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1Prof. Sergio B. MendesSpring 2018
Time Varying Electromagnetic Fields
Chapter 10 of “Modern Problems in Classical Electrodynamics” by Charles Brau
Chapter 6 and 14 of “Classical Electrodynamics” by John Jackson
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2Prof. Sergio B. MendesSpring 2018
Maxwell’s equations:
𝛁𝛁.𝑬𝑬 𝒓𝒓, 𝑡𝑡 =𝜌𝜌 𝒓𝒓, 𝑡𝑡𝜖𝜖0
𝛁𝛁.𝑩𝑩 𝒓𝒓, 𝑡𝑡 = 0
𝛁𝛁 × 𝑬𝑬 𝒓𝒓, 𝑡𝑡 = −𝜕𝜕𝑩𝑩 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
𝛁𝛁 × 𝑩𝑩 𝒓𝒓, 𝑡𝑡 = 𝜇𝜇𝑜𝑜 𝑱𝑱 𝒓𝒓, 𝑡𝑡 + 𝜇𝜇𝑜𝑜 𝜖𝜖0𝜕𝜕𝑬𝑬 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
1.
2.
3.
4.
Gauss’s law
Faraday’s law
GeneralizedAmpère’s law
Gauss’s law of magnetism
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3Prof. Sergio B. MendesSpring 2018
𝑩𝑩 𝒓𝒓, 𝒕𝒕 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓, 𝑡𝑡
Fields 𝑬𝑬 𝒓𝒓, 𝑡𝑡 & 𝑩𝑩 𝒓𝒓, 𝑡𝑡
in terms of
Potentials Φ 𝒓𝒓, 𝑡𝑡 & 𝑨𝑨 𝒓𝒓, 𝑡𝑡
𝑬𝑬 𝒓𝒓, 𝑡𝑡 = −𝛻𝛻Φ 𝒓𝒓, 𝑡𝑡 −𝜕𝜕𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
2.
3. 𝛁𝛁 × 𝑬𝑬 𝒓𝒓, 𝑡𝑡 = −𝜕𝜕𝑩𝑩 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
𝛁𝛁.𝑩𝑩 𝒓𝒓, 𝑡𝑡 = 0
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4Prof. Sergio B. MendesSpring 2018
−𝛻𝛻2Φ 𝒓𝒓, 𝑡𝑡 −𝜕𝜕𝜕𝜕𝑡𝑡
𝛻𝛻.𝑨𝑨 𝒓𝒓, 𝑡𝑡 =𝜌𝜌 𝒓𝒓, 𝑡𝑡𝜖𝜖0
−𝛻𝛻𝟐𝟐𝑨𝑨 𝒓𝒓, 𝑡𝑡 + 𝜇𝜇𝑜𝑜 𝜖𝜖0𝜕𝜕2𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡2 + 𝛻𝛻 𝛻𝛻.𝑨𝑨 𝒓𝒓, 𝑡𝑡 + 𝜇𝜇𝑜𝑜 𝜖𝜖0
𝜕𝜕𝜕𝜕𝑡𝑡 Φ 𝒓𝒓, 𝑡𝑡 = 𝜇𝜇𝑜𝑜 𝑱𝑱 𝒓𝒓, 𝑡𝑡
1.
4.
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5Prof. Sergio B. MendesSpring 2018
How to determine
the scalar Φ 𝒓𝒓, 𝑡𝑡 and vector 𝑨𝑨 𝒓𝒓, 𝑡𝑡 potentials
directly from
the charge 𝜌𝜌 𝒓𝒓, 𝑡𝑡 and current 𝑱𝑱 𝒓𝒓, 𝑡𝑡 densities ?
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6Prof. Sergio B. MendesSpring 2018
𝑨𝑨𝑨 𝒓𝒓, 𝑡𝑡 = 𝑨𝑨 𝒓𝒓, 𝑡𝑡 + 𝜵𝜵𝛬𝛬 𝒓𝒓, 𝑡𝑡
Φ′ 𝒓𝒓, 𝑡𝑡 = Φ 𝒓𝒓, 𝑡𝑡 −𝜕𝜕𝛬𝛬 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
𝛁𝛁.𝑩𝑩𝑨 𝒓𝒓, 𝑡𝑡 = 𝛁𝛁.𝑩𝑩 𝒓𝒓, 𝑡𝑡 = 0
𝑬𝑬 𝒓𝒓, 𝑡𝑡 = −𝛻𝛻Φ𝑨 𝒓𝒓, 𝑡𝑡 −𝜕𝜕𝑨𝑨𝑨 𝒓𝒓, 𝑡𝑡
𝜕𝜕𝑡𝑡= −𝛻𝛻Φ 𝒓𝒓, 𝑡𝑡 −
𝜕𝜕𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
2.
3.
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7Prof. Sergio B. MendesSpring 2018
𝛻𝛻.𝑨𝑨 𝒓𝒓, 𝑡𝑡 + 𝜇𝜇𝑜𝑜 𝜖𝜖0𝜕𝜕𝜕𝜕𝑡𝑡Φ 𝒓𝒓, 𝑡𝑡 = 0
−𝛻𝛻𝟐𝟐𝑨𝑨 𝒓𝒓, 𝑡𝑡 + 𝜇𝜇𝑜𝑜 𝜖𝜖0𝜕𝜕2𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡2
= 𝜇𝜇𝑜𝑜 𝑱𝑱 𝒓𝒓, 𝑡𝑡
−𝛻𝛻2Φ 𝒓𝒓, 𝑡𝑡 + 𝜇𝜇𝑜𝑜 𝜖𝜖0𝜕𝜕2Φ 𝒓𝒓, 𝑡𝑡
𝜕𝜕𝑡𝑡2=𝜌𝜌 𝒓𝒓, 𝑡𝑡𝜖𝜖0
In the Lorenz gauge:
𝜌𝜌 𝒓𝒓, 𝑡𝑡 determines Φ 𝒓𝒓, 𝑡𝑡
𝑱𝑱 𝒓𝒓, 𝑡𝑡 determines 𝑨𝑨 𝒓𝒓, 𝑡𝑡
𝜕𝜕𝜇𝜇𝐴𝐴𝜇𝜇 = 0
𝜕𝜕𝛼𝛼𝜕𝜕𝛼𝛼𝐴𝐴𝛽𝛽 = 𝜇𝜇0 𝐽𝐽𝛽𝛽
𝐽𝐽𝜇𝜇 ≡
𝑐𝑐 𝜌𝜌𝐽𝐽𝑥𝑥𝐽𝐽𝑦𝑦𝐽𝐽𝑧𝑧
𝐴𝐴𝜇𝜇 ≡
Φ/𝑐𝑐𝐴𝐴𝑥𝑥𝐴𝐴𝑦𝑦𝐴𝐴𝑧𝑧
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𝑓𝑓(𝑡𝑡) =1
2 𝜋𝜋�−∞
∞𝐹𝐹 𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
𝐹𝐹 𝜔𝜔 = �−∞
+∞𝑓𝑓 𝑡𝑡𝑨 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡′ 𝑑𝑑𝑡𝑡′
8
Fourier Analysis
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9Prof. Sergio B. MendesSpring 2018
Φ 𝒓𝒓, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞Φ 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
Φ 𝒓𝒓,𝜔𝜔 = �−∞
+∞Φ 𝒓𝒓, 𝑡𝑡𝑨 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡′ 𝑑𝑑𝑡𝑡′
𝜌𝜌 𝒓𝒓, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞𝜌𝜌 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
𝜌𝜌 𝒓𝒓,𝜔𝜔 = �−∞
+∞𝜌𝜌 𝒓𝒓, 𝑡𝑡𝑨 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡′ 𝑑𝑑𝑡𝑡′
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10Prof. Sergio B. MendesSpring 2018
𝛻𝛻2Φ 𝒓𝒓, 𝑡𝑡 = −𝜌𝜌 𝒓𝒓, 𝑡𝑡𝜖𝜖0
+1𝑐𝑐2𝜕𝜕2Φ 𝒓𝒓, 𝑡𝑡
𝜕𝜕𝑡𝑡2
𝛻𝛻2Φ 𝒓𝒓,𝜔𝜔 = −𝜌𝜌 𝒓𝒓,𝜔𝜔𝜖𝜖0
−𝜔𝜔2
𝑐𝑐2Φ 𝒓𝒓,𝜔𝜔
𝛻𝛻2Φ 𝒓𝒓, 𝑡𝑡 = 𝛻𝛻21
2 𝜋𝜋�−∞
∞Φ 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔 =
12 𝜋𝜋
�−∞
∞𝛻𝛻2Φ 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
1𝑐𝑐2𝜕𝜕2Φ 𝒓𝒓, 𝑡𝑡
𝜕𝜕𝑡𝑡2 =1𝑐𝑐2
𝜕𝜕2
𝜕𝜕𝑡𝑡21
2 𝜋𝜋�−∞
∞Φ 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔 =
12 𝜋𝜋�−∞
∞−𝜔𝜔2
𝑐𝑐2 Φ 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
−𝜌𝜌 𝒓𝒓, 𝑡𝑡𝜖𝜖0
= −1𝜖𝜖0
12 𝜋𝜋�−∞
∞𝜌𝜌 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔 =
12 𝜋𝜋�−∞
∞−𝜌𝜌 𝒓𝒓,𝜔𝜔𝜖𝜖0
𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
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11Prof. Sergio B. MendesSpring 2018
Φ 𝒓𝒓 = �−∞
+∞
𝐺𝐺 𝒓𝒓, 𝒓𝒓′𝜌𝜌 𝒓𝒓′
𝜖𝜖0𝑑𝑑𝑑𝑑𝑨
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨 =1
4 𝜋𝜋 𝒓𝒓 − 𝒓𝒓′
𝛻𝛻2𝐺𝐺 𝒓𝒓, 𝒓𝒓′ = − 𝛿𝛿3 𝒓𝒓 − 𝒓𝒓′
= 𝐺𝐺 𝓻𝓻
Remember the time independent problem:
𝛻𝛻2Φ 𝒓𝒓 = −𝜌𝜌 𝒓𝒓𝜖𝜖0
𝓻𝓻 ≡ 𝒓𝒓 − 𝒓𝒓′because:
𝛻𝛻2Φ 𝒓𝒓 = 𝛻𝛻2�−∞
+∞
𝐺𝐺 𝒓𝒓, 𝒓𝒓′𝜌𝜌 𝒓𝒓′
𝜖𝜖0𝑑𝑑𝑑𝑑𝑨 = �
−∞
+∞
− 𝛿𝛿3 𝒓𝒓 − 𝒓𝒓′𝜌𝜌 𝒓𝒓′
𝜖𝜖0𝑑𝑑𝑑𝑑𝑨 = −
𝜌𝜌 𝒓𝒓𝜖𝜖0
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12Prof. Sergio B. MendesSpring 2018
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 = �−∞
+∞𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨, 𝑡𝑡𝑨 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡′ 𝑑𝑑𝑡𝑡′
Back to the time varying problem:
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13Prof. Sergio B. MendesSpring 2018
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 = ? ? ?
Φ 𝒓𝒓,𝜔𝜔 = �−∞
+∞
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔𝜖𝜖0
𝑑𝑑𝑑𝑑𝑨
𝛻𝛻2Φ 𝒓𝒓,𝜔𝜔 = −𝜌𝜌 𝒓𝒓,𝜔𝜔𝜖𝜖0
−𝜔𝜔2
𝑐𝑐2Φ 𝒓𝒓,𝜔𝜔
𝛻𝛻2𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 = − 𝛿𝛿3 𝒓𝒓 − 𝒓𝒓′ −𝜔𝜔2
𝑐𝑐2𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 must satisfy the following equation:
𝛻𝛻2Φ 𝒓𝒓,𝜔𝜔 = �−∞
+∞
− 𝛿𝛿3 𝒓𝒓 − 𝒓𝒓′ −𝜔𝜔2
𝑐𝑐2𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔
𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔𝜖𝜖0
𝑑𝑑𝑑𝑑′
because:
= −𝜌𝜌 𝒓𝒓,𝜔𝜔𝜖𝜖0
−𝜔𝜔2
𝑐𝑐2Φ 𝒓𝒓,𝜔𝜔
= 𝛻𝛻2�−∞
+∞
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔𝜖𝜖0
𝑑𝑑𝑑𝑑′
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14Prof. Sergio B. MendesSpring 2018
𝓇𝓇 ≡ 𝒓𝒓 − 𝒓𝒓′𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 = 𝐺𝐺 𝓇𝓇,𝜔𝜔
𝛻𝛻2𝐺𝐺 𝓇𝓇,𝜔𝜔 = − 𝛿𝛿3 𝓇𝓇 −𝜔𝜔2
𝑐𝑐2𝐺𝐺 𝓇𝓇,𝜔𝜔
𝛻𝛻2 =1𝓇𝓇2
𝜕𝜕𝜕𝜕𝓇𝓇 𝓇𝓇2 𝜕𝜕
𝜕𝜕𝓇𝓇 +1
𝓇𝓇2 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃𝜕𝜕𝜕𝜕𝜃𝜃 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃
𝜕𝜕𝜕𝜕𝜃𝜃
+1
𝓇𝓇2 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃𝜕𝜕2
𝜕𝜕𝜑𝜑2
𝛻𝛻2𝐺𝐺 𝓇𝓇,𝜔𝜔 =1𝓇𝓇2
𝜕𝜕𝜕𝜕𝓇𝓇
𝓇𝓇2 𝜕𝜕𝜕𝜕𝓇𝓇
𝐺𝐺 𝓇𝓇,𝜔𝜔
𝛻𝛻2𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 = − 𝛿𝛿3 𝒓𝒓 − 𝒓𝒓′ −𝜔𝜔2
𝑐𝑐2𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔
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15Prof. Sergio B. MendesSpring 2018
𝐺𝐺 𝓇𝓇,𝜔𝜔 =𝑒𝑒± 𝑖𝑖 𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇
𝛻𝛻2𝐺𝐺 𝓇𝓇,𝜔𝜔 =1𝓇𝓇2
𝜕𝜕𝜕𝜕𝓇𝓇
𝓇𝓇2 𝜕𝜕𝜕𝜕𝓇𝓇
𝑒𝑒± 𝑖𝑖 𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝓇𝓇 ≠ 0
𝓇𝓇 = 0 𝛻𝛻2𝐺𝐺 𝓇𝓇,𝜔𝜔 = 𝛻𝛻21
4 𝜋𝜋 𝓇𝓇
𝛻𝛻2𝐺𝐺 𝓇𝓇,𝜔𝜔 +𝜔𝜔2
𝑐𝑐2𝐺𝐺 𝓇𝓇,𝜔𝜔 = − 𝛿𝛿3 𝓇𝓇
𝐺𝐺 𝓇𝓇,𝜔𝜔 = ? ? ?
in general:
= −𝜔𝜔2
𝑐𝑐2𝐺𝐺 𝓇𝓇,𝜔𝜔
= − 𝛿𝛿3 𝓇𝓇
𝛻𝛻2𝐺𝐺 𝓇𝓇,𝜔𝜔 =1𝓇𝓇2
𝜕𝜕𝜕𝜕𝓇𝓇
𝓇𝓇2 𝜕𝜕𝜕𝜕𝓇𝓇
𝐺𝐺 𝓇𝓇,𝜔𝜔 = − 𝛿𝛿3 𝓇𝓇 −𝜔𝜔2
𝑐𝑐2𝐺𝐺 𝓇𝓇,𝜔𝜔
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16Prof. Sergio B. MendesSpring 2018
Φ 𝒓𝒓,𝜔𝜔 = �−∞
+∞
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔𝜖𝜖0
𝑑𝑑𝑑𝑑𝑨
= �−∞
+∞𝑒𝑒± 𝑖𝑖 𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔𝜖𝜖0
𝑑𝑑𝑑𝑑𝑨
𝐺𝐺 𝓇𝓇,𝜔𝜔 =𝑒𝑒± 𝑖𝑖 𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇
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17Prof. Sergio B. MendesSpring 2018
𝑨𝑨 𝒓𝒓,𝜔𝜔 = 𝜇𝜇𝑜𝑜�−∞
+∞
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
= 𝜇𝜇𝑜𝑜�−∞
+∞𝑒𝑒± 𝑖𝑖 𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
𝐺𝐺 𝓇𝓇,𝜔𝜔 =𝑒𝑒± 𝑖𝑖 𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇
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18Prof. Sergio B. MendesSpring 2018
Φ 𝒓𝒓, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞Φ 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
Φ 𝒓𝒓,𝜔𝜔 = �−∞
+∞𝑒𝑒± 𝑖𝑖 𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔𝜖𝜖0
𝑑𝑑𝑑𝑑𝑨
=1
2 𝜋𝜋�−∞
∞�−∞
+∞𝑒𝑒± 𝑖𝑖 𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔𝜖𝜖0
𝑑𝑑𝑑𝑑𝑨 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
= �−∞
+∞1
4 𝜋𝜋 𝜖𝜖0 𝓇𝓇𝑑𝑑𝑑𝑑𝑨
12 𝜋𝜋
�−∞
∞𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 ∓ 𝓇𝓇
𝑐𝑐 𝑑𝑑𝜔𝜔
=1
4 𝜋𝜋 𝜖𝜖0�−∞
+∞𝜌𝜌 𝒓𝒓′, 𝑡𝑡′ = 𝑡𝑡 ∓ 𝓇𝓇𝑐𝑐
𝓇𝓇𝑑𝑑𝑑𝑑𝑨
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19Prof. Sergio B. MendesSpring 2018
Φ 𝒓𝒓, 𝑡𝑡 =1
4 𝜋𝜋 𝜖𝜖0�−∞
+∞𝜌𝜌 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 − 𝓇𝓇𝑐𝑐
𝓇𝓇𝑑𝑑𝑑𝑑𝑨
𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 −𝓇𝓇𝑐𝑐
𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 −𝓇𝓇𝑐𝑐
𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎 ≡ 𝑡𝑡 +𝓇𝓇𝑐𝑐
𝓻𝓻 ≡ 𝒓𝒓 − 𝒓𝒓′
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20Prof. Sergio B. MendesSpring 2018
𝛻𝛻2𝑨𝑨 𝒓𝒓, 𝑡𝑡 = −𝜇𝜇𝑜𝑜 𝑱𝑱 𝒓𝒓, 𝑡𝑡 +1𝑐𝑐2𝜕𝜕2𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡2
𝛻𝛻2𝑨𝑨 𝒓𝒓,𝜔𝜔 = −𝜇𝜇𝑜𝑜 𝑱𝑱 𝒓𝒓,𝜔𝜔 −𝜔𝜔2
𝑐𝑐2𝑨𝑨 𝒓𝒓,𝜔𝜔
𝑨𝑨 𝒓𝒓, 𝑡𝑡 =𝜇𝜇𝑜𝑜4 𝜋𝜋
�−∞
+∞ 𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 − 𝓇𝓇𝑐𝑐
𝓇𝓇𝑑𝑑𝑑𝑑𝑨
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21Prof. Sergio B. MendesSpring 2018
𝛻𝛻2Φ 𝒓𝒓, 𝑡𝑡 = −𝜌𝜌 𝒓𝒓, 𝑡𝑡𝜖𝜖0
+1𝑐𝑐2𝜕𝜕2Φ 𝒓𝒓, 𝑡𝑡
𝜕𝜕𝑡𝑡2
𝛻𝛻2Φ 𝒓𝒓,𝜔𝜔 = −𝜌𝜌 𝒓𝒓,𝜔𝜔𝜖𝜖0
−𝜔𝜔2
𝑐𝑐2 Φ 𝒓𝒓,𝜔𝜔
Φ 𝒓𝒓,𝜔𝜔 = �−∞
+∞
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔𝜖𝜖0
𝑑𝑑𝑑𝑑𝑨
𝛻𝛻2𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 = − 𝛿𝛿3 𝒓𝒓 − 𝒓𝒓′ −𝜔𝜔2
𝑐𝑐2 𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔
𝐺𝐺 𝓇𝓇,𝜔𝜔 =𝑒𝑒± 𝑖𝑖 𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇
Φ 𝒓𝒓, 𝑡𝑡 =1
4 𝜋𝜋 𝜖𝜖0�−∞
+∞𝜌𝜌 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 − 𝓇𝓇𝑐𝑐
𝓇𝓇 𝑑𝑑𝑑𝑑𝑨
Φ 𝒓𝒓, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞Φ 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
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22Prof. Sergio B. MendesSpring 2018
𝛻𝛻2𝑨𝑨 𝒓𝒓, 𝑡𝑡 = −𝜇𝜇𝑜𝑜 𝑱𝑱 𝒓𝒓, 𝑡𝑡 +1𝑐𝑐2𝜕𝜕2𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡2
𝛻𝛻2𝑨𝑨 𝒓𝒓,𝜔𝜔 = −𝜇𝜇𝑜𝑜 𝑱𝑱 𝒓𝒓,𝜔𝜔 −𝜔𝜔2
𝑐𝑐2 𝑨𝑨 𝒓𝒓,𝜔𝜔
𝑨𝑨 𝒓𝒓,𝜔𝜔 = �−∞
+∞
𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 𝜇𝜇𝑜𝑜 𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
𝛻𝛻2𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔 = − 𝛿𝛿3 𝒓𝒓 − 𝒓𝒓′ −𝜔𝜔2
𝑐𝑐2 𝐺𝐺 𝒓𝒓, 𝒓𝒓𝑨,𝜔𝜔
𝐺𝐺 𝓇𝓇,𝜔𝜔 =𝑒𝑒± 𝑖𝑖 𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇
𝑨𝑨 𝒓𝒓, 𝑡𝑡 =𝜇𝜇𝑜𝑜4 𝜋𝜋�
−∞
+∞ 𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 − 𝓇𝓇𝑐𝑐
𝓇𝓇 𝑑𝑑𝑑𝑑𝑨
𝑨𝑨 𝒓𝒓, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞𝑨𝑨 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
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23Prof. Sergio B. MendesSpring 2018
Φ 𝒓𝒓, 𝑡𝑡 =1
4 𝜋𝜋 𝜖𝜖0�−∞
+∞𝜌𝜌 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 − 𝓇𝓇𝑐𝑐
𝓇𝓇𝑑𝑑𝑑𝑑𝑨
𝑨𝑨 𝒓𝒓, 𝑡𝑡 =𝜇𝜇𝑜𝑜4 𝜋𝜋
�−∞
+∞ 𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 − 𝓇𝓇𝑐𝑐
𝓇𝓇𝑑𝑑𝑑𝑑𝑨
Retarded Potentials
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24Prof. Sergio B. MendesSpring 2018
𝑩𝑩 𝒓𝒓, 𝒕𝒕 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓, 𝑡𝑡
𝑬𝑬 𝒓𝒓, 𝑡𝑡 = −𝛻𝛻Φ 𝒓𝒓, 𝑡𝑡 −𝜕𝜕𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
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25Prof. Sergio B. MendesSpring 2018
Generalized Coulomb Law
Φ 𝒓𝒓, 𝑡𝑡 =1
4 𝜋𝜋 𝜖𝜖0�−∞
+∞𝜌𝜌 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡
𝓇𝓇𝑑𝑑𝑑𝑑𝑨
𝑬𝑬 𝒓𝒓, 𝑡𝑡 = −𝛻𝛻Φ 𝒓𝒓, 𝑡𝑡 −𝜕𝜕𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
𝑨𝑨 𝒓𝒓, 𝑡𝑡 =𝜇𝜇𝑜𝑜4 𝜋𝜋
�−∞
+∞𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡
𝓇𝓇𝑑𝑑𝑑𝑑𝑨
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =1
4 𝜋𝜋 𝜖𝜖0�−∞
+∞𝓻𝓻𝓇𝓇3 𝜌𝜌 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 +
𝓻𝓻𝑐𝑐 𝓇𝓇2
𝜕𝜕𝜌𝜌 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝜕𝜕𝑡𝑡′
−1
𝑐𝑐2 𝓇𝓇𝜕𝜕𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡
𝜕𝜕𝑡𝑡′𝑑𝑑𝑑𝑑𝑨
𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 −𝓇𝓇𝑐𝑐
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26Prof. Sergio B. MendesSpring 2018
𝑩𝑩 𝒓𝒓, 𝑡𝑡 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓, 𝑡𝑡
𝑨𝑨 𝒓𝒓, 𝑡𝑡 =𝜇𝜇𝑜𝑜4 𝜋𝜋
�−∞
+∞𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡
𝓇𝓇𝑑𝑑𝑑𝑑𝑨
𝑩𝑩 𝒓𝒓, 𝑡𝑡 =𝜇𝜇𝑜𝑜4 𝜋𝜋
�−∞
+∞
𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ×𝓻𝓻𝓇𝓇3 +
𝜕𝜕𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝜕𝜕𝑡𝑡
×𝓻𝓻
𝑐𝑐 𝓇𝓇𝟐𝟐 𝑑𝑑𝑑𝑑𝑨
𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 −𝓇𝓇𝑐𝑐
Generalized Biot-Savart Law
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27Prof. Sergio B. MendesSpring 2018
Jefimenko’s Relations
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =1
4 𝜋𝜋 𝜖𝜖0�−∞
+∞𝓻𝓻𝓇𝓇3 𝜌𝜌 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 +
𝓻𝓻𝑐𝑐 𝓇𝓇2
𝜕𝜕𝜌𝜌 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝜕𝜕𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡
−1
𝑐𝑐2 𝓇𝓇𝜕𝜕𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝜕𝜕𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡
𝑑𝑑𝑑𝑑𝑨
𝑩𝑩 𝒓𝒓, 𝒕𝒕 =𝜇𝜇𝑜𝑜4 𝜋𝜋
�−∞
+∞
𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ×𝓻𝓻𝓇𝓇3 +
𝜕𝜕𝑱𝑱 𝒓𝒓′, 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝜕𝜕𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡
×𝓻𝓻𝑐𝑐 𝓇𝓇𝟐𝟐 𝑑𝑑𝑑𝑑𝑨
𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 −𝓇𝓇𝑐𝑐𝓻𝓻 ≡ 𝒓𝒓 − 𝒓𝒓′
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28Prof. Sergio B. MendesSpring 2018
A Single Moving Charge
path of the moving charge:
𝜌𝜌 𝒓𝒓′, 𝑡𝑡′ = 𝑞𝑞 𝛿𝛿3 𝒓𝒓′ − 𝒓𝒓𝒐𝒐(𝑡𝑡′)
𝑱𝑱 𝒓𝒓′, 𝑡𝑡′ = 𝑞𝑞𝑑𝑑𝒓𝒓𝒐𝒐(𝑡𝑡′)𝑑𝑑𝑡𝑡′
𝛿𝛿3 𝒓𝒓′ − 𝒓𝒓𝒐𝒐(𝑡𝑡′)
charge density:
𝒓𝒓𝒐𝒐(𝑡𝑡′)
current density:
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29Prof. Sergio B. MendesSpring 2018
𝜌𝜌 𝒓𝒓′, 𝑡𝑡′ = 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡
= �−∞
∞
𝑞𝑞 𝛿𝛿3 𝒓𝒓′ − 𝒓𝒓𝒐𝒐(𝑡𝑡′) 𝛿𝛿 𝑡𝑡′ − 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 𝑑𝑑𝑡𝑡𝑨
𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 −𝒓𝒓 − 𝒓𝒓′
𝑐𝑐
𝜌𝜌 𝒓𝒓′, 𝑡𝑡′ = 𝑞𝑞 𝛿𝛿3 𝒓𝒓′ − 𝒓𝒓𝒐𝒐(𝑡𝑡′)
= �−∞
∞
𝜌𝜌 𝒓𝒓′, 𝑡𝑡′ 𝛿𝛿 𝑡𝑡′ − 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 𝑑𝑑𝑡𝑡𝑨
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30Prof. Sergio B. MendesSpring 2018
Φ 𝒓𝒓, 𝑡𝑡 =1
4 𝜋𝜋 𝜖𝜖0�−∞
+∞
𝜌𝜌 𝒓𝒓′, 𝑡𝑡′ = 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡1
𝒓𝒓 − 𝒓𝒓′𝑑𝑑𝑑𝑑𝑨
𝜌𝜌 𝒓𝒓′, 𝑡𝑡′ = 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 = �−∞
∞
𝑞𝑞 𝛿𝛿3 𝒓𝒓′ − 𝒓𝒓𝒐𝒐(𝑡𝑡′) 𝛿𝛿 𝑡𝑡′ − 𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 𝑑𝑑𝑡𝑡𝑨
=1
4 𝜋𝜋 𝜖𝜖0�−∞
+∞
�−∞
∞
𝑞𝑞 𝛿𝛿3 𝒓𝒓′ − 𝒓𝒓𝒐𝒐(𝑡𝑡′) 𝛿𝛿 𝑡𝑡′ − 𝑡𝑡 +𝒓𝒓 − 𝒓𝒓′
𝑐𝑐 𝑑𝑑𝑡𝑡𝑨1
𝒓𝒓 − 𝒓𝒓′ 𝑑𝑑𝑑𝑑𝑨
Φ 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖0�−∞
∞
𝛿𝛿 𝑡𝑡′ − 𝑡𝑡 +𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′)
𝑐𝑐𝑑𝑑𝑡𝑡𝑨
1𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′)
By performing the volume integral, we get:
𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡 ≡ 𝑡𝑡 −𝒓𝒓 − 𝒓𝒓′
𝑐𝑐
Scalar Potential created by a charge in motion
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31Prof. Sergio B. MendesSpring 2018
𝛿𝛿 𝑓𝑓 𝑥𝑥 𝑔𝑔 𝑥𝑥 𝑑𝑑𝑥𝑥 = ? ?
�𝑓𝑓 𝑥𝑥0 − 𝑎𝑎𝑓𝑓
𝑎𝑎𝑥𝑥 𝜖𝜖
𝑓𝑓 𝑥𝑥0 + 𝑎𝑎𝑓𝑓𝑎𝑎𝑥𝑥 𝜖𝜖
𝛿𝛿 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑥𝑥0 𝑔𝑔 𝑥𝑥 𝑑𝑑𝑓𝑓 = 𝑔𝑔 𝑥𝑥0
𝑓𝑓 𝑥𝑥0 = 0
𝛿𝛿 𝑓𝑓 𝑥𝑥 𝑔𝑔 𝑥𝑥 =1𝑑𝑑𝑓𝑓𝑑𝑑𝑥𝑥
𝛿𝛿 𝑥𝑥 − 𝑥𝑥0 𝑔𝑔 𝑥𝑥
�𝑥𝑥0− 𝜖𝜖
𝑥𝑥0+ 𝜖𝜖𝛿𝛿 𝑥𝑥 − 𝑥𝑥0 𝑔𝑔 𝑥𝑥 𝑑𝑑𝑥𝑥 = 𝑔𝑔 𝑥𝑥0
𝛿𝛿 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑥𝑥0 𝑔𝑔 𝑥𝑥 𝑑𝑑𝑓𝑓 = 𝛿𝛿 𝑥𝑥 − 𝑥𝑥0 𝑔𝑔 𝑥𝑥 𝑑𝑑𝑥𝑥
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32Prof. Sergio B. MendesSpring 2018
�−∞
+∞𝛿𝛿 𝑓𝑓 𝑥𝑥 𝑔𝑔 𝑥𝑥 𝑑𝑑𝑥𝑥 = �
−∞
+∞ 1𝑑𝑑𝑓𝑓𝑑𝑑𝑥𝑥
𝛿𝛿 𝑥𝑥 − 𝑥𝑥0 𝑔𝑔 𝑥𝑥 𝑑𝑑𝑥𝑥
𝛿𝛿 𝑓𝑓 𝑥𝑥 𝑔𝑔 𝑥𝑥 =1𝑑𝑑𝑓𝑓𝑑𝑑𝑥𝑥
𝛿𝛿 𝑥𝑥 − 𝑥𝑥0 𝑔𝑔 𝑥𝑥
=𝑔𝑔 𝑥𝑥0𝑑𝑑𝑓𝑓𝑑𝑑𝑥𝑥 𝑥𝑥 = 𝑥𝑥0
𝑓𝑓 𝑥𝑥0 = 0when
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33Prof. Sergio B. MendesSpring 2018
𝛿𝛿 𝑡𝑡′ − 𝑡𝑡 +𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′)
𝑐𝑐= 𝛿𝛿 𝑓𝑓 𝑡𝑡′
𝑓𝑓 𝑡𝑡′ ≡ 𝑡𝑡′ − 𝑡𝑡 +𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′)
𝑐𝑐
𝑑𝑑𝑓𝑓 𝑡𝑡′
𝑑𝑑𝑡𝑡𝑨= 1 −
𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝑡𝑡′
𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝑡𝑡′.𝜷𝜷(𝑡𝑡′)= 1 −
𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝑡𝑡′ .𝑑𝑑𝒓𝒓𝒐𝒐 𝑡𝑡′
𝑑𝑑𝑡𝑡′𝑐𝑐 𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝑡𝑡′
𝜷𝜷 𝑡𝑡′ ≡1𝑐𝑐𝑑𝑑𝒓𝒓𝒐𝒐(𝑡𝑡′)𝑑𝑑𝑡𝑡′
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34Prof. Sergio B. MendesSpring 2018
=𝑞𝑞
4 𝜋𝜋 𝜖𝜖01
𝑑𝑑𝑓𝑓 𝑡𝑡′𝑑𝑑𝑡𝑡′ 𝑡𝑡′= 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
1𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′ = 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
=𝑞𝑞
4 𝜋𝜋 𝜖𝜖0 𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′)1
1 − 𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′)𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′) .𝜷𝜷(𝑡𝑡′)
𝑡𝑡′ = 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
Φ 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖0�−∞
∞
𝛿𝛿 𝑡𝑡′ − 𝑡𝑡 +𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′)
𝑐𝑐1
𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′)𝑑𝑑𝑡𝑡𝑨
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35Prof. Sergio B. MendesSpring 2018
Φ 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖0 𝓇𝓇1
1 − �𝓻𝓻.𝜷𝜷 𝑡𝑡′ =𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝓻𝓻 ≡ 𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′)
Simplifying the Notation:
Φ 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖0 𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡′)1
1 − 𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝑡𝑡′𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝑡𝑡′
.𝜷𝜷(𝑡𝑡′)𝑡𝑡′ = 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
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36Prof. Sergio B. MendesSpring 2018
𝑨𝑨 𝒓𝒓, 𝑡𝑡 =𝜇𝜇0 𝑞𝑞
4 𝜋𝜋 𝓇𝓇𝑐𝑐 𝜷𝜷
1 − �𝓻𝓻.𝜷𝜷 𝑡𝑡′ = 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
Vector Potential created by a charge in motion
𝑨𝑨 𝒓𝒓, 𝑡𝑡 = Φ 𝒓𝒓, 𝑡𝑡1𝑐𝑐𝜷𝜷 𝑡𝑡′ = 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
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37Prof. Sergio B. MendesSpring 2018
Lienard-Wiechert Potentials
Φ 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖0 𝓇𝓇1
1 − �𝓻𝓻.𝜷𝜷 𝑡𝑡′ =𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝑨𝑨 𝒓𝒓, 𝑡𝑡 =𝜇𝜇0 𝑞𝑞
4 𝜋𝜋 𝓇𝓇𝑐𝑐 𝜷𝜷
1 − �𝓻𝓻.𝜷𝜷 𝑡𝑡′ = 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
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38Prof. Sergio B. MendesSpring 2018
𝑩𝑩 𝒓𝒓, 𝒕𝒕 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓, 𝑡𝑡
𝑬𝑬 𝒓𝒓, 𝑡𝑡 = −𝛻𝛻Φ 𝒓𝒓, 𝑡𝑡 −𝜕𝜕𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
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39Prof. Sergio B. MendesSpring 2018
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖01
1 − �𝓻𝓻.𝜷𝜷 31
𝛾𝛾2 𝓇𝓇2 �𝓻𝓻 − 𝜷𝜷 +1𝑐𝑐 𝓇𝓇
�𝓻𝓻 × �𝓻𝓻 − 𝜷𝜷 × �̇�𝜷𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝑩𝑩 𝒓𝒓, 𝒕𝒕 = �𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 ×𝑬𝑬 𝒓𝒓, 𝑡𝑡𝑐𝑐
Electric and Magnetic Fieldscreated by a charge in motion
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40Prof. Sergio B. MendesSpring 2018
Poynting Vector fora charge in motion
𝑺𝑺 𝒓𝒓, 𝑡𝑡 =1𝜇𝜇𝑜𝑜𝑬𝑬 𝒓𝒓, 𝑡𝑡 × 𝑩𝑩 𝒓𝒓, 𝒕𝒕
=1𝜇𝜇𝑜𝑜𝑬𝑬 𝒓𝒓, 𝑡𝑡 × �𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 ×
𝑬𝑬 𝒓𝒓, 𝑡𝑡𝑐𝑐
=1𝑐𝑐 𝜇𝜇𝑜𝑜
�𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 𝑬𝑬 𝒓𝒓, 𝑡𝑡 2 − 𝑬𝑬 𝒓𝒓, 𝑡𝑡 �𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 .𝑬𝑬 𝒓𝒓, 𝑡𝑡
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41Prof. Sergio B. MendesSpring 2018
𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 = 𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
𝑬𝑬 𝒓𝒓, 𝑡𝑡 2 �𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟𝑬𝑬 𝒓𝒓, 𝑡𝑡 2 𝑐𝑐𝑐𝑐𝑠𝑠 𝛿𝛿 �𝒆𝒆
𝑺𝑺 𝒓𝒓, 𝑡𝑡 =1𝑐𝑐 𝜇𝜇𝑜𝑜
�𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 𝑬𝑬 𝒓𝒓, 𝑡𝑡 2 − 𝑬𝑬 𝒓𝒓, 𝑡𝑡 �𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 .𝑬𝑬 𝒓𝒓, 𝑡𝑡
𝒪𝒪𝒓𝒓𝒐𝒐(𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
𝒓𝒓
𝑡𝑡
𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
𝛿𝛿
𝑺𝑺 𝒓𝒓, 𝑡𝑡
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42Prof. Sergio B. MendesSpring 2018
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖01
1 − �𝓻𝓻.𝜷𝜷 31
𝛾𝛾2 𝓇𝓇2 �𝓻𝓻 − 𝜷𝜷𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
�̇�𝜷 = 𝟎𝟎
(1) Special case: no acceleration
𝜷𝜷 = 𝒄𝒄𝒐𝒐𝒄𝒄𝒄𝒄𝒕𝒕𝒄𝒄𝒄𝒄𝒕𝒕
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖01
1 − �𝓻𝓻.𝜷𝜷 31
𝛾𝛾2 𝓇𝓇2 �𝓻𝓻 − 𝜷𝜷 +1𝑐𝑐 𝓇𝓇
�𝓻𝓻 × �𝓻𝓻 − 𝜷𝜷 × �̇�𝜷𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
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43Prof. Sergio B. MendesSpring 2018
𝒓𝒓𝒐𝒐(𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
𝒓𝒓
𝒓𝒓𝒐𝒐(𝑡𝑡)
𝓻𝓻 = 𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖01
1 − �𝓻𝓻.𝜷𝜷 31
𝛾𝛾2 𝓇𝓇2 �𝓻𝓻 − 𝜷𝜷𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡)
𝜷𝜷 𝑐𝑐 (𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
= �𝓻𝓻 𝑐𝑐 𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡 − 𝜷𝜷 𝑐𝑐 (𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)= �𝓻𝓻 𝑐𝑐 𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
= �𝓻𝓻 − 𝜷𝜷 𝑐𝑐 𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
= �𝓻𝓻 − 𝜷𝜷 𝓇𝓇
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =𝑞𝑞 𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡)
4 𝜋𝜋 𝜖𝜖0 𝛾𝛾21
𝓇𝓇 1 − �𝓻𝓻.𝜷𝜷 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
3
𝒪𝒪
𝑡𝑡
𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡 𝑡𝑡
= 𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡 − 𝒓𝒓𝒐𝒐 𝑡𝑡 − 𝒓𝒓𝒐𝒐(𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
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44Prof. Sergio B. MendesSpring 2018
𝒓𝒓𝒐𝒐(𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
𝒓𝒓
𝒓𝒓𝒐𝒐(𝑡𝑡)
𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡 = 𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡) = �𝓻𝓻 − 𝜷𝜷 𝓇𝓇
𝜷𝜷 𝑐𝑐 (𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
�𝓻𝓻 𝑐𝑐 𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =𝑞𝑞 𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡)
4 𝜋𝜋 𝜖𝜖0 𝛾𝛾21
𝓇𝓇 1 − �𝓻𝓻.𝜷𝜷 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
3
𝓇𝓇 1 − �𝓻𝓻.𝜷𝜷 = 𝓇𝓇 �𝓻𝓻. �𝓻𝓻 − 𝜷𝜷 = �𝓻𝓻. 𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡)
= 𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡) 𝑐𝑐𝑐𝑐𝑠𝑠 𝛿𝛿
𝑠𝑠𝑠𝑠𝑠𝑠 𝛿𝛿𝛽𝛽 𝑐𝑐 𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
=𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃
𝑐𝑐 𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
𝑠𝑠𝑠𝑠𝑠𝑠 𝛿𝛿 = 𝛽𝛽 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃
𝑐𝑐𝑐𝑐𝑠𝑠 𝛿𝛿 = 1 − 𝛽𝛽2 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 1/2
= 𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡) 1 − 𝛽𝛽2 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 1/2
𝛿𝛿
𝜃𝜃
𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡 𝑡𝑡
𝑡𝑡
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45Prof. Sergio B. MendesSpring 2018
𝒓𝒓𝒐𝒐(𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
𝒓𝒓
𝒓𝒓𝒐𝒐(𝑡𝑡)
𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡)
𝜷𝜷 𝑐𝑐 (𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖01 − 𝛽𝛽2
1 − 𝛽𝛽2 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 3/2𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡)𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡) 3
𝜃𝜃
𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡 𝑡𝑡
𝑡𝑡
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46Prof. Sergio B. MendesSpring 2018
𝑺𝑺 𝒓𝒓, 𝑡𝑡 =1𝑐𝑐 𝜇𝜇𝑜𝑜
�𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 𝑬𝑬 𝒓𝒓, 𝑡𝑡 2 − 𝑬𝑬 𝒓𝒓, 𝑡𝑡 �𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 .𝑬𝑬 𝒓𝒓, 𝑡𝑡
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖01 − 𝛽𝛽2
1 − 𝛽𝛽2𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 3/2𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡)𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡) 3
𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡𝒓𝒓 − 𝒓𝒓𝒐𝒐(𝑡𝑡)
𝜃𝜃
𝑬𝑬 𝒓𝒓, 𝑡𝑡 2 �𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝛿𝛿
𝑬𝑬 𝒓𝒓, 𝑡𝑡 2 𝑐𝑐𝑐𝑐𝑠𝑠 𝛿𝛿 �𝒆𝒆
𝑺𝑺 𝒓𝒓, 𝑡𝑡 =1𝑐𝑐 𝜇𝜇𝑜𝑜
𝑠𝑠𝑠𝑠𝑠𝑠 𝛿𝛿 𝑬𝑬 𝒓𝒓, 𝑡𝑡 2
𝑺𝑺 𝒓𝒓, 𝑡𝑡 = 𝑺𝑺 𝒓𝒓, 𝑡𝑡 �𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 − �𝒆𝒆
=1𝑐𝑐 𝜇𝜇𝑜𝑜
𝛽𝛽 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑬𝑬 𝒓𝒓, 𝑡𝑡 2
𝑟𝑟 → ∞
𝑟𝑟2 𝑺𝑺 𝒓𝒓, 𝑡𝑡 ∝1𝑟𝑟2
→ 0
𝜷𝜷 𝑐𝑐 (𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
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47Prof. Sergio B. MendesSpring 2018
𝒓𝒓 → ∞
(2) Special case: far from charge
𝑬𝑬 𝒓𝒓, 𝑡𝑡 ≅ 𝑬𝑬𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 ≡𝑞𝑞
4 𝜋𝜋 𝜖𝜖01
1 − �𝓻𝓻.𝜷𝜷 31𝑐𝑐 𝓇𝓇
�𝓻𝓻 × �𝓻𝓻 − 𝜷𝜷 × �̇�𝜷𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖01
1 − �𝓻𝓻.𝜷𝜷 31
𝛾𝛾2 𝓇𝓇2 �𝓻𝓻 − 𝜷𝜷 +1𝑐𝑐 𝓇𝓇
�𝓻𝓻 × �𝓻𝓻 − 𝜷𝜷 × �̇�𝜷𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
In the presence of a non-zero acceleration and for distances far away from the charge, 𝒓𝒓 → ∞, the second term will dominate. This term is known as the radiation term because (as we will see) it describes energy moving away from the accelerated electric charge.
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48Prof. Sergio B. MendesSpring 2018
𝑬𝑬𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖01
1 − �𝓻𝓻.𝜷𝜷 31𝑐𝑐 𝓇𝓇
�𝓻𝓻 × �𝓻𝓻 − 𝜷𝜷 × �̇�𝜷𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 = 𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
𝑬𝑬𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡
𝒪𝒪𝒓𝒓𝒐𝒐(𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
𝒓𝒓
𝑡𝑡
𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
𝑬𝑬𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 ⊥ �𝓻𝓻
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49Prof. Sergio B. MendesSpring 2018
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 =1𝑐𝑐 𝜇𝜇𝑜𝑜
�𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 𝑬𝑬𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 2 − 𝑬𝑬𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 �𝓻𝓻𝑡𝑡𝜏𝜏𝑟𝑟𝑟𝑟 .𝑬𝑬𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡
=1𝑐𝑐 𝜇𝜇𝑜𝑜
�𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 𝑬𝑬𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 2
=𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�𝓻𝓻 × �𝓻𝓻 − 𝜷𝜷 × �̇�𝜷
2 �𝓻𝓻1 − �𝓻𝓻.𝜷𝜷 6 𝓇𝓇𝟐𝟐
𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟 = 𝒓𝒓 − 𝒓𝒓𝒐𝒐 𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
𝒪𝒪𝒓𝒓𝒐𝒐(𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡)
𝒓𝒓
𝑡𝑡
𝜏𝜏𝑟𝑟𝑟𝑟𝑡𝑡
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡
𝑬𝑬𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 =𝑞𝑞
4 𝜋𝜋 𝜖𝜖01
1 − �𝓻𝓻.𝜷𝜷 31𝑐𝑐 𝓇𝓇
�𝓻𝓻 × �𝓻𝓻 − 𝜷𝜷 × �̇�𝜷𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
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50Prof. Sergio B. MendesSpring 2018
(2.a) Non-relativistic velocities 𝜷𝜷 ≪ 1
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 =𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�𝓻𝓻 × �𝓻𝓻 − 𝜷𝜷 × �̇�𝜷
2 �𝓻𝓻1 − �𝓻𝓻.𝜷𝜷 6 𝓇𝓇𝟐𝟐
𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
�̇�𝜷
𝜃𝜃
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎=
𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�𝓻𝓻 × �𝓻𝓻 × �̇�𝜷
2 �𝓻𝓻𝓇𝓇𝟐𝟐 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
=𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�𝓻𝓻 �𝓻𝓻. �̇�𝜷 − �̇�𝜷
2 �𝓻𝓻𝓇𝓇𝟐𝟐 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
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51Prof. Sergio B. MendesSpring 2018
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 =𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�𝓻𝓻 �𝓻𝓻. �̇�𝜷 − �̇�𝜷
2 �𝓻𝓻𝓇𝓇𝟐𝟐 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
=𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�̇�𝜷
2𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 �𝓻𝓻𝓇𝓇𝟐𝟐 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
�𝓻𝓻 �𝓻𝓻. �̇�𝜷 − �̇�𝜷 = �̇�𝜷 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃�𝓻𝓻
�̇�𝜷
�𝓻𝓻 �̇�𝜷 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃
𝜃𝜃
All quantities calculated (and drawn) at the retarded time
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52Prof. Sergio B. MendesSpring 2018
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 =𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�̇�𝜷
2𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 �𝓻𝓻𝓇𝓇𝟐𝟐 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
�𝓻𝓻
�̇�𝜷
𝜃𝜃
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎
𝑃𝑃𝑟𝑟𝑎𝑎𝑎𝑎 = �𝓇𝓇𝟐𝟐 𝑑𝑑Ω 𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡
Larmor formula
=𝑞𝑞2 �̇�𝜷
2
6 𝜋𝜋 𝑐𝑐 𝜖𝜖0
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53Prof. Sergio B. MendesSpring 2018
(2.b) Velocity and Acceleration are collinear: 𝜷𝜷 ∥ �̇�𝜷
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 =𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�𝓻𝓻 × �𝓻𝓻 − 𝜷𝜷 × �̇�𝜷
2 �𝓻𝓻1 − �𝓻𝓻.𝜷𝜷 6 𝓇𝓇𝟐𝟐
𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
=𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�̇�𝜷
2𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 �𝓻𝓻
1 − 𝛽𝛽 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 6 𝓇𝓇𝟐𝟐𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
=𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�𝓻𝓻 × �𝓻𝓻 × �̇�𝜷
2 �𝓻𝓻1 − �𝓻𝓻.𝜷𝜷 6 𝓇𝓇𝟐𝟐
𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝓻𝓻𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝜃𝜃
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎
𝜷𝜷 ∥ �̇�𝜷
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54Prof. Sergio B. MendesSpring 2018
�̇�𝜷
𝛽𝛽 ≅ 0
�̇�𝜷�̇�𝜷 �̇�𝜷
𝛽𝛽 ≅ 0.5 𝛽𝛽 ≅ 0.9𝛽𝛽 ≅ 0.99
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 =𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�̇�𝜷
2𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 �𝓻𝓻
1 − 𝛽𝛽 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 6 𝓇𝓇𝟐𝟐𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
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55Prof. Sergio B. MendesSpring 2018
(2.c) Velocity and Acceleration are perpendicular: 𝜷𝜷 ⊥ �̇�𝜷
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 =𝑞𝑞2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖0�𝓻𝓻 × �𝓻𝓻 − 𝜷𝜷 × �̇�𝜷
2 �𝓻𝓻1 − �𝓻𝓻.𝜷𝜷 6 𝓇𝓇𝟐𝟐
𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
�𝓻𝓻 ≡ 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑐𝑐𝑐𝑐𝑠𝑠 𝜑𝜑 , 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑 , 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃
𝜷𝜷 ≡ 0,0,𝛽𝛽
�̇�𝜷 ≡ �̇�𝛽, 0,0
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56Prof. Sergio B. MendesSpring 2018
𝜷𝜷 ⊥ �̇�𝜷
𝑺𝑺𝑟𝑟𝑎𝑎𝑎𝑎 𝒓𝒓, 𝑡𝑡 =𝑞𝑞2 �̇�𝜷
2
16 𝜋𝜋2 𝑐𝑐 𝜖𝜖01
1 − 𝛽𝛽 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 4 −1 − 𝛽𝛽2 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 𝑐𝑐𝑐𝑐𝑠𝑠2 𝜑𝜑
1 − 𝛽𝛽 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 6�𝓻𝓻𝓇𝓇𝟐𝟐
𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟
𝜷𝜷 = 𝛽𝛽 �𝒆𝒆𝒛𝒛
�̇�𝜷 = �̇�𝛽 �𝒆𝒆𝒙𝒙
𝜃𝜃
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57Prof. Sergio B. MendesSpring 2018
Radiation from a Collection of ChargesMore than a single charge, back to 𝜌𝜌 and 𝑱𝑱
𝜵𝜵.𝑨𝑨 𝒓𝒓, 𝑡𝑡 +1𝑐𝑐2
𝜕𝜕𝜕𝜕𝑡𝑡Φ 𝒓𝒓, 𝑡𝑡 = 0
𝜵𝜵.𝑨𝑨 𝒓𝒓,𝜔𝜔 −𝑠𝑠 𝜔𝜔𝑐𝑐2
Φ 𝒓𝒓,𝜔𝜔 = 0
Φ 𝒓𝒓, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞Φ 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
𝑨𝑨 𝒓𝒓, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞𝑨𝑨 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
Φ 𝒓𝒓,𝜔𝜔 =𝑐𝑐2
𝑠𝑠 𝜔𝜔𝜵𝜵.𝑨𝑨 𝒓𝒓,𝜔𝜔
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58Prof. Sergio B. MendesSpring 2018
𝑬𝑬 𝒓𝒓, 𝑡𝑡 = −𝛻𝛻Φ 𝒓𝒓, 𝑡𝑡 −𝜕𝜕𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
𝑬𝑬 𝒓𝒓,𝜔𝜔 = −𝛻𝛻Φ 𝒓𝒓,𝜔𝜔 + 𝑠𝑠 𝜔𝜔 𝑨𝑨 𝒓𝒓,𝜔𝜔
= −𝛻𝛻𝑐𝑐2
𝑠𝑠 𝜔𝜔𝜵𝜵.𝑨𝑨 𝒓𝒓,𝜔𝜔 + 𝑠𝑠 𝜔𝜔 𝑨𝑨 𝒓𝒓,𝜔𝜔
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59Prof. Sergio B. MendesSpring 2018
𝑩𝑩 𝒓𝒓, 𝑡𝑡 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓, 𝑡𝑡
𝑩𝑩 𝒓𝒓,𝜔𝜔 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓,𝜔𝜔
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60Prof. Sergio B. MendesSpring 2018
𝑨𝑨 𝒓𝒓,𝜔𝜔 = 𝜇𝜇𝑜𝑜�−∞
+∞𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
𝓇𝓇 ≡ 𝒓𝒓 − 𝒓𝒓𝑨
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61Prof. Sergio B. MendesSpring 2018
𝑨𝑨 𝒓𝒓,𝜔𝜔 = 𝜇𝜇𝑜𝑜�−∞
+∞𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
𝜔𝜔𝑐𝑐𝓇𝓇 ≪ 1
=𝜇𝜇𝑜𝑜4 𝜋𝜋
�−∞
+∞𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔𝓇𝓇
𝑑𝑑𝑑𝑑𝑨
Near-field approximation
𝓇𝓇 ≪ 𝜆𝜆
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62Prof. Sergio B. MendesSpring 2018
𝑨𝑨 𝒓𝒓,𝜔𝜔 = 𝜇𝜇𝑜𝑜�−∞
+∞𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
𝜵𝜵.𝑨𝑨 𝒓𝒓,𝜔𝜔
=𝜇𝜇𝑜𝑜4 𝜋𝜋
�−∞
+∞
𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 . �𝓻𝓻 −1𝓇𝓇2 + 𝑠𝑠
𝜔𝜔𝑐𝑐 𝓇𝓇
𝑒𝑒𝑖𝑖𝜔𝜔𝑐𝑐 𝓇𝓇𝑑𝑑𝑑𝑑𝑨
Far-field approximation:
=𝜇𝜇𝑜𝑜4 𝜋𝜋
�−∞
+∞
𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 .𝜵𝜵𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝓇𝓇
𝓇𝓇𝑑𝑑𝑑𝑑𝑨
𝜔𝜔𝑐𝑐𝓇𝓇 ≫ 1 𝓇𝓇 ≫ 𝜆𝜆 never satisfied for
the static problem !!
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63Prof. Sergio B. MendesSpring 2018
𝜵𝜵.𝑨𝑨 𝒓𝒓,𝜔𝜔 =𝜇𝜇𝑜𝑜4 𝜋𝜋
�−∞
+∞
𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 . �𝓻𝓻 −1𝓇𝓇2 + 𝑠𝑠
𝜔𝜔𝑐𝑐 𝓇𝓇
𝑒𝑒𝑖𝑖𝜔𝜔𝑐𝑐 𝓇𝓇𝑑𝑑𝑑𝑑𝑨
𝜔𝜔𝑐𝑐𝓇𝓇 ≫ 1
𝜵𝜵.𝑨𝑨 𝒓𝒓,𝜔𝜔 ≅ 𝜇𝜇𝑜𝑜�−∞
+∞
𝑠𝑠𝜔𝜔𝑐𝑐�𝓻𝓻 .
𝑒𝑒𝑖𝑖𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
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64Prof. Sergio B. MendesSpring 2018
𝜵𝜵.𝑨𝑨 𝒓𝒓,𝜔𝜔 = 𝜇𝜇𝑜𝑜�−∞
+∞
𝑠𝑠𝜔𝜔𝑐𝑐�𝓻𝓻 .
𝑒𝑒𝑖𝑖𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
𝜵𝜵 𝜵𝜵.𝑨𝑨 𝒓𝒓,𝜔𝜔
= −𝜔𝜔2
𝑐𝑐2�𝓻𝓻 �𝓻𝓻.𝑨𝑨 𝒓𝒓,𝜔𝜔
= 𝜇𝜇𝑜𝑜�−∞
+∞
𝑠𝑠𝜔𝜔𝑐𝑐�𝓻𝓻 𝑠𝑠
𝜔𝜔𝑐𝑐�𝓻𝓻 .
𝑒𝑒𝑖𝑖𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
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65Prof. Sergio B. MendesSpring 2018
𝑬𝑬 𝒓𝒓,𝜔𝜔 = −𝑐𝑐2
𝑠𝑠 𝜔𝜔𝛻𝛻 𝜵𝜵.𝑨𝑨 𝒓𝒓,𝜔𝜔 + 𝑠𝑠 𝜔𝜔 𝑨𝑨 𝒓𝒓,𝜔𝜔
=𝑐𝑐2
𝑠𝑠 𝜔𝜔𝜔𝜔2
𝑐𝑐2�𝓻𝓻 �𝓻𝓻.𝑨𝑨 𝒓𝒓,𝜔𝜔 + 𝑠𝑠 𝜔𝜔 𝑨𝑨 𝒓𝒓,𝜔𝜔
= 𝑠𝑠 𝜔𝜔 𝑨𝑨 𝒓𝒓,𝜔𝜔 − �𝓻𝓻 �𝓻𝓻.𝑨𝑨 𝒓𝒓,𝜔𝜔
= 𝑠𝑠 𝜔𝜔 𝑨𝑨⊥ 𝒓𝒓,𝜔𝜔
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66Prof. Sergio B. MendesSpring 2018
𝑩𝑩 𝒓𝒓,𝜔𝜔 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓,𝜔𝜔
= 𝑠𝑠𝜔𝜔𝑐𝑐�𝓻𝓻 × 𝑨𝑨 𝒓𝒓,𝜔𝜔
= 𝑠𝑠𝜔𝜔𝑐𝑐�𝓻𝓻 × 𝑨𝑨⊥ 𝒓𝒓,𝜔𝜔 + 𝑨𝑨∥ 𝒓𝒓,𝜔𝜔
= 𝑠𝑠𝜔𝜔𝑐𝑐�𝓻𝓻 × 𝑨𝑨⊥ 𝒓𝒓,𝜔𝜔
= �𝓻𝓻 ×𝑬𝑬 𝒓𝒓,𝜔𝜔
𝑐𝑐
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67Prof. Sergio B. MendesSpring 2018
𝑺𝑺 𝒓𝒓,𝜔𝜔 =1
2 𝜇𝜇𝑜𝑜𝑬𝑬 𝒓𝒓,𝜔𝜔 × 𝑩𝑩∗ 𝒓𝒓,𝜔𝜔
=1
2 𝜇𝜇𝑜𝑜𝑬𝑬 𝒓𝒓,𝜔𝜔 × �𝓻𝓻 ×
𝑬𝑬∗ 𝒓𝒓,𝜔𝜔𝑐𝑐
=1
2 𝜇𝜇𝑜𝑜𝑠𝑠 𝜔𝜔 𝑨𝑨⊥ 𝒓𝒓,𝜔𝜔 × �𝓻𝓻 ×
−𝑠𝑠 𝜔𝜔 𝑨𝑨⊥∗ 𝒓𝒓,𝜔𝜔𝑐𝑐
=𝜔𝜔2
2 𝜇𝜇𝑜𝑜 𝑐𝑐𝑨𝑨⊥ 𝒓𝒓,𝜔𝜔 2 �𝓻𝓻
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68Prof. Sergio B. MendesSpring 2018
=𝜇𝜇𝑜𝑜 𝜔𝜔2
32 𝜋𝜋2 𝑐𝑐�−∞
+∞𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝓇𝓇
𝓇𝓇𝑱𝑱⊥ 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
2
�𝓻𝓻
𝑨𝑨 𝒓𝒓,𝜔𝜔 = 𝜇𝜇𝑜𝑜�−∞
+∞𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
𝑺𝑺 𝒓𝒓,𝜔𝜔 =𝜔𝜔2
2 𝜇𝜇𝑜𝑜 𝑐𝑐𝑨𝑨⊥ 𝒓𝒓,𝜔𝜔 2 �𝓻𝓻
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69Prof. Sergio B. MendesSpring 2018
𝓇𝓇 = 𝒓𝒓 − 𝒓𝒓𝑨 = 𝑟𝑟2 + 𝑟𝑟𝑨2 − 2 𝑟𝑟 𝑟𝑟′𝑐𝑐𝑐𝑐𝑠𝑠 𝛾𝛾 1/2
= 𝑟𝑟 1 +𝑟𝑟𝑨2
𝑟𝑟2− 2
𝑟𝑟′
𝑟𝑟𝑐𝑐𝑐𝑐𝑠𝑠 𝛾𝛾
1/2
≅ 𝑟𝑟 − 𝑟𝑟′ 𝑐𝑐𝑐𝑐𝑠𝑠 𝛾𝛾
𝑟𝑟′
𝑟𝑟 ≪ 1
�𝒓𝒓. �𝒓𝒓𝑨 ≡ 𝑐𝑐𝑐𝑐𝑠𝑠 𝛾𝛾
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70Prof. Sergio B. MendesSpring 2018
𝑺𝑺 𝒓𝒓,𝜔𝜔 =𝜇𝜇𝑜𝑜 𝜔𝜔2
32 𝜋𝜋2 𝑐𝑐�−∞
+∞𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝓇𝓇
𝓇𝓇𝑱𝑱⊥ 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
2
�𝓻𝓻
𝓇𝓇 ≅ 𝑟𝑟 − 𝑟𝑟′𝑐𝑐𝑐𝑐𝑠𝑠 𝛾𝛾
=𝜇𝜇𝑜𝑜 𝜔𝜔2
32 𝜋𝜋2 𝑟𝑟2 𝑐𝑐�−∞
+∞
𝑒𝑒−𝑖𝑖𝜔𝜔𝑐𝑐 𝑟𝑟
′𝑐𝑐𝑜𝑜𝑐𝑐 𝛾𝛾 𝑱𝑱⊥ 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
2
�𝓻𝓻
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71Prof. Sergio B. MendesSpring 2018
Example: Antenna
𝜃𝜃
𝜃𝜃 = 𝛾𝛾
𝑱𝑱⊥ 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑′ = −𝐼𝐼 𝑧𝑧′,𝜔𝜔 𝑑𝑑𝑧𝑧′𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 �𝒆𝒆𝜃𝜃�𝒓𝒓
�𝒓𝒓𝑨
𝑟𝑟
�−∞
+∞
𝑒𝑒−𝑖𝑖𝜔𝜔𝑐𝑐 𝑟𝑟
′𝑐𝑐𝑜𝑜𝑐𝑐 𝛾𝛾 𝑱𝑱⊥ 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨 = −�𝒆𝒆𝜃𝜃 �−∞
∞
𝑒𝑒−𝑖𝑖𝜔𝜔𝑐𝑐 𝑧𝑧
′𝑐𝑐𝑜𝑜𝑐𝑐 𝜃𝜃 𝐼𝐼 𝑧𝑧′,𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑑𝑑𝑧𝑧′
𝐿𝐿
−𝐿𝐿
𝑧𝑧′ �𝒆𝒆𝜃𝜃
𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔
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72Prof. Sergio B. MendesSpring 2018
𝐼𝐼 𝑧𝑧′,𝜔𝜔 = 𝐼𝐼𝑜𝑜 −𝐿𝐿 ≤ 𝑧𝑧′≤ 𝐿𝐿
= 0 otherwise
−�𝒆𝒆𝜃𝜃 �−∞
∞
𝑒𝑒−𝑖𝑖𝜔𝜔𝑐𝑐 𝑧𝑧
′𝑐𝑐𝑜𝑜𝑐𝑐 𝜃𝜃 𝐼𝐼 𝑧𝑧′,𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑑𝑑𝑧𝑧′ = −�𝒆𝒆𝜃𝜃 𝐼𝐼𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 �−𝐿𝐿
𝐿𝐿
𝑒𝑒−𝑖𝑖𝜔𝜔𝑐𝑐 𝑧𝑧
′𝑐𝑐𝑜𝑜𝑐𝑐 𝜃𝜃 𝑑𝑑𝑧𝑧′
= −�𝒆𝒆𝜃𝜃 𝐼𝐼𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 2𝐿𝐿 𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐𝜔𝜔𝑐𝑐 𝐿𝐿 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃
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73Prof. Sergio B. MendesSpring 2018
𝑺𝑺 𝒓𝒓,𝜔𝜔 =𝜇𝜇𝑜𝑜 𝜔𝜔2
32 𝜋𝜋2 𝑟𝑟2 𝑐𝑐�−∞
+∞
𝑒𝑒−𝑖𝑖𝜔𝜔𝑐𝑐 𝑟𝑟
′𝑐𝑐𝑜𝑜𝑐𝑐 𝛾𝛾 𝑱𝑱⊥ 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
2
�𝓻𝓻
=𝜇𝜇𝑜𝑜 𝜔𝜔2 𝐼𝐼𝑜𝑜2
32 𝜋𝜋2 𝑟𝑟2 𝑐𝑐𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 4 𝐿𝐿2 𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐2
𝜔𝜔𝑐𝑐𝐿𝐿 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 �𝓻𝓻
𝜔𝜔𝑐𝑐 𝐿𝐿 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 = 𝑚𝑚 𝜋𝜋 𝜃𝜃𝑚𝑚 = 𝑐𝑐𝑐𝑐𝑠𝑠−1 𝑚𝑚
𝜆𝜆2 𝐿𝐿 > 0
𝑚𝑚 = 0, ±1, ±2, ±3, …
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74Prof. Sergio B. MendesSpring 2018
𝜃𝜃𝑚𝑚 = 𝑐𝑐𝑐𝑐𝑠𝑠−1 𝑚𝑚𝜆𝜆
2 𝐿𝐿> 0
𝑚𝑚 = 0, ±1, ±2, ±3, …
• If 𝐿𝐿 < 𝜆𝜆2
then only 𝑚𝑚 = 0
• If 𝐿𝐿 > 𝜆𝜆2
then higher orders are possible
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75Prof. Sergio B. MendesSpring 2018
Example: Electric Dipole
𝒑𝒑 𝜔𝜔 = �−∞
∞
𝒓𝒓′ 𝜌𝜌 𝒓𝒓′,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
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76Prof. Sergio B. MendesSpring 2018
Charge Conservation
−𝜕𝜕𝜌𝜌 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡
= 𝛁𝛁. 𝑱𝑱 𝒓𝒓, 𝑡𝑡
𝑠𝑠 𝜔𝜔 𝜌𝜌 𝒓𝒓,𝜔𝜔 = 𝛁𝛁. 𝑱𝑱 𝒓𝒓,𝜔𝜔
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77Prof. Sergio B. MendesSpring 2018
𝑨𝑨 𝒓𝒓,𝜔𝜔 = 𝜇𝜇𝑜𝑜�−∞
+∞𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
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78Prof. Sergio B. MendesSpring 2018
𝓇𝓇 = 𝒓𝒓 − 𝒓𝒓𝑨 = 𝑟𝑟2 + 𝑟𝑟𝑨2 − 2 𝑟𝑟 𝑟𝑟′𝑐𝑐𝑐𝑐𝑠𝑠 𝛾𝛾 1/2
= 𝑟𝑟 1 +𝑟𝑟𝑨2
𝑟𝑟2− 2
𝑟𝑟′
𝑟𝑟𝑐𝑐𝑐𝑐𝑠𝑠 𝛾𝛾
1/2
𝜔𝜔𝑐𝑐𝒓𝒓 − 𝒓𝒓𝑨 ≅
𝜔𝜔𝑐𝑐𝑟𝑟
𝜔𝜔𝑐𝑐𝑟𝑟′ ≪ 1
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79Prof. Sergio B. MendesSpring 2018
𝑨𝑨 𝒓𝒓,𝜔𝜔 = 𝜇𝜇𝑜𝑜�−∞
+∞𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝓇𝓇
4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
≅ 𝜇𝜇𝑜𝑜𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝑟𝑟
4 𝜋𝜋 𝑟𝑟�−∞
+∞
𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
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80Prof. Sergio B. MendesSpring 2018
�−∞
+∞
𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
= −�−∞
+∞
𝒓𝒓′ 𝑠𝑠 𝜔𝜔 𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
= −𝑠𝑠 𝜔𝜔 𝒑𝒑 𝜔𝜔
= −𝑠𝑠 𝜔𝜔�−∞
+∞
𝒓𝒓′ 𝜌𝜌 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
= −�−∞
+∞
𝒓𝒓′ 𝛁𝛁. 𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
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81Prof. Sergio B. MendesSpring 2018
𝑨𝑨 𝒓𝒓,𝜔𝜔 ≅ 𝜇𝜇𝑜𝑜𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝑟𝑟
4 𝜋𝜋 𝑟𝑟�−∞
+∞
𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨
≅ −𝜇𝜇𝑜𝑜𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝑟𝑟
4 𝜋𝜋 𝑟𝑟𝑠𝑠 𝜔𝜔 𝒑𝒑 𝜔𝜔
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82Prof. Sergio B. MendesSpring 2018
𝑺𝑺 𝒓𝒓,𝜔𝜔 =𝜔𝜔2
2 𝜇𝜇𝑜𝑜 𝑐𝑐𝑨𝑨⊥ 𝒓𝒓,𝜔𝜔 2 �𝓻𝓻
𝑨𝑨 𝒓𝒓,𝜔𝜔 ≅ −𝜇𝜇𝑜𝑜𝑒𝑒𝑖𝑖
𝜔𝜔𝑐𝑐 𝑟𝑟
4 𝜋𝜋 𝑟𝑟𝑠𝑠 𝜔𝜔 𝒑𝒑 𝜔𝜔
=𝜇𝜇𝑜𝑜 𝜔𝜔4
32 𝜋𝜋2 𝑐𝑐 𝑟𝑟2𝒑𝒑⊥ 𝜔𝜔 2 �𝓻𝓻
𝑨𝑨⊥ 𝒓𝒓,𝜔𝜔 2 =𝜇𝜇𝑜𝑜2 𝜔𝜔2
16 𝜋𝜋2 𝑟𝑟2𝒑𝒑⊥ 𝜔𝜔 2
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83Prof. Sergio B. MendesSpring 2018
𝑺𝑺 𝒓𝒓, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞𝑺𝑺 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
𝑺𝑺 𝒓𝒓,𝜔𝜔 = �−∞
∞𝑺𝑺 𝒓𝒓, 𝑡𝑡 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝑡𝑡
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84Prof. Sergio B. MendesSpring 2018
𝑬𝑬 𝒓𝒓, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞𝑬𝑬 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
𝑩𝑩 𝒓𝒓, 𝑡𝑡 =1
2 𝜋𝜋�−∞
∞𝑩𝑩 𝒓𝒓,𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔
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85Prof. Sergio B. MendesSpring 2018
=1𝜇𝜇𝑜𝑜�−∞
∞ 12 𝜋𝜋
�−∞
∞𝑬𝑬 𝒓𝒓,𝜔𝜔𝑨 𝑒𝑒− 𝑖𝑖 𝜔𝜔′ 𝑡𝑡 𝑑𝑑𝜔𝜔𝑨 ×
12 𝜋𝜋
�−∞
∞𝑩𝑩 𝒓𝒓,𝜔𝜔" 𝑒𝑒− 𝑖𝑖 𝜔𝜔" 𝑡𝑡 𝑑𝑑𝜔𝜔" 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝑡𝑡
𝑺𝑺 𝒓𝒓,𝜔𝜔 = �−∞
∞𝑺𝑺 𝒓𝒓, 𝑡𝑡 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝑡𝑡
=1𝜇𝜇𝑜𝑜�−∞
∞𝑬𝑬 𝒓𝒓, 𝑡𝑡 × 𝑩𝑩 𝒓𝒓, 𝑡𝑡 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝑡𝑡
= 1𝜇𝜇𝑜𝑜
12 𝜋𝜋 ∫−∞
∞ 𝑬𝑬 𝒓𝒓,𝜔𝜔𝑨 𝑑𝑑𝜔𝜔𝑨 × 12 𝜋𝜋 ∫−∞
∞ 𝑩𝑩 𝒓𝒓,𝜔𝜔𝜔 𝑑𝑑𝜔𝜔𝜔𝛿𝛿 𝜔𝜔 − 𝜔𝜔′ − 𝜔𝜔𝜔
=1𝜇𝜇𝑜𝑜
14 𝜋𝜋2
�−∞
∞𝑬𝑬 𝒓𝒓,𝜔𝜔𝑨 × 𝑩𝑩 𝒓𝒓,𝜔𝜔 − 𝜔𝜔′ 𝑑𝑑𝜔𝜔𝑨
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86Prof. Sergio B. MendesSpring 2018
𝑺𝑺 𝒓𝒓,𝜔𝜔 =1𝜇𝜇𝑜𝑜
14 𝜋𝜋2
�−∞
∞𝑬𝑬 𝒓𝒓,𝜔𝜔𝑨 × 𝑩𝑩 𝒓𝒓,𝜔𝜔 −𝜔𝜔′ 𝑑𝑑𝜔𝜔𝑨
𝑬𝑬 𝒓𝒓,𝜔𝜔𝑨 = 𝑬𝑬𝑜𝑜 𝛿𝛿 𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜 + 𝑬𝑬𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜
𝑩𝑩 𝒓𝒓,𝜔𝜔𝑨 = 𝑩𝑩𝑜𝑜 𝛿𝛿 𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜 + 𝑩𝑩𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜
𝑩𝑩 𝒓𝒓,𝜔𝜔 − 𝜔𝜔′ = 𝑩𝑩𝑜𝑜 𝛿𝛿 𝜔𝜔 − 𝜔𝜔′ − 𝜔𝜔𝑜𝑜 + 𝑩𝑩𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔 + 𝜔𝜔′ − 𝜔𝜔𝑜𝑜
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87Prof. Sergio B. MendesSpring 2018
�−∞
∞𝑬𝑬𝑜𝑜 𝛿𝛿 𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜 × 𝑩𝑩𝑜𝑜 𝛿𝛿 𝜔𝜔 − 𝜔𝜔′ − 𝜔𝜔𝑜𝑜 𝑑𝑑𝜔𝜔𝑨
𝑬𝑬 𝒓𝒓,𝜔𝜔𝑨 = 𝑬𝑬𝑜𝑜 𝛿𝛿 𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜 + 𝑬𝑬𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜
𝑩𝑩 𝒓𝒓,𝜔𝜔 − 𝜔𝜔′ = 𝑩𝑩𝑜𝑜 𝛿𝛿 𝜔𝜔 − 𝜔𝜔′ − 𝜔𝜔𝑜𝑜 + 𝑩𝑩𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔 + 𝜔𝜔′ − 𝜔𝜔𝑜𝑜
�−∞
∞𝑬𝑬𝑜𝑜 𝛿𝛿 𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜 × 𝑩𝑩𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔 + 𝜔𝜔′ − 𝜔𝜔𝑜𝑜 𝑑𝑑𝜔𝜔𝑨
�−∞
∞𝑬𝑬𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜 × 𝑩𝑩𝑜𝑜 𝛿𝛿 𝜔𝜔 − 𝜔𝜔′ − 𝜔𝜔𝑜𝑜 𝑑𝑑𝜔𝜔𝑨
�−∞
∞𝑬𝑬𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜 × 𝑩𝑩𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔 + 𝜔𝜔′ − 𝜔𝜔𝑜𝑜 𝑑𝑑𝜔𝜔𝑨
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88Prof. Sergio B. MendesSpring 2018
�−∞
∞𝑬𝑬𝑜𝑜 𝛿𝛿 𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜 × 𝑩𝑩𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔 + 𝜔𝜔′ − 𝜔𝜔𝑜𝑜 𝑑𝑑𝜔𝜔′ = 𝑬𝑬𝑜𝑜 × 𝑩𝑩𝑜𝑜∗ 𝛿𝛿 𝜔𝜔
�−∞
∞𝑬𝑬𝑜𝑜∗ 𝛿𝛿 −𝜔𝜔𝑨 − 𝜔𝜔𝑜𝑜 × 𝑩𝑩𝑜𝑜 𝛿𝛿 𝜔𝜔 − 𝜔𝜔′ − 𝜔𝜔𝑜𝑜 𝑑𝑑𝜔𝜔′ = 𝑬𝑬𝑜𝑜∗ × 𝑩𝑩𝑜𝑜 𝛿𝛿 𝜔𝜔