Faraday s Law (continued)

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1 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 = dl = d da = dφ Remembe fo a long staight wie of length l, V = l. Note: Fo electic fields fom static chages, the voltage diffeence aound a closed path is always zeo. Not tue hee! Thee ae two souces fo electic fields!

2 Induced lectic Fields Suppose we have electomagnetic that has an inceasing magnetic field Using Faaday s Law we pedict, dl = d dφ da = If we take a cicula path inside and centeed on the magnet cente axis, the electic field will be tangent to the cicle. ( field lines ae cicles.) NOT such an field can neve be made by static chages Cylindical symmety hee gives cicula field lines, but Faaday s Law geneal. field lines will look like an onion slice (w/o smell!!) N S NOT thee ae no wie loops, the fields can appea w/o loops If we place a loop thee, a cuent would flow in the loop

3 Field Inside a Long Wie Suppose a total cuent I flows though the wie of adius a into the sceen as shown. Calculate field as a function of, the distance fom the cente of the wie. field diection tangent to cicles. x x x x x x x x x x x x x dl x x x x x x x x x x x x x x x x x a x x x x x field is only a function of take path to be cicle of adius : dl = (2 π ) Cuent passing though cicle: I dl = μ o I Ampee's Law: enclosed a 2 I = enclosed 2 μ I 2 π a 0 = 2

4 Induced lectic 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 field and thee ae no wies o loops. Note the fields ae pedicted by Faaday s Law. dl ( 2π ) d = da d = A = dφ

5 ddy Cuents Changing magnetic fields in metal induce eddy cuents. xample: negy loss in tansfomes. To educe, use laminations. ut eddy cuents often useful.

6 Maxwell s quations (integal fom) Name quation Desciption Gauss Law fo lecticity Gauss Law fo Magnetism Faaday s Law Ampee s Law Q da = ε0 da = 0 dl dφ = dl = μ0 i Needs to be modified. +? Chage and electic fields Magnetic fields lectical effects fom changing field Magnetic effects fom cuent Thee is a seious asymmety.

7 Remaks on Gauss Law s with diffeent closed sufaces da da = = Q 0 enclosed ε 0 Gauss Law s woks fo ANY CLOSD SURFAC cylinde Sufaces fo integation of flux squae sphee bagel

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

9 Genealized Ampee s Law and displacement cuent dl = μ0i enclose Ampee s oiginal law,, 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 /, in analogy to Faaday

10 Genealized Ampee s Law and displacement cuent Maxwell solved dilemma by adding an addition tem called displacement cuent, i D = ε dφ /, in analogy to Faaday s Law. dφ dl = μ0 ( ic + id ) = μ0 ic + ε 0 Cuent is once moe continuous: i D between the plates = i C in the wie. q = CV εa = d = εa ( d) = εφ dq dφ = ic = ε

11 Summay of Faaday s Law 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 field. Note; it does not matte if thee is a wie loop o an imaginay closed path, an field will be induced. Potential has no meaning in this non-consevative field.

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

13 Maxwell s quations 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 quations.

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

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations Phys-7 Lectue 17 Motional Electomotive Foce (emf) Induced Electic Fields Displacement Cuents Maxwell s Equations Fom Faaday's Law to Displacement Cuent AC geneato Magnetic Levitation Tain Review of Souces