Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations

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1 Phys-7 Lectue 17 Motional Electomotive Foce (emf) Induced Electic Fields Displacement Cuents Maxwell s Equations

2 Fom Faaday's Law to Displacement Cuent AC geneato Magnetic Levitation Tain

3 Review of Souces of B fields Clicke A long wie caies a cuent I. The B field at distance fom the wie I Question I a) B µ I / π o b) c) B µ I / π o B µ I / π o Veify using Ampee s Law Question II Diection is (a)adial, pep to wie (b) tangential (c) along the wie (d) anti-paallel to the wie

4 Motional Electomotive Foce ε d B da dφ B In Faaday s Law, we can induce EMF in the loop when the magnetic flux, Φ B, changes as a function of time. Thee ae two Cases when Φ B is changing, 1) Change the magnetic field (non-constant ove time) ) Change o move the loop in a constant magnetic field The slide wie geneato is an example of # and the induction of EMF via moving pats of the loop is called, motional EMF.

5 Slide Wie Geneato; evisited again Suppose we move a conducting ba in a constant B field, then a foce Fq v B moves + chage up and chage down. The chage distibution poduces an electic field and EMF, Ɛ, between a & b. This continues until equilibium is eached. E F q qv B q b v B ε E dl vbl In effect the ba s motional EMF is an equivalent to a battey EMF a

6 Slide Wie Geneato; evisited again If the od is on the U shaped conducto, the chages don t build up at the ends but move though the U shaped potion. They poduce an electic field in the cicuit. The wie acts as a souce of EMF just like a battey. Called motional electomotive foce. ε b a E dl vbl

7 Diect Cuent Homopola Geneato invented by Faaday Rotate a metal disk in a constant pependicula magnetic field. The chages in the disk when moving eceive a adial foce. The causes cuent to flow fom cente to point b. R ε ω B d 0 1 ω BR

8 Faaday s Law (continued) What causes cuent to flow in wie? Answe: an E field in the wie. A changing magnetic flux not only causes an EMF aound a loop but an induced electic field. Can wite Faaday s Law: ε E dl d B da dφ B Remembe fo a long staight wie of length l, V El. Note: Fo electic fields fom static chages, the EMF fom a closed path is always zeo. Not tue hee. Thee ae two possible souces fo electic fields!

9 Induced Electic Fields Suppose we have electomagnetic that has an inceasing magnetic field Using Faaday s Law we pedict, E dl d dφ B da If we take a cicula path inside and centeed on the magnet cente axis, the electic field will be tangent to the cicle. (E field lines ae cicles.) NOTE such an E field can neve be made by static chages B N S B E E field lines will look like an onion slice N.B. thee ae no wie loops, E fields can appea w/o loops If we place a loop thee, a cuent would flow in the loop

10 Induced Electic Fields; example If we have a solenoid coil with changing cuent thee will be cicula electic fields ceated outside the solenoid. It looks vey much like the mag. field aound a cuent caying wie, but it is an E field and thee ae no wies o loops. E Note the E fields ae pedicted by Faaday eqn. E dl d B da dφ B

11 Eddy Cuents Changing magnetic fields in metal induce eddy cuents. Example: Enegy loss in tansfomes. To educe use laminations. But eddy cuents often useful.

12 Maxwell s Equations (integal fom) Name Gauss Law fo Electicity Gauss Law fo Magnetism Faaday s Law Ampee s Law Equation Q E da ε0 B da 0 E dl dφ B dl µ 0 i B Needs to be modified. +? Desciption Chage and electic fields Magnetic fields Electical effects fom changing B field Magnetic effects fom cuent Thee is a seious asymmety.

13 Remaks on Gauss Law s with diffeent closed sufaces E B da da Q 0 enclosed ε 0 Gauss Law s woks fo ANY CLOSED SURFACE cylinde Sufaces fo integation of E flux squae sphee bagel

14 Remaks on Faaday s Law with diffeent attached sufaces E dl d B da Faaday s Law woks fo any closed Loop and ANY attached suface aea Line integal defines the Closed loop Suface aea integation fo B flux disk cylinde Fish bowl This is poved in Vecto Calculus with Stokes Theoem

15 Genealized Ampee s Law and displacement cuent Ampee s oiginal law, B dl µ 0I enclose, is incomplete. Conside the paallel plate capacito and suppose a cuent i c is flowing chaging up the plate. If Ampee s law is applied fo the given path in eithe the plane suface o the bulging suface we we should get the same esults, but the bulging suface has i c 0, so something is missing.

Genealized Ampee s Law and displacement cuent Maxwell solved dilemma by adding an addition tem called displacement cuent, i D ε dφ E /, in analogy to

16 Genealized Ampee s Law and displacement cuent Maxwell solved dilemma by adding an addition tem called displacement cuent, i D ε dφ E /, in analogy to Faaday s Law. dφ E B dl µ 0 ( ic + id ) µ 0 ic + ε 0 Cuent is once moe continuous: i D between the plates i C in the wie. q CV εa d εea ( Ed) εφ E dq dφ ic ε E

17 Summay of Faaday s Law E dl dφ If we fom any closed loop, the line integal of the electic field equals the time ate change of magnetic flux though the suface enclosed by the loop. If thee is a changing magnetic field, then thee will be electic fields induced in closed paths. The electic fields diection will tend to educe the changing B field. B B E Note; it does not matte if thee is a wie loop o an imaginay closed path, an E field will be induced. Potential has no meaning in this non-consevative E field.

18 I Chage is flowing onto this paallel plate capacito at a ate dq/ A II III What is the displacement cuent in egions I and III? A) A B) 1 A C) 0 D) -A

19 I Chage is flowing onto this paallel plate capacito at a ate dq/ A II III What is the displacement cuent in egion II? A) -/3A B) 1 A C) A D) O A

20 Summay of Ampee s Genealized Law B dl µ 0 i + ε c 0 dφ E Cuent i c If we fom any closed loop, the line integal of the B field is nonzeo if thee is (constant o changing) cuent though the loop. B If thee is a changing electic field though the loop, then thee will be magnetic fields induced about a closed loop path. E B

21 Maxwell s Equations James Clek Maxwell ( ) genealized Ampee s Law made equations symmetic: a changing magnetic field poduces an electic field a changing electic field poduces a magnetic field Showed that Maxwell s equations pedicted electomagnetic waves and c 1/ ε 0 µ 0 Unified electicity and magnetism and light. All of electicity and magnetism can be summaized by Maxwell s Equations.

22 Moe impotant applications of Faaday s Law

23 Mutual Inductance If we have a constant cuent i 1 in coil 1, a constant magnetic field is ceated and this poduces a constant magnetic flux in coil. Since the Φ B is constant, thee NO induced cuent in coil. If cuent i 1 is time vaying, then the Φ B flux is vaying and this induces an emf ε in coil, the emf is ε N dφ B We intoduce a atio, called mutual inductance, of flux in coil divided by the cuent in coil 1. NΦ M 1 B i1

24 Mutual Inductance N mutual inductance, Φ M 1 B, can now be used in Faaday s eqn. i M i N Φ 1 1 B 1 di d di M 1 Φ N B M 1 1 ε ; ε 1 We can also the vaying cuent i which ceates a changing flux Φ B1 in coil 1 and induces an emf ε 1. This is given by a simila eqn. ε1 di M 1 It can be shown (we do not pove hee) that, M M 1 The units of mutual inductance is T m /A Webe/A Heny (afte Joseph Heny, who nealy discoveed Faaday s Law) 1 M

25 Mutual Inductance The induced emf, has the following featues; ε di 1 M The induced emf opposes the magnetic flux change (Lenz s Law) The induced emf inceases if the cuent changes vey fast The induced emf depends on M, which depends only the geomety of the two coils and not the cuent. Fo a few simple cases, we can calculate M, but usually it is just measued.

26 Poblem 30.1 Two coils have mutual inductance of H. The cuent in the fist coil inceases at a unifom ate of 830 A/s. A) What is the magnitude of induced emf in the nd coil? Is it constant? B) suppose that the cuent is instead in the nd coil, what is the magnitude of the induced emf in the 1 st coil? ε M di 1 ( H)(830 A ) s 0.7V di ε1 M 0. 7V

27 Magnetic field due to coil 1 is B 1 µ 0 n 1 i 1 µ 0 N 1 i 1 / l Mutual inductance is, l A N N l i A i N N i A B N i N M B µ µ Φ The induced emf in coil1 fom coil is di l A N N di M µ ε Tesla Coil Example

Faraday s Law (continued)

Faraday s Law (continued) Faaday s Law (continued) What causes cuent to flow in wie? Answe: an field in the wie. A changing magnetic flux not only causes an MF aound a loop but an induced electic field. Can wite Faaday s Law: ε