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Chapter 2 – Electrostatics 𝑭𝑭 = 1 𝑞𝑞𝑞𝑞 4𝜋𝜋𝜖𝜖∘ 𝓇𝓇 2 � 𝓻𝓻 𝑭𝑭 = 𝑄𝑄𝑬𝑬 𝒓𝒓 𝑉𝑉(𝒓𝒓) ≡ − ∫∞ 𝑬𝑬(𝒓𝒓′ ) ⋅ 𝑑𝑑𝒍𝒍′ = 𝑊𝑊 = 1 8𝜋𝜋𝜖𝜖∘ 𝑞𝑞𝑖𝑖 𝑞𝑞𝑗𝑗 ∑𝑛𝑛𝑖𝑖=1,𝑗𝑗≠𝑖𝑖 𝑟𝑟𝑖𝑖𝑖𝑖 1 𝑄𝑄 𝐶𝐶 ≡ 𝑉𝑉 1 4𝜋𝜋𝜖𝜖∘ 𝐶𝐶∥ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 1 4𝜋𝜋𝜖𝜖∘ ∫ 𝜌𝜌(𝒓𝒓′ ) 𝓇𝓇 𝓇𝓇 𝜖𝜖∘ 2 1 𝐴𝐴𝜖𝜖∘ ∫ 4𝜋𝜋𝜖𝜖∘ 𝜆𝜆�𝒓𝒓′ � � 𝑑𝑑𝑙𝑙 ′ 𝓻𝓻 𝜎𝜎 � 𝑬𝑬 = 𝜖𝜖 𝒏𝒏 ∘ 𝑑𝑑 𝑅𝑅 𝑅𝑅 2 𝑟𝑟 ′ 𝑟𝑟 𝑉𝑉𝑑𝑑𝑑𝑑𝑑𝑑 (𝒓𝒓) = 1 𝒑𝒑⋅𝒓𝒓� 4𝜋𝜋𝜖𝜖∘ 𝑟𝑟 2 = Chapter 5 – Magnetostatics 𝑭𝑭 = 𝑄𝑄(𝑬𝑬 + 𝒗𝒗 × 𝑩𝑩) 𝜇𝜇∘ 4𝜋𝜋 𝑩𝑩𝑑𝑑𝑑𝑑𝑑𝑑 (𝒓𝒓) = 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 ∫ 𝑱𝑱(𝒓𝒓′ ) 𝓇𝓇 𝜇𝜇∘ 𝑚𝑚 4𝜋𝜋𝑟𝑟 3 𝑄𝑄𝑄𝑄𝑄𝑄 = 𝑩𝑩(𝒓𝒓) = 𝑑𝑑3 𝑥𝑥 𝜇𝜇∘ 4𝜋𝜋 ∫ 𝑚𝑚𝑣𝑣 2 𝑅𝑅 � 𝑱𝑱×𝓻𝓻 Δ𝐵𝐵⊥ = 0 𝓇𝓇 2 𝜇𝜇∘ 4𝜋𝜋𝑟𝑟 3 𝜇𝜇 𝑱𝑱𝑏𝑏 �𝒓𝒓′ � 𝓇𝓇 𝑑𝑑3 𝑥𝑥 ′ + ∮𝒮𝒮 Δ𝐵𝐵∥ = 𝜇𝜇∘ 𝐾𝐾 [3(𝒎𝒎 ∙ 𝒓𝒓�)𝒓𝒓� − 𝒎𝒎] 𝑈𝑈 = −𝒎𝒎 ∙ 𝑩𝑩 𝑲𝑲𝑏𝑏 �𝒓𝒓′ � 𝓇𝓇 𝑑𝑑𝑎𝑎′ � 𝑩𝑩 = 𝜇𝜇𝑯𝑯; 𝜇𝜇 ≡ 𝜇𝜇∘ (1 + 𝜒𝜒𝑚𝑚 ) 𝑴𝑴 = 𝜒𝜒𝑚𝑚 𝑯𝑯 Chapter 7 – Electrodynamics ℰ = −𝑀𝑀 𝑉𝑉 = 𝐼𝐼𝐼𝐼 𝑑𝑑𝐼𝐼1 𝑑𝑑𝑑𝑑 1 𝑃𝑃 = 𝐼𝐼𝐼𝐼 = 𝐼𝐼 2 𝑅𝑅 𝑑𝑑𝑑𝑑 Φ = 𝐿𝐿𝐿𝐿 ⇒ ℰ = −𝐿𝐿 𝑑𝑑𝑑𝑑 1 2𝜋𝜋𝜖𝜖 𝐿𝐿 1 4𝜋𝜋𝜖𝜖∘ 𝒑𝒑 𝑣𝑣𝑣𝑣𝑣𝑣. ∑∞ 𝑛𝑛=0 Δ 𝜕𝜕𝑨𝑨 𝜕𝜕𝜕𝜕 = −𝜇𝜇∘ 𝑲𝑲 ∘ 1 𝑝𝑝 4𝜋𝜋𝜖𝜖∘ 𝑟𝑟 3 𝒎𝒎 𝑴𝑴 ≡ 𝑣𝑣𝑣𝑣𝑣𝑣. 1 �� = �2 cos 𝜃𝜃 𝒓𝒓� + sin 𝜃𝜃 𝜽𝜽 𝜕𝜕𝜕𝜕 �� 𝑨𝑨𝑑𝑑𝑑𝑑𝑑𝑑 (𝒓𝒓) = 𝜌𝜌𝑏𝑏 ≡ −𝛁𝛁 ∙ 𝑷𝑷 𝜇𝜇∘ 𝐼𝐼 2𝜋𝜋𝜋𝜋 � 𝝓𝝓 𝑲𝑲 ≡ 𝑩𝑩 = 𝛁𝛁 × 𝑨𝑨 𝜇𝜇∘ 𝒎𝒎×𝒓𝒓� 𝑚𝑚𝒛𝒛� 𝜇𝜇∘ 𝑚𝑚 sin 𝜃𝜃 4𝜋𝜋 𝑟𝑟 2 �� 4𝜋𝜋 𝐵𝐵(𝑧𝑧)𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 𝑱𝑱𝑏𝑏 = 𝛁𝛁 × 𝑴𝑴 1 𝜇𝜇∘ 𝜖𝜖∘ 2 𝐸𝐸 2 𝑄𝑄 2 𝑑𝑑Φ 𝑑𝑑𝑑𝑑 𝜇𝜇∘ 𝐼𝐼 𝑟𝑟 2 𝑑𝑑𝑰𝑰 = 𝜎𝜎𝒗𝒗 𝑑𝑑𝑙𝑙⊥ 𝛁𝛁 ∙ 𝐀𝐀 = 0 � 𝝓𝝓 𝜎𝜎∘ (𝜃𝜃) =− 𝑅𝑅 2 1 [3(𝒑𝒑 ⋅ 𝒓𝒓�)𝒓𝒓� − 𝒑𝒑] 1 4𝜋𝜋𝜖𝜖∘ ∫ � 𝑷𝑷(𝒓𝒓′ )∙𝓻𝓻 𝓇𝓇 2 𝑑𝑑3 𝑥𝑥 𝐶𝐶 = 𝜖𝜖𝑟𝑟 𝐶𝐶𝑣𝑣𝑣𝑣𝑣𝑣 𝑱𝑱 ≡ 𝑑𝑑𝑰𝑰 𝑑𝑑𝑎𝑎⊥ = 𝜌𝜌𝒗𝒗 ∇2 𝑨𝑨 = −𝜇𝜇∘ 𝑱𝑱 𝟏𝟏 𝑩𝑩𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝜇𝜇∘ 𝑛𝑛𝑛𝑛𝒛𝒛� � 𝑲𝑲𝑏𝑏 = 𝑴𝑴 × 𝒏𝒏 𝛁𝛁 × 𝑬𝑬 = − 𝜕𝜕𝜕𝜕 Δ𝐻𝐻 ⊥ = −Δ𝑀𝑀⊥ � Δ𝐻𝐻 ∥ = 𝑲𝑲𝑓𝑓 × 𝒏𝒏 Φ2 = ∫ 𝑩𝑩1 ∙ 𝑑𝑑𝒂𝒂2 = M21 𝐼𝐼1 𝜕𝜕𝑬𝑬 𝛁𝛁 × 𝑩𝑩 = 𝜇𝜇∘ 𝑱𝑱 + 𝜇𝜇∘ 𝜖𝜖∘ 𝜕𝜕𝜕𝜕 → ∮ 𝑩𝑩 ∙ 𝑑𝑑𝒍𝒍 = 𝜇𝜇∘ 𝐼𝐼𝑒𝑒𝑒𝑒𝑒𝑒 + 𝜇𝜇∘ 𝜖𝜖∘ ∫ � 𝜕𝜕𝜕𝜕 � ∙ 𝑑𝑑𝒂𝒂 ∞ 1 ∫ 𝐵𝐵2 𝑑𝑑3 𝑥𝑥 2𝜇𝜇∘ O 𝜖𝜖∘ 𝒎𝒎 ≡ 𝐼𝐼𝒂𝒂 = ∫(𝒓𝒓 × 𝑱𝑱)𝑑𝑑3 𝑥𝑥 𝟐𝟐 2 (𝑅𝑅 2 +𝑧𝑧 2 )3⁄2 𝜕𝜕𝑩𝑩 𝜕𝜕𝑬𝑬 � 𝒏𝒏 𝒑𝒑𝑛𝑛𝑛𝑛𝑛𝑛 = 𝒑𝒑 − 𝑄𝑄𝒓𝒓∘ 𝑉𝑉(𝒓𝒓) = 𝐼𝐼𝑏𝑏 = ∫ 𝑱𝑱𝑏𝑏 ∙ 𝑑𝑑𝒂𝒂 + ∫ 𝑲𝑲𝑏𝑏 ∙ 𝑑𝑑𝒍𝒍 ℰ = ∮ 𝒇𝒇𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∙ 𝑑𝑑𝒍𝒍 = − 𝑟𝑟=𝑅𝑅 𝑭𝑭 = 𝛁𝛁𝑊𝑊 = −𝛁𝛁U 4𝜋𝜋𝜖𝜖∘ 𝑟𝑟 3 𝑱𝑱𝑓𝑓 = 𝛁𝛁 × 𝑯𝑯 ≡ 𝛁𝛁 × � 𝑩𝑩 − 𝑴𝑴� → ∮ 𝑯𝑯 ∙ 𝑑𝑑𝒍𝒍 = 𝐼𝐼𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒 𝑱𝑱𝑏𝑏 = 𝜒𝜒𝑚𝑚 𝑱𝑱𝑓𝑓 𝜕𝜕𝜕𝜕 𝜎𝜎 2𝜖𝜖∘ 𝑈𝑈 = 2 𝐶𝐶𝑉𝑉 2 = 2𝐶𝐶 = 2 𝑄𝑄 ∫(𝑟𝑟 ′ )𝑛𝑛 𝑃𝑃𝑛𝑛 (cos 𝜃𝜃)𝜌𝜌(𝒓𝒓′ )𝑑𝑑𝑉𝑉 ′ 𝛁𝛁 ∙ 𝑩𝑩 = 0 𝑩𝑩𝑙𝑙.𝑠𝑠.𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 = �Δ 𝑷𝑷 = 𝜖𝜖∘ 𝜒𝜒𝑒𝑒 𝑬𝑬 → 𝑫𝑫 = 𝜖𝜖𝑬𝑬; 𝜖𝜖 ≡ 𝜖𝜖∘ (1 + 𝜒𝜒𝑒𝑒 ) = 𝜖𝜖∘ 𝜖𝜖𝑟𝑟 𝑊𝑊 = 𝐿𝐿𝐼𝐼 2 = ∫𝒱𝒱 (𝑨𝑨 ∙ 𝑱𝑱)𝑑𝑑3 𝑥𝑥 = �∫𝒱𝒱 𝐵𝐵2 𝑑𝑑3 𝑥𝑥 − ∮𝒮𝒮 (𝑨𝑨 × 𝑩𝑩) ∙ 𝑑𝑑𝒂𝒂� = 2 2 2𝜇𝜇 ∘ 1 𝑟𝑟 𝑛𝑛+1 � 𝜎𝜎𝑏𝑏 ≡ 𝑷𝑷 ∙ 𝒏𝒏 Δ𝑫𝑫∥ = Δ𝑷𝑷∥ 𝛁𝛁 × 𝐁𝐁 = μ∘ 𝑱𝑱 � 𝒏𝒏 𝑬𝑬𝑖𝑖𝑖𝑖𝑖𝑖.𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = � ⇒ 𝑃𝑃𝑒𝑒.𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝒇𝒇𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 2𝜖𝜖 𝒏𝒏 𝑰𝑰∝𝑑𝑑𝒍𝒍 Chapter 6 – Magnetic Fields in Matter 𝝉𝝉 = 𝒎𝒎 × 𝑩𝑩 → 𝑭𝑭 = 𝜵𝜵(𝒎𝒎 ∙ 𝑩𝑩) 𝑷𝑷 = 𝜎𝜎 𝜖𝜖∘ 𝜌𝜌 𝜖𝜖∘ 𝑊𝑊𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑊𝑊1 + 𝑊𝑊2 + 𝜖𝜖∘ ∫ 𝑬𝑬1 ⋅ 𝑬𝑬2 𝑑𝑑 3 𝑥𝑥 𝑭𝑭𝑚𝑚𝑚𝑚𝑚𝑚 = ∫(𝑰𝑰 × 𝑩𝑩)𝑑𝑑𝑑𝑑 �� 𝐼𝐼 ∫(𝑑𝑑𝒍𝒍 × 𝑩𝑩) 𝑰𝑰 = 𝜆𝜆𝒗𝒗 Δ𝑨𝑨 = 0 ; Δ𝐸𝐸 ∥ = 0 ⇒ Δ𝑬𝑬 = 𝜖𝜖∘ 𝑬𝑬𝑑𝑑𝑑𝑑𝑑𝑑 (𝑟𝑟, 𝜃𝜃) = 4𝜋𝜋𝜖𝜖∘ 𝑟𝑟 2 ∮ 𝑩𝑩 ∙ 𝑑𝑑𝒍𝒍 = 𝜇𝜇∘ 𝐼𝐼𝑒𝑒𝑒𝑒𝑒𝑒 �� = �2 cos 𝜃𝜃 𝒓𝒓� + sin 𝜃𝜃 𝜽𝜽 𝑨𝑨(𝒓𝒓) = 4𝜋𝜋∘ �∫𝒱𝒱 1 𝛁𝛁 ⋅ 𝑬𝑬 = 𝜖𝜖∘ 𝐶𝐶𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = ln(𝑏𝑏⁄∘𝑎𝑎) 𝑝𝑝 cos 𝜃𝜃 Δ𝐷𝐷 ⊥ = 𝜎𝜎𝑓𝑓 𝜎𝜎 𝑄𝑄𝑒𝑒𝑒𝑒𝑒𝑒 𝜎𝜎2 𝜕𝜕𝜕𝜕 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏.𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒. (1+𝜖𝜖)1/2 𝑟𝑟 ′ 𝑟𝑟 Δ𝐸𝐸 ⊥ = ∞ ∫ 𝐸𝐸 2 𝑑𝑑 3 𝑥𝑥 2 𝒪𝒪 𝜖𝜖∘ 𝑑𝑑𝑉𝑉 ′ ; 𝓇𝓇 = 𝑟𝑟√1 + 𝜖𝜖, 𝜖𝜖 ≡ � � � − 2 cos 𝜃𝜃� �⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯� 𝜌𝜌𝑓𝑓 = 𝛁𝛁 ∙ 𝑫𝑫 ≡ 𝛁𝛁 ∙ (𝜖𝜖∘ 𝑬𝑬 + 𝑷𝑷) → ∮ 𝑫𝑫 ∙ 𝑑𝑑𝒂𝒂 = 𝑄𝑄𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒 𝛁𝛁 ∙ 𝑱𝑱 = − 𝜌𝜌 𝜖𝜖∘ 𝜎𝜎 = −𝜖𝜖∘ 𝜕𝜕𝜕𝜕 𝝉𝝉 = 𝒑𝒑 × 𝑬𝑬 → 𝑭𝑭 = (𝒑𝒑 ∙ 𝛁𝛁)𝑬𝑬 𝑈𝑈 = −𝒑𝒑 ∙ 𝑬𝑬 𝒑𝒑 = 𝛼𝛼𝑬𝑬 1 ∇2 𝑉𝑉 = − �∫𝒱𝒱 𝐸𝐸 2 𝑑𝑑 3 𝑥𝑥 + ∮𝒮𝒮 𝑉𝑉𝑬𝑬 ⋅ 𝑑𝑑𝒂𝒂� = Chapter 4 – Electric Fields in Matter 𝑱𝑱 = 𝜎𝜎𝒇𝒇 Φ𝐸𝐸 ≡ ∫𝒮𝒮 𝑬𝑬 ⋅ 𝑑𝑑𝒂𝒂 ∮ 𝑬𝑬 ⋅ 𝑑𝑑𝒂𝒂 = 𝐶𝐶𝑠𝑠𝑠𝑠ℎ𝑒𝑒𝑒𝑒𝑒𝑒 = 4𝜋𝜋𝜖𝜖∘ 𝑅𝑅 1−𝑅𝑅2 𝒑𝒑 ≡ ∫ 𝒓𝒓′ 𝜌𝜌(𝒓𝒓′ )𝑑𝑑𝑉𝑉 ′ ⇒ ∑𝑛𝑛𝑖𝑖=1 𝑞𝑞𝑖𝑖 𝒓𝒓′𝑖𝑖 𝑨𝑨(𝒓𝒓) = 𝓇𝓇 2 𝑑𝑑𝑉𝑉 ′ 𝑬𝑬 = −𝛁𝛁𝑉𝑉 𝜌𝜌𝑖𝑖𝑖𝑖 = 0 Chapter 3 – Potentials 𝑉𝑉(𝒓𝒓) = 𝜌𝜌�𝒓𝒓′ � ∫ = ∫ 𝜌𝜌(𝒙𝒙)𝑉𝑉(𝒙𝒙)𝑑𝑑3 𝑥𝑥 = 2 𝑬𝑬𝑖𝑖𝑖𝑖 = 0 Conductors: 𝑬𝑬(𝒓𝒓) = 𝛁𝛁 ∙ 𝑫𝑫 = 𝜌𝜌𝑓𝑓 𝛁𝛁 × 𝑯𝑯 = 𝑱𝑱𝑓𝑓 + 𝜕𝜕𝑫𝑫 𝜕𝜕𝜕𝜕 Chapter 8 – Conservation Laws 1 1 𝑢𝑢 = �𝜖𝜖∘ 𝐸𝐸 2 + 2 𝜇𝜇∘ 𝐵𝐵2 � 𝑺𝑺 ≡ 1 𝑇𝑇𝑖𝑖𝑖𝑖 ≡ 𝜖𝜖∘ �𝐸𝐸𝑖𝑖 𝐸𝐸𝑗𝑗 − 𝛿𝛿𝑖𝑖𝑖𝑖 𝐸𝐸 2 � + 2 1 1 𝜇𝜇∘ 1 √𝜖𝜖∘ 𝜇𝜇∘ = 𝑐𝑐 𝜔𝜔 𝑘𝑘 𝑣𝑣 = ; 𝑛𝑛 ≡ � 1 1 𝑐𝑐 𝜖𝜖𝜖𝜖 𝛼𝛼−𝛽𝛽 � 𝐸𝐸∘𝐼𝐼 𝛼𝛼+𝛽𝛽 𝛼𝛼 ≡ 2 � 𝐸𝐸∘𝐼𝐼 𝛼𝛼+𝛽𝛽 ∇2 𝑉𝑉 + 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 (𝛁𝛁 ∙ 𝑨𝑨) = − 𝛁𝛁 ∙ 𝑨𝑨 = 0 Coulomb gauge �∇2 𝑉𝑉 = − 𝜌𝜌 𝑡𝑡𝑟𝑟 ≡ 𝑡𝑡 − 𝑐𝑐 𝑉𝑉𝐿𝐿.𝑊𝑊. (𝒓𝒓, 𝑡𝑡) = 1 𝑉𝑉(𝑟𝑟, 𝑡𝑡) = 4𝜋𝜋𝜖𝜖 ∫ 1 𝑞𝑞𝑞𝑞 4𝜋𝜋𝜖𝜖∘ (𝓇𝓇𝓇𝓇−𝓻𝓻∙𝒗𝒗) 𝜌𝜌 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.𝑣𝑣 �⎯⎯⎯� 1 ∘ 𝜌𝜌�𝒓𝒓′ ,𝑡𝑡𝑟𝑟 � 𝓇𝓇 𝑞𝑞 𝑞𝑞 𝓇𝓇 4𝜋𝜋𝜖𝜖∘ (𝓻𝓻∙𝒖𝒖)3 𝒂𝒂=0 𝑬𝑬 = − 𝜇𝜇∘ 𝑝𝑝∘ 𝜔𝜔2 4𝜋𝜋 Mdip rad.: 𝑩𝑩 = − � sin 𝜃𝜃 𝑟𝑟 Edip rad.: � � cos(𝜔𝜔𝑡𝑡𝑟𝑟 ) 𝜽𝜽 𝑨𝑨(𝑟𝑟, 𝜃𝜃, 𝑡𝑡) = − 𝑉𝑉(𝒓𝒓, 𝑡𝑡) ≅ 𝜇𝜇 ∘ [𝒓𝒓� × 𝒑𝒑̈ ] 𝑩𝑩(𝒓𝒓, 𝑡𝑡) ≅ − 4𝜋𝜋𝜋𝜋𝜋𝜋 L.W. Point Charge: 𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑎𝑎 ∫0 sin � 𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎 � sin � 1 𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎 1 𝑄𝑄 � + 4𝜋𝜋𝜖𝜖∘ 𝑟𝑟 𝒓𝒓�∙𝒑𝒑(𝑡𝑡𝑟𝑟 ) 𝑟𝑟 2 0, 𝑛𝑛 ≠ 𝑚𝑚 � 𝑑𝑑𝑑𝑑 = �𝑎𝑎 , 𝑛𝑛 = 𝑚𝑚 2 𝒓𝒓�∙𝒑𝒑̇ (𝑡𝑡𝑟𝑟 ) 𝑺𝑺(𝒓𝒓, 𝑡𝑡) ≅ 𝑞𝑞 𝓇𝓇 [𝓻𝓻 4𝜋𝜋𝜖𝜖∘ (𝓻𝓻∙𝒖𝒖)3 1 4𝜋𝜋𝜋𝜋 6 cos 𝜃𝜃 𝑟𝑟 𝑟𝑟𝑟𝑟 1 𝜖𝜖 𝐸𝐸 2 𝒛𝒛� 2𝑐𝑐 ∘ ∘ 𝑛𝑛 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 � 𝑟𝑟 𝑑𝑑3 𝑥𝑥 � 𝑨𝑨(𝒓𝒓, 𝑡𝑡) ≅ 𝑟𝑟 2 𝜇𝜇∘ 2 sin 𝜃𝜃 � [𝑝𝑝̈ ( )] 𝑡𝑡 � � 𝒓𝒓 𝑟𝑟 16𝜋𝜋2 𝑐𝑐 𝑟𝑟 2 × (𝒖𝒖 × 𝒂𝒂)] ∫𝒱𝒱 (𝛁𝛁 ⋅ 𝒗𝒗 )𝑑𝑑𝑑𝑑 = ∮𝒮𝒮 𝒗𝒗 ⋅ 𝑑𝑑𝒂𝒂 𝐼𝐼 𝑐𝑐 sin 𝜃𝜃𝑇𝑇 𝜃𝜃𝐼𝐼 = 𝜃𝜃𝑅𝑅 𝛼𝛼−𝛽𝛽 2 � ; 𝛼𝛼+𝛽𝛽 =� = 𝛁𝛁 ∙ ⃖⃗ 𝑻𝑻 sin 𝜃𝜃𝐼𝐼 𝑨𝑨(𝑟𝑟, 𝜃𝜃, 𝑡𝑡) = − ⟨𝑺𝑺⟩ = � 𝑉𝑉 ′ = 𝑉𝑉 − 𝒗𝒗 𝑉𝑉(𝒓𝒓, 𝑡𝑡) 𝑐𝑐 2 = 𝜇𝜇∘ 𝑝𝑝∘ 𝜔𝜔 4𝜋𝜋𝜋𝜋 𝒂𝒂=0 1 (𝒗𝒗 × 𝑐𝑐 2 sin(𝜔𝜔𝑡𝑡𝑟𝑟 ) 𝒛𝒛� � 𝑟𝑟 2 𝒓𝒓� 𝜇𝜇∘ 𝑚𝑚∘ 𝜔𝜔2 sin 𝜃𝜃 � � � cos(𝜔𝜔𝑡𝑡𝑟𝑟 ) 𝝓𝝓 4𝜋𝜋𝜋𝜋 𝑟𝑟 𝜇𝜇∘ 𝑚𝑚∘2 𝜔𝜔4 12𝜋𝜋𝑐𝑐 3 𝑬𝑬(𝒓𝒓, 𝑡𝑡) ≅ 𝑺𝑺𝑟𝑟𝑟𝑟𝑟𝑟 = 𝜇𝜇∘ 4𝜋𝜋𝜋𝜋 [(𝒓𝒓� ∙ 𝒑𝒑̈ )𝒓𝒓� − 𝒑𝒑̈ ] = 𝜇𝜇 𝜇𝜇∘ 4𝜋𝜋𝜋𝜋 ∘ [𝑝𝑝̈ (𝑡𝑡𝑟𝑟 )]2 𝑃𝑃𝑟𝑟𝑟𝑟𝑟𝑟 (𝑡𝑡𝑟𝑟 ) ≅ 6𝜋𝜋𝜋𝜋 1 � 𝐸𝐸 2 𝓻𝓻 𝜇𝜇∘ 𝑐𝑐 𝑟𝑟𝑟𝑟𝑟𝑟 ∮𝒮𝒮 (𝛁𝛁 × 𝒗𝒗) ⋅ 𝑑𝑑𝒂𝒂 = ∮𝒞𝒞 𝒗𝒗 ⋅ 𝑑𝑑𝒍𝒍 1 𝑓𝑓(𝑥𝑥 − 𝑎𝑎) ≈ ∑𝑛𝑛=0 � 𝑑𝑑𝑛𝑛 𝑛𝑛! 𝑑𝑑𝑥𝑥 𝑛𝑛 𝒓𝒓� ⟨𝑃𝑃⟩ = 𝑃𝑃 = 𝑓𝑓(𝑎𝑎)� (𝑥𝑥 − 𝑎𝑎)𝑛𝑛 12𝜋𝜋𝜋𝜋 𝜇𝜇∘ 𝑞𝑞2 𝑎𝑎 2 6𝜋𝜋𝜋𝜋 𝛁𝛁 ⋅ � 2 � = 4𝜋𝜋𝛿𝛿 3 (𝒓𝒓) 𝑟𝑟 𝜇𝜇∘ 𝑝𝑝∘2 𝜔𝜔4 [𝒓𝒓� × (𝒓𝒓� × 𝒑𝒑̈ )] 𝑛𝑛2 𝜕𝜕𝜕𝜕 𝜕𝜕2 𝓻𝓻moving q = 𝒓𝒓 − 𝒘𝒘(𝑡𝑡𝑟𝑟 ) 𝑞𝑞𝑞𝑞𝒗𝒗 (𝓇𝓇𝓇𝓇−𝓻𝓻∙𝒗𝒗) 𝑛𝑛1 𝜕𝜕𝜕𝜕 □2 ≡ ∇2 − 𝜇𝜇∘ 𝜖𝜖∘ 𝜕𝜕𝑡𝑡 2 𝜇𝜇∘ 𝑝𝑝∘2 𝜔𝜔4 sin2 𝜃𝜃 32𝜋𝜋2 𝑐𝑐 = 2 2 � 𝛼𝛼+𝛽𝛽 𝑇𝑇 = 𝛼𝛼𝛼𝛼 � � × 𝑬𝑬) �⎯� 𝑩𝑩(𝒓𝒓, 𝑡𝑡) = (𝓻𝓻 ⟨𝑃𝑃⟩ = 𝜇𝜇∘ 𝒑𝒑̇ (𝑡𝑡𝑟𝑟 ) 4𝜋𝜋 𝜌𝜌 𝜇𝜇 𝜕𝜕𝒈𝒈 𝜕𝜕𝜕𝜕 1 𝑐𝑐 � 𝑹𝑹 𝑬𝑬 = − 𝐼𝐼𝑅𝑅 𝐼𝐼𝐼𝐼 𝜇𝜇2 𝑛𝑛1 𝜖𝜖∘ 𝐴𝐴𝐿𝐿.𝑊𝑊. (𝑟𝑟, 𝑡𝑡) = 4𝜋𝜋∘ 𝒇𝒇 = 𝜌𝜌𝑬𝑬 + 𝑱𝑱 × 𝑩𝑩 𝑨𝑨′ = 𝑨𝑨 + 𝛁𝛁𝜆𝜆 ⇒� □2 𝑨𝑨 = −𝜇𝜇∘ 𝑱𝑱 𝓇𝓇 𝜇𝜇1 𝑛𝑛2 𝛽𝛽 ≡ � = −𝜇𝜇∘ 𝑱𝑱 □2 𝑉𝑉 = − 𝑱𝑱�𝒓𝒓′ ,𝑡𝑡𝑟𝑟 � � � cos(𝜔𝜔𝑡𝑡𝑟𝑟 ) 𝝓𝝓 𝑅𝑅 ≡ 1 = 0) 𝑃𝑃𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 2 tan 𝜃𝜃𝐵𝐵 ≅ 𝑛𝑛2 � sin(𝜔𝜔𝑡𝑡𝑟𝑟 ) 𝜇𝜇∘ 𝑚𝑚∘2 𝜔𝜔4 sin2 𝜃𝜃 � 2 𝒓𝒓� 32𝜋𝜋2 𝑐𝑐 3 𝑟𝑟 (1 + 𝑥𝑥)±𝑛𝑛 ≈ 1 ± 𝑛𝑛𝑛𝑛 + 𝑛𝑛(𝑛𝑛 ∓ 1)𝑥𝑥 2 ± 𝑛𝑛(𝑛𝑛 ∓ 1)(𝑛𝑛 ∓ 2)𝑥𝑥 3 + 𝒪𝒪(𝑥𝑥 4 ) 2 � 𝜇𝜇∘ 𝑝𝑝∘ 𝜔𝜔2 sin 𝜃𝜃 ⟨𝑺𝑺⟩ = � + 𝜕𝜕𝜕𝜕 1−𝑣𝑣 2 ⁄𝑐𝑐 2 𝜇𝜇∘ 𝑚𝑚∘ 𝜔𝜔 sin 𝜃𝜃 � � � sin(𝜔𝜔𝑡𝑡𝑟𝑟 ) 𝝓𝝓 4𝜋𝜋𝜋𝜋 𝑟𝑟 𝜇𝜇∘ 𝑚𝑚∘ 𝜔𝜔2 sin 𝜃𝜃 � � � cos(𝜔𝜔𝑡𝑡𝑟𝑟 ) 𝜽𝜽 4𝜋𝜋𝑐𝑐 2 𝑟𝑟 Arbitrary source: Math: 𝑩𝑩 = − 4𝜋𝜋𝜖𝜖∘ 𝑐𝑐 𝑑𝑑𝑑𝑑 1 4𝜋𝜋𝜖𝜖∘ [1−(𝑣𝑣 2 ⁄𝑐𝑐 2 ) sin2 𝜃𝜃]3⁄2 𝑅𝑅 2 𝑝𝑝∘ 𝜔𝜔 𝑉𝑉(𝑟𝑟, 𝜃𝜃, 𝑡𝑡) = − 𝜕𝜕𝜕𝜕 � − 𝒗𝒗 𝑢𝑢 ≡ 𝑐𝑐𝓻𝓻 𝑞𝑞 𝑑𝑑𝑑𝑑 𝒈𝒈 = 𝜇𝜇∘ 𝜖𝜖∘ 𝑺𝑺 𝐸𝐸∘𝐼𝐼 − 𝐸𝐸∘𝑅𝑅 = 𝛽𝛽𝐸𝐸∘𝑇𝑇 � − 𝛁𝛁 �𝛁𝛁 ∙ 𝑨𝑨 + 𝜇𝜇∘ 𝜖𝜖∘ 𝑹𝑹 ≡ 𝒓𝒓 − 𝒗𝒗𝑡𝑡 4𝜋𝜋𝜖𝜖∘ 𝑅𝑅�1−(𝑣𝑣 2 ⁄𝑐𝑐 2 ) sin2 𝜃𝜃 Chapter 11 – Radiation ⟨𝒈𝒈⟩ = 𝑨𝑨(𝑟𝑟, 𝑡𝑡) = 4𝜋𝜋∘ ∫ [(𝑐𝑐 2 − 𝑣𝑣 2 )𝒖𝒖 + 𝓻𝓻 × (𝒖𝒖 × 𝒂𝒂)] �⎯� 𝑃𝑃(𝑟𝑟, 𝑡𝑡) = ∮ 𝑺𝑺 ∙ 𝑑𝑑𝒂𝒂 𝜕𝜕𝑡𝑡 2 𝜇𝜇 𝑑𝑑 3 𝑥𝑥 𝓇𝓇𝓇𝓇 − 𝓻𝓻 ∙ 𝒗𝒗 = �(𝑐𝑐 2 𝑡𝑡 − 𝒓𝒓 ∙ 𝒗𝒗)2 + (𝑐𝑐 2 − 𝑣𝑣 2 )(𝑟𝑟 2 − 𝑐𝑐 2 𝑡𝑡 2 ) 𝑬𝑬(𝒓𝒓, 𝑡𝑡) = 2 𝜇𝜇1 ≅𝜇𝜇2 1−𝛽𝛽 2 �� 2 2 1 ⁄𝑛𝑛2 ) −𝛽𝛽 𝜕𝜕2 𝑨𝑨 = −𝛁𝛁 ∙ 𝑺𝑺 ( 𝐼𝐼 ≡ ⟨𝑆𝑆⟩ = 𝑐𝑐𝜖𝜖∘ 𝐸𝐸∘2 cos 𝜃𝜃𝐼𝐼 �∇2 𝑨𝑨 − 𝜇𝜇∘ 𝜖𝜖∘ 𝜖𝜖∘ 1 �1−[(𝑛𝑛1 ⁄𝑛𝑛2 ) sin 𝜃𝜃𝐼𝐼 ]2 = Lorenz gauge: 𝛁𝛁 ∙ 𝑨𝑨 = −𝜇𝜇∘ 𝜖𝜖∘ 𝜖𝜖∘ 𝓇𝓇 cos 𝜃𝜃𝐼𝐼 𝜕𝜕𝜕𝜕 ⟨𝑢𝑢⟩ = 𝜖𝜖∘ 𝐸𝐸∘2 �=0 � ∙ 𝒌𝒌 𝒏𝒏 sin2 𝜃𝜃𝐵𝐵 = (𝑛𝑛 𝐸𝐸∘𝑇𝑇 = � 𝜕𝜕𝑨𝑨 cos 𝜃𝜃𝑇𝑇 𝜕𝜕𝜕𝜕 𝒑𝒑 = 𝜇𝜇∘ 𝜖𝜖∘ ∫𝒱𝒱 𝑺𝑺𝑑𝑑 3 𝑥𝑥 𝜕𝜕𝜕𝜕 𝑐𝑐 Chapter 10 – Potentials and Fields 𝑬𝑬 = −𝛁𝛁𝑉𝑉 − ⃖⃗ − 𝜖𝜖∘ 𝜇𝜇∘ 𝜕𝜕𝑺𝑺 𝒇𝒇 = 𝛁𝛁 ∙ 𝑻𝑻 2 𝐸𝐸∘𝐼𝐼 + 𝐸𝐸∘𝑅𝑅 = 𝛼𝛼𝐸𝐸∘𝑇𝑇 𝐸𝐸∘𝑅𝑅 = � 𝑑𝑑 = − 𝑑𝑑𝑑𝑑 �∫𝒱𝒱 𝑢𝑢𝑑𝑑3 𝑥𝑥� − ∮𝒮𝒮 𝑺𝑺 ∙ 𝑑𝑑𝒂𝒂 � 𝑬𝑬(𝒓𝒓, 𝑡𝑡) = 𝐸𝐸∘ 𝑒𝑒 𝑖𝑖(𝒌𝒌∙𝒓𝒓−𝜔𝜔𝜔𝜔) 𝒏𝒏 1 � 𝑩𝑩(𝒓𝒓, 𝑡𝑡) = 𝒌𝒌 × 𝑬𝑬 𝐵𝐵∘ = 𝐸𝐸∘ 𝜖𝜖∘ 𝜇𝜇∘ 𝑛𝑛 𝑑𝑑𝑑𝑑 �𝐵𝐵𝑖𝑖 𝐵𝐵𝑗𝑗 − 𝛿𝛿𝑖𝑖𝑖𝑖 𝐵𝐵2 � 𝜇𝜇∘ Chapter 9 – Electromagnetic Waves 𝑐𝑐 = 𝑑𝑑𝑑𝑑 (𝑬𝑬 × 𝑩𝑩) 𝑬𝑬)

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