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      "author": "albaca",
      "body": "The Lorentz force for a charged particle is\n\nhttps://quicklatex.com/cache3/c6/ql_3aa42a7d89f8890c24dd84a38c25aec6_l3.png,\n\nhttps://quicklatex.com/cache3/dc/ql_08d4ca6454999f8fab6a14e0df5093dc_l3.png the charge of the particle, https://quicklatex.com/cache3/39/ql_5b96dc5593defb3d87eb0b94f9212f39_l3.png mass, https://quicklatex.com/cache3/ae/ql_4f26d805643696cd7e263a710a1cecae_l3.png the proper time,  https://quicklatex.com/cache3/1c/ql_67b5d1a5b6796e3e760d108399ccb61c_l3.png the electromagnetic tensor and https://quicklatex.com/cache3/27/ql_0f3660f5bd7a5c88d85fe0eb11f6e927_l3.png the 4-velocity.\n\nOk... now is where the magic of the physics begins. That equation seems like a horrible monster but I'm going to try to show a beauty face of it. \n\nThink in a particle (e.g electron) moving forward in a region where you have electric or magnetic field, the first and natural question is how do these fields change the direction of the  particle? find out.\n\nLetting \n\n https://quicklatex.com/cache3/a4/ql_ff48ee026c488fd73801c2bf18f37ca4_l3.png\n\n\nthese fields are constants and uniforms with that said we can start on it.\n\nFrom the Lorentz force, the 0-component is\n\nhttps://quicklatex.com/cache3/0e/ql_d49b631fd8ced729e97d7f9f3940a00e_l3.png,\n\nthe i-component is \n\nhttps://quicklatex.com/cache3/63/ql_f9a47bb0634c20be25d4a68b0c312563_l3.png.\n\nWhen we've done with all the maths, will have this pair of equations\n\nhttps://quicklatex.com/cache3/a2/ql_2384dabdd38a1775da04517a4569aca2_l3.png.\n\nFor the problem of interest the pair of equations will reduce to \n\nhttps://quicklatex.com/cache3/55/ql_c22ca122d3f4ea7a011ad4bc28dd6f55_l3.png\n\nrecall that upper notation just means components https://quicklatex.com/cache3/f5/ql_c291aef9baf9f1a95eb8428746b123f5_l3.png of contravariant vector.\n\nwe can solve it just taking the derivative of https://quicklatex.com/cache3/42/ql_6631de30a4f2e1f5b5dcdf2dfa498a42_l3.png respect to https://quicklatex.com/cache3/ae/ql_4f26d805643696cd7e263a710a1cecae_l3.png again and substituting https://quicklatex.com/cache3/b8/ql_28bc919604436083a4c01f18f31f01b8_l3.png\n\nWe're doing three new definitions just in case... we're calling  https://quicklatex.com/cache3/f8/ql_1af12f6f6c30b31c4fd87262f0bf1bf8_l3.png as https://quicklatex.com/cache3/7c/ql_9b5b719cc33ab3335dc1951816baaa7c_l3.png (because of the second derivative), https://quicklatex.com/cache3/72/ql_8d2fbfb6b3af863cd612332e5a479f72_l3.png and https://quicklatex.com/cache3/cb/ql_d6bd2b8f6a0088cd63681138a8a529cb_l3.png.\n\nSo the final equations for the component 2 of the 4-velocity is \n\nhttps://quicklatex.com/cache3/15/ql_5273ab07cbc6e1ebb2c532a5b1ded515_l3.png.\n\nWe can check 2 cases, 1st one https://quicklatex.com/cache3/e0/ql_6f6dbbec4fab9fbdbad9474ed9faf3e0_l3.png it leaves that https://quicklatex.com/cache3/f8/ql_99545b5e18cf71d6e79bcfd32552e1f8_l3.png and the second one https://quicklatex.com/cache3/a4/ql_4114db259acc461269c3b8fd352745a4_l3.png which is https://quicklatex.com/cache3/24/ql_e210bd6f18962e40ff3aa34286af0924_l3.png. These both cases just are going to implicate that the solution for the equation of https://quicklatex.com/cache3/7c/ql_9b5b719cc33ab3335dc1951816baaa7c_l3.png will be a complex exponential or a real one. In this post I'm going to begin with first one.\n\n\nAssuming https://quicklatex.com/cache3/e0/ql_6f6dbbec4fab9fbdbad9474ed9faf3e0_l3.png, we have that the solution for https://quicklatex.com/cache3/7c/ql_9b5b719cc33ab3335dc1951816baaa7c_l3.png is\n\nhttps://quicklatex.com/cache3/ec/ql_9d6014ed702119f10c68ddf9637ae3ec_l3.png,\n\nwe're calling https://quicklatex.com/cache3/47/ql_e6808696c8ee88fdf66e0438ee366247_l3.png.\n\nJust substituting the last result in the equations for https://quicklatex.com/cache3/16/ql_02d445f6978417fd1de0d1eb2a976a16_l3.png. We have this two results \n\nhttps://quicklatex.com/cache3/a7/ql_f88904f01aa102c8b36f0c1acf5094a7_l3.png\n\nSetting the initial conditions \n\nhttps://quicklatex.com/cache3/9a/ql_40fff772a8813d3ebc9c64bb69677d9a_l3.png\n\n\n\nThe world line  will be,\n\n\nhttps://quicklatex.com/cache3/88/ql_ffc6bf9eea60157b46d56a6db126db88_l3.png,\n\nhttps://quicklatex.com/cache3/b8/ql_5097753940fcd86b669abe39c84852b8_l3.png\n\n\n\n\nI've omitted the 3 component because it's a constant(straight line).\n\n \nWe can notice if https://quicklatex.com/cache3/41/ql_bc1b5b7b7cb2ca4473a541b1ab1a0b41_l3.png the component 0 of the 4-velocity and the Lorentz factor https://quicklatex.com/cache3/03/ql_ff32d20418d6a86b23695837bae82203_l3.png are just a constants, so the energy https://quicklatex.com/cache3/e7/ql_93ee3c84eca38658f4c75b1fbf8ba4e7_l3.png is a constant as well.\n\nBy another hand when https://quicklatex.com/cache3/81/ql_8b46ca4f7b7359b7d32f4ec3ace9cb81_l3.png the solution is very similar (setting all the constants just like the last case) but in a complex exponential form, we have\n\nhttps://quicklatex.com/cache3/9f/ql_0029942363529d8cd590914d6191a09f_l3.png,\nhttps://quicklatex.com/cache3/93/ql_0a5576b7794f6fb5590f6c001a9d3a93_l3.png\n\nAt the end the velocity is of course changed because of the electromagnetic field. \n\nThe next and last step is just plot how is the movement of a charged particle due to a electromagnetic field.\n\nWe're going to back to the no-relativistic limit, just putting https://quicklatex.com/cache3/93/ql_0c1628506b7350e644d4ea19eb9ab693_l3.png.\n\nThe harmonic motion for  https://quicklatex.com/cache3/81/ql_8b46ca4f7b7359b7d32f4ec3ace9cb81_l3.png, looks like.\n\n\n![1.png](https://cdn.steemitimages.com/DQmcCUydpusMtaJxueLt8FY4jxjaeDQKDLGoyx2WH7G633E/1.png)\n\nThe hyperbolic motion looks\n\n![1.png](https://cdn.steemitimages.com/DQmfH5VBLePc847E8BMsugka1z2mmfomKyQPmHeG4DL36od/1.png)\n\nThe dominance of the electric field kills the harmonic motion.",
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      "body": "@@ -649,12 +649,9 @@\n egin\n-ning\n+s\n . Th\n",
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      "author": "albaca",
      "body": "The Lorentz force for a charged particle is\n\nhttps://quicklatex.com/cache3/c6/ql_3aa42a7d89f8890c24dd84a38c25aec6_l3.png,\n\nhttps://quicklatex.com/cache3/dc/ql_08d4ca6454999f8fab6a14e0df5093dc_l3.png the charge of the particle, https://quicklatex.com/cache3/39/ql_5b96dc5593defb3d87eb0b94f9212f39_l3.png mass, https://quicklatex.com/cache3/ae/ql_4f26d805643696cd7e263a710a1cecae_l3.png the proper time,  https://quicklatex.com/cache3/1c/ql_67b5d1a5b6796e3e760d108399ccb61c_l3.png the electromagnetic tensor and https://quicklatex.com/cache3/27/ql_0f3660f5bd7a5c88d85fe0eb11f6e927_l3.png the 4-velocity.\n\nOk... now is where the magic of the physics beginning. That equation seems like a horrible monster but I'm going to try to show a beauty face of it. \n\nThink in a particle (e.g electron) moving forward in a region where you have electric or magnetic field, the first and natural question is how do these fields change the direction of the  particle? find out.\n\nLetting \n\n https://quicklatex.com/cache3/a4/ql_ff48ee026c488fd73801c2bf18f37ca4_l3.png\n\n\nthese fields are constants and uniforms with that said we can start on it.\n\nFrom the Lorentz force, the 0-component is\n\nhttps://quicklatex.com/cache3/0e/ql_d49b631fd8ced729e97d7f9f3940a00e_l3.png,\n\nthe i-component is \n\nhttps://quicklatex.com/cache3/63/ql_f9a47bb0634c20be25d4a68b0c312563_l3.png.\n\nWhen we've done with all the maths, will have this pair of equations\n\nhttps://quicklatex.com/cache3/a2/ql_2384dabdd38a1775da04517a4569aca2_l3.png.\n\nFor the problem of interest the pair of equations will reduce to \n\nhttps://quicklatex.com/cache3/55/ql_c22ca122d3f4ea7a011ad4bc28dd6f55_l3.png\n\nrecall that upper notation just means components https://quicklatex.com/cache3/f5/ql_c291aef9baf9f1a95eb8428746b123f5_l3.png of contravariant vector.\n\nwe can solve it just taking the derivative of https://quicklatex.com/cache3/42/ql_6631de30a4f2e1f5b5dcdf2dfa498a42_l3.png respect to https://quicklatex.com/cache3/ae/ql_4f26d805643696cd7e263a710a1cecae_l3.png again and substituting https://quicklatex.com/cache3/b8/ql_28bc919604436083a4c01f18f31f01b8_l3.png\n\nWe're doing three new definitions just in case... we're calling  https://quicklatex.com/cache3/f8/ql_1af12f6f6c30b31c4fd87262f0bf1bf8_l3.png as https://quicklatex.com/cache3/7c/ql_9b5b719cc33ab3335dc1951816baaa7c_l3.png (because of the second derivative), https://quicklatex.com/cache3/72/ql_8d2fbfb6b3af863cd612332e5a479f72_l3.png and https://quicklatex.com/cache3/cb/ql_d6bd2b8f6a0088cd63681138a8a529cb_l3.png.\n\nSo the final equations for the component 2 of the 4-velocity is \n\nhttps://quicklatex.com/cache3/15/ql_5273ab07cbc6e1ebb2c532a5b1ded515_l3.png.\n\nWe can check 2 cases, 1st one https://quicklatex.com/cache3/e0/ql_6f6dbbec4fab9fbdbad9474ed9faf3e0_l3.png it leaves that https://quicklatex.com/cache3/f8/ql_99545b5e18cf71d6e79bcfd32552e1f8_l3.png and the second one https://quicklatex.com/cache3/a4/ql_4114db259acc461269c3b8fd352745a4_l3.png which is https://quicklatex.com/cache3/24/ql_e210bd6f18962e40ff3aa34286af0924_l3.png. These both cases just are going to implicate that the solution for the equation of https://quicklatex.com/cache3/7c/ql_9b5b719cc33ab3335dc1951816baaa7c_l3.png will be a complex exponential or a real one. In this post I'm going to begin with first one.\n\n\nAssuming https://quicklatex.com/cache3/e0/ql_6f6dbbec4fab9fbdbad9474ed9faf3e0_l3.png, we have that the solution for https://quicklatex.com/cache3/7c/ql_9b5b719cc33ab3335dc1951816baaa7c_l3.png is\n\nhttps://quicklatex.com/cache3/ec/ql_9d6014ed702119f10c68ddf9637ae3ec_l3.png,\n\nwe're calling https://quicklatex.com/cache3/47/ql_e6808696c8ee88fdf66e0438ee366247_l3.png.\n\nJust substituting the last result in the equations for https://quicklatex.com/cache3/16/ql_02d445f6978417fd1de0d1eb2a976a16_l3.png. We have this two results \n\nhttps://quicklatex.com/cache3/a7/ql_f88904f01aa102c8b36f0c1acf5094a7_l3.png\n\nSetting the initial conditions \n\nhttps://quicklatex.com/cache3/9a/ql_40fff772a8813d3ebc9c64bb69677d9a_l3.png\n\n\n\nThe world line  will be,\n\n\nhttps://quicklatex.com/cache3/88/ql_ffc6bf9eea60157b46d56a6db126db88_l3.png,\n\nhttps://quicklatex.com/cache3/b8/ql_5097753940fcd86b669abe39c84852b8_l3.png\n\n\n\n\nI've omitted the 3 component because it's a constant(straight line).\n\n \nWe can notice if https://quicklatex.com/cache3/41/ql_bc1b5b7b7cb2ca4473a541b1ab1a0b41_l3.png the component 0 of the 4-velocity and the Lorentz factor https://quicklatex.com/cache3/03/ql_ff32d20418d6a86b23695837bae82203_l3.png are just a constants, so the energy https://quicklatex.com/cache3/e7/ql_93ee3c84eca38658f4c75b1fbf8ba4e7_l3.png is a constant as well.\n\nBy another hand when https://quicklatex.com/cache3/81/ql_8b46ca4f7b7359b7d32f4ec3ace9cb81_l3.png the solution is very similar (setting all the constants just like the last case) but in a complex exponential form, we have\n\nhttps://quicklatex.com/cache3/9f/ql_0029942363529d8cd590914d6191a09f_l3.png,\nhttps://quicklatex.com/cache3/93/ql_0a5576b7794f6fb5590f6c001a9d3a93_l3.png\n\nAt the end the velocity is of course changed because of the electromagnetic field. \n\nThe next and last step is just plot how is the movement of a charged particle due to a electromagnetic field.\n\nWe're going to back to the no-relativistic limit, just putting https://quicklatex.com/cache3/93/ql_0c1628506b7350e644d4ea19eb9ab693_l3.png.\n\nThe harmonic motion for  https://quicklatex.com/cache3/81/ql_8b46ca4f7b7359b7d32f4ec3ace9cb81_l3.png, looks like.\n\n\n![1.png](https://cdn.steemitimages.com/DQmcCUydpusMtaJxueLt8FY4jxjaeDQKDLGoyx2WH7G633E/1.png)\n\nThe hyperbolic motion looks\n\n![1.png](https://cdn.steemitimages.com/DQmfH5VBLePc847E8BMsugka1z2mmfomKyQPmHeG4DL36od/1.png)\n\nThe dominance of the electric field kills the harmonic motion.",
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      "permlink": "lorentz-force-for-a-relativistic-charge",
      "title": "Lorentz force for a relativistic charge."
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      "author": "albaca",
      "body": "@@ -361,72 +361,9 @@\n s?.%0A\n-The Lorentz transforms of this problem can be analysed for it, g\n+G\n iven\n@@ -1697,26 +1697,16 @@\n f light,\n- with this\n  we have\n",
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      "body": "@@ -412,25 +412,22 @@\n nalysed \n-with this\n+for it\n , given \n",
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{
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    {
      "author": "albaca",
      "body": "Today I was trying to figure out what's the mean about a spacelike interval... Here we go.\n\nThe first thing I think about it... I didn't have a clear idea about the real meaning of it and that's ok but right now I think I can understand it.\n\nFirst we're going to discuss about this question, Is there a frame in which spatially separated events are simultaneous?.\nThe Lorentz transforms of this problem can be analysed with this, given two events P and Q, we are going to fix P at (0,0) that's just https://quicklatex.com/cache3/8d/ql_b61fdce8a39ddada7f9fe835d1b2028d_l3.png and https://quicklatex.com/cache3/0c/ql_2a305c61d12d9308343538993916c80c_l3.png, with this I just wanted to say we have a point P in a place of space-time. The point Q at https://quicklatex.com/cache3/26/ql_a7f0a849351d3df3b7b116495e969126_l3.png. If that place exists the time measured will be https://quicklatex.com/cache3/3e/ql_4cf41591ebb928ad1a4c0b652a10833e_l3.png  and it should be 0.  We're going to talk about it. The way to work with frames in relativity is with the Lorentz transformations, a frame with a relative velocity https://quicklatex.com/cache3/09/ql_b5ed781b36a14772c442fbaebb47bb09_l3.png of P could be described by\n\nhttps://quicklatex.com/cache3/55/ql_7a7249d323699fe061408f0c14a55255_l3.png, \n\njust set https://quicklatex.com/cache3/f1/ql_92580983cf4d7b25fcd57121e713a0f1_l3.png (natural units).  We want those events to be simultaneous (https://quicklatex.com/cache3/b5/ql_9c0a8b44604124f7902ee4c4b9a05bb5_l3.png).\nSetting https://quicklatex.com/cache3/b5/ql_9c0a8b44604124f7902ee4c4b9a05bb5_l3.png at the Lorentz transformation, we'll get\n\nhttps://quicklatex.com/cache3/be/ql_d9eeb3478dfe0cf82f07789d72c43ebe_l3.png,\n\nthis velocity can't overtake the velocity of light, with this we have\n\nhttps://quicklatex.com/cache3/46/ql_da8a6005bacdf7dea4ae794eaed04346_l3.png.\n\nWe just have to check what's a spacelike interval. This kind of intervals are defined by\n\nhttps://quicklatex.com/cache3/df/ql_d9f5e6a427f224ce54025da36431d0df_l3.png.\n\nIn 1-D the interval will be\n\nhttps://quicklatex.com/cache3/37/ql_bd9222d029de25801de7cfbe7371ca37_l3.png,\n\n\nthis is just that we've derived. \n\nWe can conclude that in a spacelike interval, there is a frame where you can have two events simultaneous.",
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