Dynamic Fields, Maxwell s Equations (Chapter 6)

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1 Dynamic Fields, Maxwell s Equations (Chapter 6) So far, we have studied static electric and magnetic fields. In the real world, however, nothing is static. Static fields are only approximations when the fields change very slowly, and slow is in a relative sense here. To really understand electromagnetic fields, we need to study the dynamic fields. You will see the E & M fields are coupled to each other. Four visual pictures to help you understand the four Maxwell s equations Two remain the same for dynamic and static fields. Two are different. (1) E ds D ds Q dv 0 0 dv Q E D This holds for dynamic fields even when changes with time. Quiz: how can change with time?

2 (2) B ds 0 B 0 What goes in must come out: no monopoles. Always true, static or dynamic. B (3) The electrostatic field is conservative E dl 0 E 0 This is why we can define potential. E B Edl ds t Pay attention to this negative sign. Faraday s law: B t E B t This electric field induced by a changing magnetic field is not conservative! It s not an electrostatic field even when is a constant. Cannot define a potential! E B

3 (4) Ampere s law (static) H dl J ds I I H J H Ampere s law (dynamic) D H dl ( J ) ds t I D ds t H D J t Displacement current We have covered the static case in pretty much detail. Here, in the dynamic case the current could include the displacement current. (3) and (4) are about the coupling between E & M fileds. They are the foundations of electromagnetic waves, to be discussed in Ch. 7. i D H

4 In Ch.6, we focus on Eq. (3), Faraday s law: B Edl ds t Pay attention to this negative sign. Plan view: E B t E B Viewed from another perspective: b a Now we placed a wire loop in this field. This electric field will drive a current. This gap is so small that b a E dl The voltmeter will measure a voltage This voltage is due to the non electrostatic field. It is a electromotive force. Just like that of a battery, which is due to chemistry. V emf = emf = b a E dl

5 This voltage is due to the non electrostatic field. It is a electromotive force. Just like that of a battery, which is due to chemistry. b a V emf = emf = b a E dl Notice that for an electrostatic field b V = a b + V E a Let s use an analogy to explain the subtle difference between an emf and a voltage: V = V emf = emf The pump works against gravity. The battery works against the electrostatic force.

6 V = V emf = emf = a b Edl B ds t = b a = Define magnetic flux B is therefore called the "magnetic flux density." We may also have a coil of N turns instead of a loop of just one turn: What if you replace the voltmeter with a load resistor? Magnetic flux linkage

7 Now let s see what happens if we feed a current to the coil, when there is no external magnetic filed.

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17 F B DC Motor No torque, but coil keeps rotating due to inertia F F F If current flows in same direction F F F F See also:

18 AC Motor Rotates at the frequency of the sine wave: synchronous motor. Random rotation direction.

19 Generator

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21 Magnetism of Materials H M lim V 0 i m V i M m H J M M B 0 J total 0( J J M ) 0J 0M 0 H 0M 0( H M) 0(1 m) H 0 r H H, where 0 ( 1 m) 0 r P lim V 0 i p V i P E e 0 1 E 0 total 1 E P 0 ( )E 0 where 0 P P 1 ( P ) D, e 0 ( 1 e) 0 r, D 0 re E P

22 Dielectric polarization always works against the external electric field. The magnetization M, however, may be parallel or anti-parallel to the external magnetic field H. Paramagnetic: m > 0, r = 1 + m > 1 Diamagnetic: m < 0, r = 1 + m < 1 r 1 0 Ferromagnetic: r >> 1, nonlinearity, hysteresis The description we give here is phenomenological no real understanding. Now that we have tried to give you a qualitative explanation of diamagnetism and paramagnetism, we must correct ourselves and say that it is not possible to understand the magnetic effects of materials in any honest way from the point of view of classical physics. Such magnetic effects are a completely quantummechanical phenomenon. It is, however, possible to make some phoney classical arguments and to get some idea of what is going on. -- Richar Feynman

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