time varying electromagnetic fields 611 spring 19... · 2020. 3. 17. · time varying...

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

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

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

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.

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 ?

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.

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

𝑓𝑓(𝑡𝑡) =1

2 𝜋𝜋�−∞

∞𝐹𝐹 𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔

𝐹𝐹 𝜔𝜔 = �−∞

+∞𝑓𝑓 𝑡𝑡𝑨 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡′ 𝑑𝑑𝑡𝑡′

8

Fourier Analysis

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′

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

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

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:

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′

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

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

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

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

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

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′

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

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

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

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

24Prof. Sergio B. MendesSpring 2018

𝑩𝑩 𝒓𝒓, 𝒕𝒕 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓, 𝑡𝑡

𝑬𝑬 𝒓𝒓, 𝑡𝑡 = −𝛻𝛻Φ 𝒓𝒓, 𝑡𝑡 −𝜕𝜕𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡

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

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

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′

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:

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

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

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

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

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′

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

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

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

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

38Prof. Sergio B. MendesSpring 2018

𝑩𝑩 𝒓𝒓, 𝒕𝒕 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓, 𝑡𝑡

𝑬𝑬 𝒓𝒓, 𝑡𝑡 = −𝛻𝛻Φ 𝒓𝒓, 𝑡𝑡 −𝜕𝜕𝑨𝑨 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡

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

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

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

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

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)

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

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

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)

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.

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

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

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

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

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

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

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

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

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

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

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

59Prof. Sergio B. MendesSpring 2018

𝑩𝑩 𝒓𝒓, 𝑡𝑡 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓, 𝑡𝑡

𝑩𝑩 𝒓𝒓,𝜔𝜔 = 𝛻𝛻 × 𝑨𝑨 𝒓𝒓,𝜔𝜔

60Prof. Sergio B. MendesSpring 2018

𝑨𝑨 𝒓𝒓,𝜔𝜔 = 𝜇𝜇𝑜𝑜�−∞

+∞𝑒𝑒𝑖𝑖

𝜔𝜔𝑐𝑐 𝓇𝓇

4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨

𝓇𝓇 ≡ 𝒓𝒓 − 𝒓𝒓𝑨

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

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 !!

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

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

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

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

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

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

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

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

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

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

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, …

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

75Prof. Sergio B. MendesSpring 2018

Example: Electric Dipole

𝒑𝒑 𝜔𝜔 = �−∞

∞

𝒓𝒓′ 𝜌𝜌 𝒓𝒓′,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨

76Prof. Sergio B. MendesSpring 2018

Charge Conservation

−𝜕𝜕𝜌𝜌 𝒓𝒓, 𝑡𝑡𝜕𝜕𝑡𝑡

= 𝛁𝛁. 𝑱𝑱 𝒓𝒓, 𝑡𝑡

𝑠𝑠 𝜔𝜔 𝜌𝜌 𝒓𝒓,𝜔𝜔 = 𝛁𝛁. 𝑱𝑱 𝒓𝒓,𝜔𝜔

77Prof. Sergio B. MendesSpring 2018

𝑨𝑨 𝒓𝒓,𝜔𝜔 = 𝜇𝜇𝑜𝑜�−∞

+∞𝑒𝑒𝑖𝑖

𝜔𝜔𝑐𝑐 𝓇𝓇

4 𝜋𝜋 𝓇𝓇𝑱𝑱 𝒓𝒓𝑨,𝜔𝜔 𝑑𝑑𝑑𝑑𝑨

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

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

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

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

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

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

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

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

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

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

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