output.to from Sideway

`-=[]⟨⟩\;',./~!@#$%^&*()_+{}|:"<>? 𝑎𝑏𝑐𝑑𝑒𝑓𝑔ℎ𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝𝑞𝑟𝑠𝑡𝑢𝑣𝑤𝑥𝑦𝑧 Å − × ⋅∓±∘꞊﹦∗∙ ℯ 𝔸𝔹ℂ𝔻𝔼𝔽𝔾ℍ𝕀𝕁𝕂𝕃𝕄ℕ𝕆ℙℚℝ𝕊𝕋𝕌𝕍𝕎𝕏𝕐ℤ𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻𝐼𝐽𝐾𝐿𝑀𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈𝑉𝑊𝑋𝑌𝑍 ∼∽∾≁≂≃≄≅≆≇≈≉≌≐≠≡ ≤≥≦≧≨≩≪≫ ∈∉∊∋∌∍ ⊂⊃⊄⊅⊆⊇ 𝛼𝛽𝛾𝛿𝜀𝜁𝜂𝜃𝜄𝜅𝜆𝜇𝜈𝜉𝜊𝜋𝜌𝜎𝜏𝜐𝜑𝜒𝜓𝜔 ∀∂∃∅⦰∆∇∎∞∝∴∵ ∏∐∑⋀⋁⋂⋃ ∧∨∩∪ ∫∬∭∮∯∰∱∲∳ ∥⋮⋯⋰⋱ ‖ ′ ″ ‴ ⁄ ⁗ ʹ ʺ ‵ ‶ ‷ ﹁﹂﹃﹄︹︺︻︼ ︗ ︘ ︿﹀︽︾﹇﹈︷︸ ⏜ ⏝ ⎴ ⎵ ⏞ ⏟ ⏠ ⏡ ←↑→↓↤↦↥↧↔↕↖↗↘↙▲▼◀▶↺↻⟲⟳ ↼↽↾↿⇀⇁⇂⇃⇄⇅⇆⇇ ⇐⇑⇒⇓⇔⇌⇍⇏⇕⇖⇗⇘⇙⇙⇳⥢⥣⥤⥥⥦⥧⥨⥩⥪⥫⥬⥭⥮⥯

Draft for Information Only

Content

Ampere's Law
 Source and Reference

Ampere's Law

For a very long wire, the current in the wire create a cylindrical pattern of Magnetic field 𝐵=~~𝜇₀4𝜋~~2𝐼𝑟𝜃. The line integral along one circle is

∮ 𝐵⋅𝑑𝑙=∮ 𝜇₀4𝜋2𝐼𝑟𝜃⋅𝑑𝑙=𝜇₀4𝜋2𝐼𝑟∮ 𝜃⋅𝑑𝑙

   ∵𝑑𝑙, 𝑑𝜃 tangent to circle

       ∴𝑑𝑙=𝑟𝑑𝜃⇒∮ 𝜃⋅𝑑𝑙=𝑟∮ 𝜃𝑑𝜃=2𝜋𝑟

⇒∮ 𝐵⋅𝑑𝑙=𝜇₀4𝜋2𝐼𝑟(2𝜋𝑟)=𝜇₀𝐼_enclosed
   Ampere's Law for long wire

On each circle: magnetic field 𝐵∝1𝑟 and circumference 𝐶∝𝑟.

In any segment, the contribution from any cylindrical shell is the same.

Since magnetic field of any outer cylindrical shell surface is always the same. The magnetic field of any closed surface formed by any outer cylindrical shell surfaces is always the same.

Similarly, by limit the segment to infinite small segment, the enclosed surface can be refined to surround the current in wire with any smooth shape by following different shperes in different places with the same magnetic field.