSbaaSqaaiaa dshaaeqaaOGaey4kaSIaeqOSdi2aaSbaaSqaaiaadMeaaeqaaaGcba GaaGymaiabgUcaRiabek7aInaaBaaaleaacaWGjbaabeaakiabek7a InaaBaaaleaacaWG0baabeaaaaaakiaawIcacaGLPaaadaahaaWcbe qaaiaaikdaaaaaaaa@742A@  

(1.5)

 

After a bit of algebra we obtain the results:

 

1 ( γ Itop ' ) 2 = (1 β I 2 )(1 β t 2 ) ( 1 β I β t ) 2 1 ( γ Ibottom ' ) 2 = (1 β I 2 )(1 β t 2 ) ( 1+ β I β t ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaWaaSaaae aacaaIXaaabaWaaeWaaeaacqaHZoWzdaqhaaWcbaGaamysaiaaykW7 caWG0bGaam4BaiaadchaaeaacaGGNaaaaaGccaGLOaGaayzkaaWaaW baaSqabeaacaaIYaaaaaaakiabg2da9maalaaabaGaaiikaiaaigda cqGHsislcqaHYoGydaqhaaWcbaGaamysaaqaaiaaikdaaaGccaGGPa GaaiikaiaaigdacqGHsislcqaHYoGydaqhaaWcbaGaamiDaaqaaiaa ikdaaaGccaGGPaaabaWaaeWaaeaacaaIXaGaeyOeI0IaeqOSdi2aaS baaSqaaiaadMeaaeqaaOGaeqOSdi2aaSbaaSqaaiaadshaaeqaaaGc caGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaaaOqaamaalaaaba GaaGymaaqaamaabmaabaGaeq4SdC2aa0baaSqaaiaadMeacaaMc8Ua amOyaiaad+gacaWG0bGaamiDaiaad+gacaWGTbaabaGaai4jaaaaaO GaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaGccqGH9aqpdaWc aaqaaiaacIcacaaIXaGaeyOeI0IaeqOSdi2aa0baaSqaaiaadMeaae aacaaIYaaaaOGaaiykaiaacIcacaaIXaGaeyOeI0IaeqOSdi2aa0ba aSqaaiaadshaaeaacaaIYaaaaOGaaiykaaqaamaabmaabaGaaGymai abgUcaRiabek7aInaaBaaaleaacaWGjbaabeaakiabek7aInaaBaaa leaacaWG0baabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaa aaaaaaaaa@7E04@  

(1.6)

Using definitions of the gammas of t and I equation (1.6) becomes:

 

1 ( γ Itop ' ) 2 = 1 γ I 2 γ t 2 ( 1 β I β t ) 2 1 ( γ Ibottom ' ) 2 = 1 γ I 2 γ t 2 ( 1+ β I β t ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaWaaSaaae aacaaIXaaabaWaaeWaaeaacqaHZoWzdaqhaaWcbaGaamysaiaaykW7 caWG0bGaam4BaiaadchaaeaaaaGccaGGNaaacaGLOaGaayzkaaWaaW baaSqabeaacaaIYaaaaaaakiabg2da9maalaaabaGaaGymaaqaaiab eo7aNnaaDaaaleaacaWGjbaabaGaaGOmaaaakiabeo7aNnaaDaaale aacaWG0baabaGaaGOmaaaakmaabmaabaGaaGymaiabgkHiTiabek7a InaaBaaaleaacaWGjbaabeaakiabek7aInaaBaaaleaacaWG0baabe aaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaaakeaadaWc aaqaaiaaigdaaeaadaqadaqaaiabeo7aNnaaDaaaleaacaWGjbGaaG PaVlaadkgacaWGVbGaamiDaiaadshacaWGVbGaamyBaaqaaaaakiaa cEcaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaOGaeyypa0 ZaaSaaaeaacaaIXaaabaGaeq4SdC2aa0baaSqaaiaadMeaaeaacaaI YaaaaOGaeq4SdC2aa0baaSqaaiaadshaaeaacaaIYaaaaOWaaeWaae aacaaIXaGaey4kaSIaeqOSdi2aaSbaaSqaaiaadMeaaeqaaOGaeqOS di2aaSbaaSqaaiaadshaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabe aacaaIYaaaaaaaaaaa@738E@  

(1.7)

and then the final expression for the gamma of the charge density  becomes:

γ Itop '= γ I γ t (1 β I β t ) γ Ibottom '= γ I γ t (1+ β I β t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeq4SdC 2aaSbaaSqaaiaadMeacaaMc8UaamiDaiaad+gacaWGWbaabeaakiaa cEcacqGH9aqpcqaHZoWzdaWgaaWcbaGaamysaaqabaGccqaHZoWzda WgaaWcbaGaamiDaaqabaGccaGGOaGaaGymaiabgkHiTiabek7aInaa BaaaleaacaWGjbaabeaakiabek7aInaaBaaaleaacaWG0baabeaaki aacMcaaeaacqaHZoWzdaWgaaWcbaGaamysaiaaykW7caWGIbGaam4B aiaadshacaWG0bGaam4Baiaad2gaaeqaaOGaai4jaiabg2da9iabeo 7aNnaaBaaaleaacaWGjbaabeaakiabeo7aNnaaBaaaleaacaWG0baa beaakiaacIcacaaIXaGaey4kaSIaeqOSdi2aaSbaaSqaaiaadMeaae qaaOGaeqOSdi2aaSbaaSqaaiaadshaaeqaaOGaaiykaaaaaa@6638@  

(1.8)

The positive charge spacing is contracted by the factor 1 γ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaG ymaaqaaiabeo7aNnaaBaaaleaacaWG0baabeaaaaaaaa@3983@  and the negative charge spacing is contracted by the factor 1 γ I ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaG ymaaqaaiabeo7aNnaaBaaaleaacaWGjbaabeaakiaacEcaaaaaaa@3A0D@ .  Prior to transformation let the positive charge per unit length be + λ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgUcaRiabeU 7aSnaaDaaaleaacaaIWaaabaaaaaaa@3969@  and the negative charge per unit length be λ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabeU 7aSnaaDaaaleaacaaIWaaabaaaaaaa@3974@   .

You should recall that the negative charge was moving prior to transformation, and therefore charge spacings were contracted by 1 γ I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaG ymaaqaaiabeo7aNnaaBaaaleaacaWGjbaabeaaaaaaaa@3958@  prior to the transformation so that the factor 1 γ I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaG ymaaqaaiabeo7aNnaaBaaaleaacaWGjbaabeaaaaaaaa@3958@  must be multiplied by λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSnaaBa aaleaacqGHsislaeqaaaaa@38B9@  to get the effective negative charge density.

γ Itop '= γ I γ t (1 β I β t ) γ I γ Ibottom '= γ I γ t (1+ β I β t ) γ I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeq4SdC 2aaSbaaSqaaiaadMeacaaMc8UaamiDaiaad+gacaWGWbaabeaakiaa cEcacqGH9aqpdaWcaaqaaiabeo7aNnaaBaaaleaacaWGjbaabeaaki abeo7aNnaaBaaaleaacaWG0baabeaakiaacIcacaaIXaGaeyOeI0Ia eqOSdi2aaSbaaSqaaiaadMeaaeqaaOGaeqOSdi2aaSbaaSqaaiaads haaeqaaOGaaiykaaqaaiabeo7aNnaaBaaaleaacaWGjbaabeaaaaaa keaacqaHZoWzdaWgaaWcbaGaamysaiaaykW7caWGIbGaam4Baiaads hacaWG0bGaam4Baiaad2gaaeqaaOGaai4jaiabg2da9maalaaabaGa eq4SdC2aaSbaaSqaaiaadMeaaeqaaOGaeq4SdC2aaSbaaSqaaiaads haaeqaaOGaaiikaiaaigdacqGHRaWkcqaHYoGydaWgaaWcbaGaamys aaqabaGccqaHYoGydaWgaaWcbaGaamiDaaqabaGccaGGPaaabaGaeq 4SdC2aaSbaaSqaaiaadMeaaeqaaaaaaaaa@6BA4@  

(1.9)

 

Then the new net charges per unit length will be

λ top = γ t λ + γ Itop ' λ = γ t λ 0 ( 1(1 β I β t ) ) = γ t β I β t λ 0 λ bottom = γ t λ + γ Ibottom ' λ = γ t λ 0 ( 1(1+ β I β t ) ) = γ t β I β t λ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeq4UdW 2aaSbaaSqaaiaadshacaWGVbGaamiCaaqabaGccqGH9aqpcqaHZoWz daWgaaWcbaGaamiDaaqabaGccqaH7oaBdaWgaaWcbaGaey4kaScabe aakiabgkHiTiabeo7aNnaaBaaaleaacaWGjbGaaGPaVlaadshacaWG VbGaamiCaaqabaGccaGGNaGaeq4UdW2aaSbaaSqaaiabgkHiTaqaba GccqGH9aqpcqaHZoWzdaWgaaWcbaGaamiDaaqabaGccqaH7oaBdaWg aaWcbaGaaGimaaqabaGcdaqadaqaaiaaigdacqGHsislcaGGOaGaaG ymaiabgkHiTiabek7aInaaBaaaleaacaWGjbaabeaakiabek7aInaa BaaaleaacaWG0baabeaakiaacMcaaiaawIcacaGLPaaaaeaacqGH9a qpcqaHZoWzdaWgaaWcbaGaamiDaaqabaGccqaHYoGydaWgaaWcbaGa amysaaqabaGccqaHYoGydaWgaaWcbaGaamiDaaqabaGccqaH7oaBda WgaaWcbaGaaGimaaqabaaakeaacqaH7oaBdaWgaaWcbaGaamOyaiaa d+gacaWG0bGaamiDaiaad+gacaWGTbaabeaakiabg2da9iabeo7aNn aaBaaaleaacaWG0baabeaakiabeU7aSnaaBaaaleaacqGHRaWkaeqa aOGaeyOeI0Iaeq4SdC2aaSbaaSqaaiaadMeacaaMc8UaamOyaiaad+ gacaWG0bGaamiDaiaad+gacaWGTbaabeaakiaacEcacqaH7oaBdaWg aaWcbaGaeyOeI0cabeaakiabg2da9iabeo7aNnaaBaaaleaacaWG0b aabeaakiabeU7aSnaaBaaaleaacaaIWaaabeaakmaabmaabaGaaGym aiabgkHiTiaacIcacaaIXaGaey4kaSIaeqOSdi2aaSbaaSqaaiaadM eaaeqaaOGaeqOSdi2aaSbaaSqaaiaadshaaeqaaOGaaiykaaGaayjk aiaawMcaaaqaaiabg2da9iabgkHiTiabeo7aNnaaBaaaleaacaWG0b aabeaakiabek7aInaaBaaaleaacaWGjbaabeaakiabek7aInaaBaaa leaacaWG0baabeaakiabeU7aSnaaBaaaleaacaaIWaaabeaaaaaa@A69C@  

(1.10)

Please note in equation (1.10) that the sign of β I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaWGjbaabeaaaaa@3887@  in the top side of the loop was opposite from that in the bottom side of the loop.  Looking at equation (1.8), that fact also results in quite different charge densities in the top and bottom sides of the loop.

When the law for transformation of the force from the charge's moving frame to the laboratory frame

F testchargestationary = γ t F moving MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaa WcbaGaamiDaiaadwgacaWGZbGaamiDaiaaykW7caWGJbGaamiAaiGa cggacaGGYbGaai4zaiaadwgacaWGZbGaamiDaiaadggacaWG0bGaam yAaiaad+gacaWGUbGaamyyaiaadkhacaWG5baabeaakiabg2da9iab eo7aNnaaBaaaleaacaWG0baabeaakiaadAeadaWgaaWcbaGaamyBai aad+gacaWG2bGaamyAaiaad6gacaWGNbaabeaaaaa@55D3@  

(1.11)

is taken into account, the γ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo7aNnaaBa aaleaacaWG0bGaaGPaVdqabaaaaa@3A43@  factor in equation (1.10) is factored out and the new electric force is the same as the magnetic force before transformation.

Charge Density in a Current Loop with a Finite number of Charges

            Assume that we have linear charge distributions in all 4 legs of a current carrying loop and that we've transformed the charge velocities to make the test charge near the top leg stationary.  

 

To be definite, let's assume that the total number of negative charges in the loop is Nc and, since the loop wire has no net charge, the number of positive charges is also Nc.  Since total charge number is a scalar, the transformation discussed above cannot change this.  The height of the loop remains the same during the test particle transformation but its width is contracted by the factor 1/γt.  We will first compute the number of positive charges in the loop sides and the top and bottom of the loop.  We have the equation:

λ 0 + = N c 2(h+w) λ sides = λ 0 + λ T+B = λ 0 + γ t δ s sides = 1 λ sides = 2(h+w) N c δ s T+B = 1 λ T+B = 2(h+w) N c γ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeq4UdW 2aa0baaSqaaiaaicdaaeaacqGHRaWkaaGccqGH9aqpdaWcaaqaaiaa d6eadaWgaaWcbaGaam4yaaqabaaakeaacaaIYaGaaiikaiaadIgacq GHRaWkcaWG3bGaaiykaaaaaeaacqaH7oaBdaWgaaWcbaGaam4Caiaa dMgacaWGKbGaamyzaiaadohaaeqaaOGaeyypa0Jaeq4UdW2aa0baaS qaaiaaicdaaeaacqGHRaWkaaaakeaacqaH7oaBdaWgaaWcbaGaamiv aiabgUcaRiaadkeaaeqaaOGaeyypa0Jaeq4UdW2aa0baaSqaaiaaic daaeaacqGHRaWkaaGccqaHZoWzdaWgaaWcbaGaamiDaaqabaaakeaa cqaH0oazcaWGZbWaaSbaaSqaaiaadohacaWGPbGaamizaiaadwgaca WGZbaabeaakiabg2da9maalaaabaGaaGymaaqaaiabeU7aSnaaBaaa leaacaWGZbGaamyAaiaadsgacaWGLbGaam4CaaqabaaaaOGaeyypa0 ZaaSaaaeaacaaIYaGaaiikaiaadIgacqGHRaWkcaWG3bGaaiykaaqa aiaad6eadaWgaaWcbaGaam4yaaqabaaaaaGcbaGaeqiTdqMaam4Cam aaBaaaleaacaWGubGaey4kaSIaamOqaaqabaGccqGH9aqpdaWcaaqa aiaaigdaaeaacqaH7oaBdaWgaaWcbaGaamivaiabgUcaRiaadkeaae qaaaaakiabg2da9maalaaabaGaaGOmaiaacIcacaWGObGaey4kaSIa am4DaiaacMcaaeaacaWGobWaaSbaaSqaaiaadogaaeqaaOGaeq4SdC 2aaSbaaSqaaiaadshaaeqaaaaaaaaa@8676@  

(1.12)

 

Charge Speed in a Current Loop

It is obvious that the rate or charge flow has to be constant around the loop including at the corners where there are discontinuities in the charge per unit length.    That would dictate that the following equation be valid

v n+1 = λ n+1 λ n v n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAhadaWgaa WcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacqaH 7oaBdaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaGcbaGaeq4UdW 2aaSbaaSqaaiaad6gaaeqaaaaakiaadAhadaWgaaWcbaGaamOBaaqa baaaaa@4433@  

(1.13)

where n corresponds to the loop side number counted in the direction of charge flow. 

Recall from Figure 1 that both vt and vI in the top of the loop are both shown going to the left which is the -x direction.  Therefore, before transformation, we write:

v t =| v t | x ^ v Itop =| v I | x ^ v Ibottom =| v I | x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaCODam aaBaaaleaacaWG0baabeaakiabg2da9iabgkHiTiaacYhacaWG2bWa aSbaaSqaaiaadshaaeqaaOGaaiiFaiqacIhagaqcaaqaaiaahAhada WgaaWcbaGaamysaiaaykW7caWG0bGaam4BaiaadchaaeqaaOGaeyyp a0JaeyOeI0IaaiiFaiaadAhadaWgaaWcbaGaamysaaqabaGccaGG8b GabiiEayaajaaabaGaaCODamaaBaaaleaacaWGjbGaaGPaVlaadkga caWGVbGaamiDaiaadshacaWGVbGaamyBaaqabaGccqGH9aqpcaGG8b GaamODamaaBaaaleaacaWGjbaabeaakiaacYhaceGG4bGbaKaaaaaa @5C0A@  

(1.14)

As observed in the frame of the stationary test particle these speeds become:

v t =| v t | x ^ v Itop = ( | v t || v I | ) 1 | v I v t | c 2 x ^ v Ibottom = ( | v t |+| v I | ) 1+ | v I v t | c 2 x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaCODam aaBaaaleaacaWG0baabeaakiabg2da9iaacYhacaWG2bWaaSbaaSqa aiaadshaaeqaaOGaaiiFaiqacIhagaqcaaqaaiaahAhadaWgaaWcba GaamysaiaaykW7caWG0bGaam4BaiaadchaaeqaaOGaeyypa0ZaaSaa aeaadaqadaqaaiaacYhacaWG2bWaaSbaaSqaaiaadshaaeqaaOGaai iFaiabgkHiTiaacYhacaWG2bWaaSbaaSqaaiaadMeaaeqaaOGaaiiF aaGaayjkaiaawMcaaaqaaiaaigdacqGHsisldaWcaaqaaiaacYhaca WG2bWaaSbaaSqaaiaadMeaaeqaaOGaamODamaaBaaaleaacaWG0baa beaakiaacYhaaeaacaWGJbWaaWbaaSqabeaacaaIYaaaaaaaaaGcce GG4bGbaKaaaeaacaWH2bWaaSbaaSqaaiaadMeacaaMc8UaamOyaiaa d+gacaWG0bGaamiDaiaad+gacaWGTbaabeaakiabg2da9maalaaaba WaaeWaaeaacaGG8bGaamODamaaBaaaleaacaWG0baabeaakiaacYha cqGHRaWkcaGG8bGaamODamaaBaaaleaacaWGjbaabeaakiaacYhaai aawIcacaGLPaaaaeaacaaIXaGaey4kaSYaaSaaaeaacaGG8bGaamOD amaaBaaaleaacaWGjbaabeaakiaadAhadaWgaaWcbaGaamiDaaqaba GccaGG8baabaGaam4yamaaCaaaleqabaGaaGOmaaaaaaaaaOGabiiE ayaajaaaaaa@7AF2@  

(1.15)

 

The transformed speed of all positive charges is vt as given in equation (1.15) .  Relative to the stationary test particle, the speed of the negative charges in the left side of the loop is | v I | y ^ +| v t | x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiaacY hacaWG2bWaaSbaaSqaaiaadMeaaeqaaOGaaiiFaiqahMhagaqcaiab gUcaRiaacYhacaWG2bWaaSbaaSqaaiaadshaaeqaaOGaaiiFaiqahI hagaqcaaaa@4206@  and that in the right side of the loop is | v I | y ^ +| v t | x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacYhacaWG2b WaaSbaaSqaaiaadMeaaeqaaOGaaiiFaiqahMhagaqcaiabgUcaRiaa cYhacaWG2bWaaSbaaSqaaiaadshaaeqaaOGaaiiFaiqahIhagaqcaa aa@4119@ .   The transformed speeds of the negative charges in the top side and bottom side of the loop are given by equation (1.15).