# T_NAKÄ[uO

## Ɂuړ鎞vFMOVING CLOCKSvǂށBiQj

<<   쐬 F 2016/10/07 00:01   >>

 "Remarks on the Equivalence of Inertial and Gravitational Masses and on the Accuracy of Einstein's Theory of Gravity." MOVING CLOCKS ̑łB [ړ鎞vFMOVING CLOCKSiQ|Pj]============================================================= With these assumptions, it is easy to calculate the time before the plates collide after they are released. ̉ŁAォՓ˂܂ł̎ԂvZ邱Ƃ͊ȒPłB From the Gauss law it holds that: KEX̒藝 @$\oint _{S}\varepsilon \vec{E}\cdot \vec{S}= \mathcal{Q}\; \; \; \; \; \; (1)$ Since the enclosed Gaussian integrating surface S can be suitably selected, closely following the plate surface A, it is easily seen that for the electrical field intensity E and from it for the force F, which one plate exerts on the other, it is: ͂͂ރKEXϕʂ S  A ɂقڒǏ]ď\ɑIԂƂł̂ŁAd̋ E AāA(1̑̔ɋyڂ) Fȉ̒ʂł邱ƂȒPɊmFłB: @$F= \frac{\mathcal{Q}^{2}}{\varepsilon \cdot 2A}\; \; \; \; \; \; (2)$ The same holds for the second charge acting on the first, so the total force with which the plates attract each other is twice the value given in Eq.2. 2Ԗڂ̓dׂɑ΂Ă1ԖڂƓlɍl̂ŁAm݂Ɉ͂ (2) ŗ^l̂Q{ƂȂB Next, it is assumed that the plates have an inertial mass mi and that they are moving toward each other with a very low velocity, so that no relativistic effects apply. ɁA͊ mi AɒᑬŌ݂̕֐iłƌ􂳂ÂߑΘ_IȌʂ͓KpȂB Ordinary Newtonfs second law can thus describe the motion and the time to platefs collision can thus be calculated to be: ʏAj[g̑2@͉^̂悤ɋLq邱ƂłA̏ՓˎԂ͈ȉ̒ʂvZłB: @$t_{c}^{2}= \frac{\varepsilon \cdot m_{i}\cdot a\cdot A}{\mathcal{Q}^{2}}\; \; \; \; \; \; (3)$ The process of colliding can now be repeated at will and this device can be considered as a time-measuring clock. AՓ˃vZX͎RɌJԂ邱ƂłȂu͎ԑ莞vƍl邱ƂłB The inverse of the time to collision is the local clock rate. Փ˂܂ł̎Ԃ̋t́AǏIv݂̍łB ================================================================================================ lɑѓdʔ̍od @$E=\frac{\sigma }{2\varepsilon }= \frac{\mathcal{Q} }{2\varepsilon A }\; \; \to \; \; F= \mathcal{Q}E= \frac{\mathcal{Q}^{2} }{2\varepsilon A }$ ƂȂA(2) oĂ܂BɁAum݂Ɉ͂ (2) ŗ^l̂Q{ƂȂvƂƂAxƂ @$2F=m_{i}\alpha \; \to \; \alpha = \frac{2F}{m_{i}}= \frac{\mathcal{Q}^{2}}{m_{i}\varepsilon A}$ łAQ̔܂ł݂͂ɕ݊̂ŁAa ̔̋𓮂ƂɂȂ܂B܂ @$\frac{1}{2}a= \frac{1}{2}\alpha t_{c}^{2}\; \to \; t_{c}^{2}= \frac{a}{\alpha }$ Ȃ̂ŁA @$t_{c}^{2}= \frac{\varepsilon m_{i}aA}{\mathcal{Q}^{2} }$ (3) oĂ܂B iƌAtƂŁA{ɍĂ̂낤Hj [ړ鎞vFMOVING CLOCKSiQ|Qj]=========================================================== In the next step, the above-described clock will be observed from the laboratory frame of reference and thus it will be assumed that both parallel plates are now moving in the Z direction with a constant velocity v. ̃Xebvł́AL̎v͎nϑA̕sIȑ v łƂ Z ɓĂƉ肷B The laboratory observer at rest in the coordinate system XYZ will observe, in addition to charge, also currents that these two moving charged plates generate. nϑ҂͍Wn XYZ ɗ܂AdׂɉĂQ̓odϑł낤B The currents will cause an additional force to appear in the laboratory observerfs coordinate system. d́AȂ͂nϑ҂̍WnŌ錴ɂȂB From Amperefs law it holds that: Ay[̖@A͈ȉ̂悤ɂȂBF @$\oint _{P}\vec{H}\cdot d\vec{l}= v\: \frac{\mathcal{Q}}{L}\; \; \; \; \; \; (4)$ Again, since the integrating path P can be suitably selected, the H field is easily found to be: ܂AϕoH P K؂ɑIׂ΁AH ꂪȉ̒ʂłƂɂ킩B: @$H= v\: \frac{\mathcal{Q}}{2\cdot W\cdot L}\; \; \; \; \; \; (5)$ The mutual force, with which the plates attract each other, according to the Lorentz force equation, is: [c͂ɂA͂́Aȉ̒ʂłB: @$F=\mathcal{Q}\cdot |\vec{E}+\vec{v}\times \vec{B}|= \frac{\mathcal{Q}^{2}}{ \varepsilon \cdot A}\left ( 1-\frac{v^{2}}{c^{2}} \right )\; \; \; \; \; \; (6)$ where c is the speed of light. c ͌łB The time to collision of the moving plates as observed in the laboratory coordinate system is thus equal to: WnŊϑm̏ՓˎԂ́AȉɓB: @ @$t_{c}^{2}= \frac{\varepsilon \cdot\frac{m(rst)_{i}}{\sqrt{1-v^{2}/c^{2}}}\cdot a\cdot A(rst)\sqrt{1-v^{2}/c^{2}} }{\mathcal{Q}^{2}\cdot (1-v^{2}/c^{2})}= \frac{\varepsilon \cdot m(rst)_{i}\cdot a\cdot A(rst) }{\mathcal{Q}^{2}\cdot (1-v^{2}/c^{2})}\; \; \; \; \; \; (7)$ This result is expected since it follows from STR after the Lorentz coordinate transformation. [cϊ̓ꑊΘ_ STR ɏ]ƂǍʂ͗\zB This phenomenon is the famous time dilation effect. ̌ۂ́ALȎԒxʂłB It is nice to know that the simple clock works and produces the expected result. PȎvāA҂錋ʂ𐶂ނƂƂm邱Ƃ́Af炵B It is also important to note that the relativistic values for mi and A, as seen from the laboratory reference frame, have been substituted into Eq.7. nϑ悤ɁAmi A ̑Θ_IȒl (7) ɑウꂽ_ɒӂ邱ƂdvłB The inertial mass mi has increased and the area A has shrunk in the Z direction but both with the same coefficient of proportionality, so the effects have cancelled each other and the numerator in Eq.7 remained unchanged. mi ʐ A Z ŏk񂾂AWɂނ荇ŁAe݂͌LZA(7) ̕q͕sς̂܂܂łB The motion does not affect the distance gah, since it is perpendicular to the velocity vector v. xNg v ƒpȂ߁A^͋ "a"ɉeyڂȂB Charge Q also remains constant, since it is an absolute invariant. ΂̕sϗʂł邽߁Ad Q ̂܂܂łB =============================================================================================== @ Ay[̖@ @$\frac{1}{\mu }\oint \boldsymbol{B}\cdot d\boldsymbol{r}= \oint \boldsymbol{H}\cdot d\boldsymbol{r}= \sum_{inner}\boldsymbol{I}_{i}$ ŁAdlłA܂x̓dאx Q/L ŁAꂪPʎԂ v ߁Ad vE(Q/L) ɂȂ (4) oĂ܂B ܂AϕoH̒ 2W Ȃ̂ŁA(4) ̉Eӂ 2W Ŋ (5) oĂ܂BāA @$B=\mu H\; ,\; \varepsilon \mu =\frac{1}{c^{2}}\; \to \;B=\frac{H}{\varepsilon c^{2}}= \frac{v}{c^{2}}\frac{\mathcal{Q}}{2\varepsilon A}$ ͂͂Q{ƂƂȂ̂ŁAIɂ @$E=\frac{\mathcal{Q}}{\varepsilon A}\;, \;B= \frac{v}{c^{2}}\frac{\mathcal{Q}}{\varepsilon A}$ Ƃ傫ɂȂł傤B (6) @$F=\mathcal{Q}|\boldsymbol{E}+\boldsymbol{v}\times \boldsymbol{B}|= \mathcal{Q}\left ( \frac{\mathcal{Q}}{\varepsilon A}- \frac{v^{2}}{c^{2}}\frac{\mathcal{Q}}{\varepsilon A} \right )= \frac{\mathcal{Q}^{2}}{\varepsilon A}\left ( 1- \frac{v^{2}}{c^{2}}\right )$ ŁA̓d͈ēdׂ͐˂̂ŁAa͂dƂ̓}CiX̉eƂȂAv B ͐isin=/2jɂȂ̂ŊOς̑傫ƂĂ v |邾ɂȂ܂B (7) ͎ʂ̑ƃ[ckEĕq͕ςȂA(6) 番 (1-v2/c2) |ƂƂł傤B AΘ_IʂĂ܂LȂł傤ˁB{Ȃ玞Ԃ̒xŕ\킷łA̘_ł͂ƂɂȂĂ܂BB

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2016/10/07 09:10
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2016/10/07 17:43
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@http://teenaka.at.webry.info/201206/article_20.html
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2016/10/07 22:21
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2016/10/07 23:07
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http://www.geocities.jp/mtsugi04/relativity.html
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