Chapter 10. Potentials and fields 1 Chapter 10. Scalar and vector potentials where V is the new scalar potential 2 Scalar and vector potentials Note that the definitions of the potentials automatically take care of nonames and Faradays
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Presentation Transcript
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Chapter 10. Potentials and fields 1<br>
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Chapter 10. Scalar and vector potentials where V is the new scalar potential 2<br>
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Scalar and vector potentials Note that the definitions of the potentials automatically take care of noname’s and Faraday’s equations In fact, the potentials are defined on basis of these equations: Applying curl to √ √ 3 Why bother with the potentials? They become more useful than the fields when charges and currents move around!<br>
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What about the other two Maxwell’s equations? New Poisson’s equation: “Scalar” Laplacian: (Identity 11) 4<br>
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Scalar and vector potentials: self-reflection Two new equations contain all the information in Maxwell’s equations: 5 • The potentials are defined on basis of noname and Faraday’s equations so these two are automatically taken care of
• The other two Maxwell’s equations (Gauss’ and Ampère-Maxwell’s laws) recast into the two equations above<br>
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Gauge transformations theorem Proof: General proof: Section 10.1.2 6 Gauge transformations theorem: Potentials are not defined uniquely!<br>
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The Coulomb gauge In magnetostatics, we picked (aka the Coulomb gauge) 7<br>
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The Lorenz gauge 8 Now we pick NB1: Not Dutch H.A. Lorentz, but Danish L.V. Lorenz
NB2: In case of the static scalar potential, Coulomb and Lorenz gauges become identical d'Alembertian operator: It’s called the Lorenz gauge Let’s do a bit of transformations over the equations for potentials So electrodynamics is reduced to solving these equations<br>
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Retarded potentials In the static case with the familiar solutions The generalized solutions are called retarded potentials 9<br>
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satisfies the inhomogeneous wave equation 10<br>
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q.e.d. (Prob. 1.63) Cancel 11 Homework: check that the retarded potentials obey the Lorenz gauge (Problem 10.10)<br>
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12 See Problem 10.6 do obey How to pick?<br>
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What you should learn (Griffiths 10.2):
1. Retarded potentials
2. Efimenko’s equations
3. How slow is “slowly enough”? Lecture 30. Potentials and fields (1/2). Radiation (1/2) 13<br>
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14 The equations were first obtained by Oleg Jefimenko in 1966
(Homework: Chapter 10.2.2 and two slides at the end) If you apply the logic of retarded time to the fields, you'll get the entirely wrong answer: BTW, this is why in electrodynamics we prefer to deal with the potentials rather then with the fields Time-dependent generalization of
Coulomb's law Time-dependent generalization of the Bio-Savart law<br>
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How slow is slowly enough? 15 (It is called the near zone) We just answered a long-standing question in the course of E&M:
The quasistatic approximation can be safely used for many problems which do not involve either long distances or high frequencies – as we did many times before Let’s request the time-dependent term be much smaller than the previous one: This is the quasistatic approximation -- quasistatic approximation doesn’t work in a computer The rate of the relative change of the current is much smaller than the time it takes to get to the observation point<br>
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YouTube resources Youtube lectures by Jonathan Gardner: https://www.youtube.com/playlist?list=PLDDEED00333C1C30E
Scalar and vector potentials: https://www.youtube.com/watch?v=gc8NvnlMawY
Retarded potentials:
Part 1/3 https://www.youtube.com/watch?v=G4MdRbV-bgU
Part 3/3 https://www.youtube.com/watch?v=z9lC1rGLqDo&t=0s
Part 3/3 https://www.youtube.com/watch?v=0TXS9A6tjK4&t=49s 16<br>
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17 (was already calculated a few slides earlier)<br>
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18 Let’s find the field This is the time-dependent generalization of the Bio-Savart law Rule #7<br>
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20 Suppose the current density changes slowly enough that we can ignore all higher derivatives This is the Bio-Savart law with the current density evaluated at the non-retarded time<br>
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How slow is slowly enough? Another approach 21 (It is called the near zone) We just answered a long-standing question in the course of E&M:
The quasistatic approximation can be safely used for many problems which do not involve either long distances or high frequencies – as we did many times before For the Taylor series to converge, we need to request the next term be smaller than the previous one: This is the quasistatic approximation -- quasistatic approximation doesn’t work in a computer The rate of the change of the relative change is much smaller than the time it takes to get to the observation point<br>
