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Maxwell ¹æÁ¤½Ä°ú ÀüÀÚ±âÆÄÀÇ ¼Óµµ  🔵
    ±è°ü¼®  2018-06-22 00:35:54, Á¶È¸¼ö : 4,385
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1862³â Foucault´Â ±¤¼Ó(ÎÃáÜ)À» ÃøÁ¤ÇÏ¿´°í 1865³â MaxwellÀÌ ÀüÀÚ±âÆÄÀÇ ¼Óµµ°¡ ±¤¼Ó°ú °°À½À» ¹ß°ßÇß½À´Ï´Ù!
ÀüÀÚ±âÆÄ-±¤¼ÓÀ» ¼ö¸®¹°¸®ÇÐÀûÀ¸·Î ÀÌÇØÇÏ´Â °ÍÀÌ »ó´ë¼ºÀÌ·Ð ÀÌÇØÀÇ ¹ÙÅÁÀÌ µÇ¹Ç·Î ±× °úÁ¤À» °£°áÇϰԳª¸¶ ¿©±â¿¡ Á¤¸®ÇØ ³õ½À´Ï´Ù
EinsteinÀÇ Áß·ÂÀå ÀÌ·ÐÀº MaxwellÀÇ ÀüÀÚ±âÀå À̷п¡¼­ Ãâ¹ßÇÏ¿´À¸¹Ç·Î È¥ÀÚ¼­ ÇнÀÇÒ ¶§ ¾Æ·¡ ¼³¸íÀÌ µµ¿òÀÌ µÉ ¼ö ÀÖ½À´Ï´Ù.


À̸¦ À§Çؼ­´Â º¤ÅÍ Dot product(³»Àû), Cross product(¿ÜÀû), Del(𝛁), Gradient(±â¿ï±â), Divergence(¹ß»ê), Curl(ȸÀü)µîÀÇ °³³äÀÌ ÇÊ¿äÇÕ´Ï´Ù.
´ÙÀ½À¸·Î ¹°¸®Çп¡¼­ MaxwellÀÇ ¹æÁ¤½ÄÀ» °¢°¢ integral form(ÀûºÐÇü)°ú differential form(¹ÌºÐÇü)À¸·Î ¸ðµÎ ÀÌÇØÇÏ¿©¾ß ÇÕ´Ï´Ù.
ºûÀÇ ÆÄµ¿¹æÁ¤½Ä¿¡ ´ëÇØ¼­ Ç®¸é °è»ê»óÀ¸·Î ÀüÀÚ±âÆÄÀÇÀÇ ¼Óµµ°¡ »êÃâµÇ´Âµ¥ ±× °á°ú°¡ ¹Ù·Î ºûÀÇ ¼Óµµ¿Í µ¿ÀÏÇÑ 2.9979 *108 m/s ÀÎ °ÍÀÔ´Ï´Ù.

    [±âº» ¼öÇÐÀû Á¤ÀÇ/Á¤¸®]  

Dot product: 𝐚 ∙ 𝐛 = |𝐚||𝐛|cos𝜃                        <-  ∙ (dot), ´ë¼öÀû Á¤ÀÇ(a1b1+ a2b2+...+anbn)ÀÇ °á°ú
Cross product: 𝐚 ¡¿ 𝐛 = |𝐚||𝐛|sin𝜃                   <- ¡¿ (cross)
Del: 𝛁 ¡Õ 𝐢 ¡Ó/¡Óx + 𝐣 ¡Ó/¡Óy + 𝐤 ¡Ó/¡Óz                          <- 𝛁 (Nabla) [del]
Gradient: 𝛁 ¥× ¡Õ 𝐢 ¡Ó¥×/¡Óx + 𝐣 ¡Ó¥×/¡Óy + 𝐤 ¡Ó¥×/¡Óz       <- ¥× (psi)
    * The gradient indicates the rate of spacial change of the field at a point and the direction of steepest increase from that point.[vector result]
Divergence: 𝛁 ∙ 𝐀 ¡Õ ¡Ó𝐀x /¡Óx + ¡Ó𝐀y /¡Óy + ¡Ó𝐀z /¡Óz
    * The divergence indicates the tendency of the field to flow away from a point. [scalar result]
Curl: 𝛁 ¡¿ 𝐀 ¡Õ (¡Ó𝐀z /¡Óy - ¡Ó𝐀y /¡Óz) 𝐢  + (¡Ó𝐀x /¡Óz - ¡Ó𝐀z /¡Óx) 𝐣 + (¡Ó𝐀y /¡Óx - ¡Ó𝐀x /¡Óy) 𝐤
    * The curl indicates the tendency of the field to circulate around a point and the direction of axis of greatest circulation.[vector result]

   [°ü·Ã ¼öÇÐÀû Á¤¸®]

Curl of Gradient: 𝛁 ¡¿ 𝛁 ¥× = 0
Laplacian = Divergence of Gradient: 𝛁 ∙ 𝛁 ¥× = 𝛁©÷ ¥× = ¡Ó©÷¥×/¡Óx©÷ + ¡Ó©÷¥×/¡Óy©÷ + ¡Ó©÷¥×/¡Óz©÷
Curl of the curl:  𝛁 ¡¿ (𝛁 ¡¿ 𝐀) = 𝛁 (𝛁 ∙ 𝐀) - 𝛁©÷ 𝐀
The Divergence theorems 𝐀 ∙ ň da = ¡òv (𝛁 ∙ 𝐀) dV
    * The flux of a vector field through a closed surface S is equal to the integral of the divergence of that field over a volume V for which S is a boundary.
Stokes' theoremc 𝐀 ∙ d𝒍 = ¡òs (𝛁 ¡¿ 𝐀) ∙ ň da
    * The circulation of a vector field over a closed path C is equal to the integral of the normal component of the curl of that field over surface S for which C is boundary.

   [MaxwellÀÇ ¹æÁ¤½Ä(Maxwell's Equation)]  

1) Gauss's law for electrical fields: ¢±s 𝑬 ∙ ň da = q /¥å₀  [ň: unit normal vector, 𝑬: electric field in volts V/m, q: charge, ¥å₀: permittivity of free space(Áø°ø À¯ÀüÀ²)]
    * Electric charge produces an electric field, and the flux of that field passing through any closed surface is proportional to the total charge contained within that surface.
       applying the divergence theorem ->    𝛁 ∙ 𝑬 = ¥ñ/¥å₀  [¥ñ: charge density(ÀüÇÏ ¹Ðµµ) coulombs C/m©ø]
    * The electric field produced by electric charge diverges from positive charge and converges upon negative charge. 
                          
2) Gauss's law for magnetic fields: ¢±s 𝑩 ∙ ň da = 0 [𝑩: magnetic field in tesla T]
    * The total magnetic flux passing through any closed surface is zero.
       applying the divergence theorem ->    𝛁 ∙ 𝑩 = 0
    * The divergence of the magnetic field at any point is zero.

3) Faraday's law: ¢±c 𝑬 ∙ d𝒍  = - d/dt (¡òs 𝑩 ∙ ň da) = - ¡òs (¡Ó𝑩/¡Ót) ∙ ň da  <- emf: electromotive force(À¯µµ±âÀü·Â)
    * Changing magnetic flux through a surface induces an emf in any boundary path of that surface, and a changing magnetic field induces a circulating electric field.  
       applying Stoke's theorem ->    𝛁 ¡¿ 𝑬  = - ¡Ó𝑩/¡Ót
    * A circulating electric field is produced by a magnetic field that changes with time.

4) Ampere-Maxwell law: ¢±c 𝑩 ∙ d𝒍 = ¥ì₀ [𝑱 + ¥å₀ d/dt (¡òs 𝑬 ∙ ň da)] [¥ì0: magnetic permittivity of free space(Áø°ø ÅõÀÚÀ²),  𝑱: electric current in amperes A] <- ¥ì(mu)
    * An electric current or a changing electric flux through a surface produce a circulating magnetic field around any path that bounds that surface.
       applying Stokes' theorem ->    𝛁 ¡¿ 𝑩 = ¥ì₀ (𝑱 + ¥å₀ * ¡Ó𝑬/¡Ót) [𝑱: current density(Àü·ù ¹Ðµµ) A/m©÷]       
    * A circulating magnetic field is produced by an electric current and by an electric field that changes with time.

   [ÆÄµ¿¹æÁ¤½Ä(The Wave Equation)°ú ÀüÀÚ±âÆÄ ¼Óµµ]    

𝛁©÷ 𝑨 = (1/𝑣©÷) * (¡Ó©÷𝑨/¡Ót©÷)                                    <-  ÆÄµ¿¹æÁ¤½Ä(the wave equation),  𝑣: velocity of wave
𝛁 ¡¿ (𝛁 ¡¿ 𝑬) = 𝛁 ¡¿ (- ¡Ó𝑩/¡Ót) = - ¡Ó(𝛁 ¡¿ 𝑩)/¡Ót        <-  3) Faraday's law ¾çº¯¿¡ '𝛁 ¡¿' Àû¿ë
𝛁 ¡¿ (𝛁 ¡¿ 𝑬) = 𝛁 (𝛁 ∙ 𝑬) - 𝛁©÷𝑬                              <-  applying curl of the curl
𝛁 ∙ 𝑬 = ¥ñ/¥å₀; 𝛁 ¡¿ 𝑩 = ¥ì₀ (𝑱 + ¥å₀ * ¡Ó𝑬/¡Ót)           <-  Maxwell's equation 1) 4) ¿¡¼­
¥ñ = 0,   𝑱 = 0                                                      <-  charge- and current-free region ¼ÓÀ̹ǷÎ
𝛁©÷𝑬 = ¥ì₀ ¥å₀ * ¡Ó©÷𝑬/¡Ót©÷;  𝛁©÷𝑩 = ¥ì₀ ¥å₀ * ¡Ó©÷𝑩/¡Ót©÷
¡Å [𝑬/𝑩] 1/𝑣©÷ = ¥ì₀ ¥å₀,  𝑣 = ¡î 1/ ¥ì₀ ¥å₀ = ¡î 1/[(4¥ð *10-7 m kg/C©÷)(8.8541878*10-12 C©÷ s©÷/kg m©ø)] = 2.9979 *10⁸ m/s !

ÀÌ´Â NewtonÀÇ ¹°¸®ÇÐ ¹ýÄ¢À¸·Î´Â ÀüÇô ¼³¸íµÇÁö ¾Ê¾ÒÀ¸¹Ç·Î Çö´ë°úÇÐ ¹ßÀü¿¡ ¸·´ëÇÑ ¿µÇâ·ÂÀ» °¡Á®¿Â »ç°ÇÀ̶ó ÇϰڽÀ´Ï´Ù.
´ç½Ã¿¡ ¾Ë·ÁÁø ±¤¼Ó°ú ÀÏÄ¡ÇßÀ¸¹Ç·Î ºûÀÌ ÀüÀÚ±âÆÄÀÓÀ» ¾Ë°Ô µÇ¾úÀ¸¸ç, EinsteinÀÌ Æ¯¼ö»ó´ë¼º ÀÌ·ÐÀ» µµÃâÇÏ´Â °è±â°¡ µÇ¾ú½À´Ï´Ù.
±¹Á¦´ÜÀ§°è(SI)¿¡¼­´Â 1983³â 1¹ÌÅÍ´Â ºû(ÀüÀÚ±âÆÄ)ÀÌ Áø°ø¿¡¼­ 1/299,792,458Ãʰ£ ÁøÇàÇÑ °æ·ÎÀÇ ±æÀÌ·Î Á¤ÇÏ¿´½À´Ï´Ù.


* Âü°í¹®Çå: Daniel A. Fleisch A Student's Guide to Maxwell's Equation (Cambridge University Press 2008)

p.s. »ó´ë¼º ÀÌ·ÐÀÇ ÇнÀÀ» À§ÇÑ ÇʼöÀû ³»¿ëÀ¸·Î¼­, ÃÖ´ëÇÑ °£·«ÇÏ°Ô ±â¼úÇÏ°í µµÇظ¦ Ãß°¡ÇßÀ½.


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