October 23. Physics 272. Fall Prof. Philip von Doetinchem

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1 Physics 272 October 23 Fall Prof. Philip von Doetinchem Phys272 - Fall 14 - von Doetinchem - 170

2 Motional electromotive force Origin of electromotive force is of non-electrostatic nature (similar to battery chemical) Charges are brought to a higher potential Concept can be generalized to conductors of any shape and in any field (can be non-uniform, but not varying with time) take the perpendicular projection of the velocity with respect to the magnetic field (cross product) Use the parallel projection of the former along a line element of the conductor (scalar product) Phys272 - Fall 14 - von Doetinchem - 171

3 The Faraday disk dynamo As before for the slidewire generator: assume positive free charge carriers Positive charges accumulate at the edges Radially outward current flow Electric field builds up emf is created Difference: velocity depends on the distance to the center Phys272 - Fall 14 - von Doetinchem - 172

4 Induced electric fields We understand the concept of induction for moving charges Where is induced current coming from if the flux is changing in a stationary conductor? Phys272 - Fall 14 - von Doetinchem - 175

5 Induced electric fields Green wire loop is not in a magnetic field (magnetic field outside solenoid is negligible) Only the magnetic flux through the loop is changing Phys272 - Fall 14 - von Doetinchem - 176

6 Induced electric fields Before: charges were pushed through conductor because of magnetic forces Conclusion for stationary case: changing magnetic flux generates an induced electric field in the wire loop Furthermore: induced electric field in the loop is not conservative charges gain electric potential Stationary setup Phys272 - Fall 14 - von Doetinchem - 177

7 Induced electric fields What does the electric field look like? Cylindrical symmetry electric field has the same magnitude on the circle has to be tangential to cancel out according to Gauss's law (no net charge present inside) Line integral has to be negative when magnetic flux is increasing (Lenz's law) Phys272 - Fall 14 - von Doetinchem - 178

8 Induced electric fields The work done on an electron by the induced electric field during a complete trip around the loop is e ε energy can be removed from the electron due to the resistance of the loop The induced electric field is a non-conservative field path does matter in this case, not just the potential difference Phys272 - Fall 14 - von Doetinchem - 179

9 Nonelectrostatic electric fields Faraday's law works for two different situations: Induced current from magnetic forces when conductor moves through magnetic field Time-varying magnetic field induces electric field in a stationary conductor and induces a current The electric field of the 2nd case is also induced when no conductor is present It is not conservative Field does non-zero amount of work on charges particle on closed path This is a non-electrostatic electric field in contrast to a electrostatic electric field A change of magnetic field acts as a source of electric field that cannot be produced with a static distribution Phys272 - Fall 14 - von Doetinchem - 180

10 Eddy currents Induced currents are not necessarily confined to well-defined paths in conductors Induced eddy-like currents can form in any type of metal in changing magnetic fields or by moving through a magnetic field Applications: Currents causes heating induction furnace Eddy currents causes braking effect trains Metal detector at the airport: Magnetic field creates eddy current in objects Eddy current creates induced magnetic field Induced magnetic field creates eddy currents in receiver coil Phys272 - Fall 14 - von Doetinchem - 184

11 Direction of eddy currents upper current: falls through region of increased magnetic field builds up induced magnetic field against external field (counter-clockwise current) metallic disk falling through magnetic field - currents to the right are induced - induced currents feel upward magnetic force slow down velocity stationary magnetic field only disk is moving magnetic field is stationary lower current: falls through region of decreased magnetic field builds up induced magnetic field trying to maintain the external field (clockwise current) Phys272 - Fall 14 - von Doetinchem - 185

12 Eddy currents solid copper pendulum mounted between poles of an electromagnet pendulum is set into motion then the magnets are turned on magnets induce eddy currents in the copper opposing the motion of the pendulum pendulum quickly slows to a stop eddy current braking copper pendulum with strips cut into it is not slowed nearly as much as the solid pendulum cuts in the copper prevent large eddy currents from forming only eddy currents smaller than strips of copper can be formed Phys272 - Fall 14 - von Doetinchem - 186

13 Displacement current and Maxwell's equations A varying magnetic field creates an induced electric field Varying electric fields also create magnetic fields Essential feature to understand electromagnetic waves To understand relationship: look at charging of capacitor Phys272 - Fall 14 - von Doetinchem - 187

14 Displacement current and Maxwell's equations Look at charging of capacitor: Conducting current ic charges capacitor and builds up electric field No conducting current between plates Applying Ampere's law to both situations reveals contradiction: Phys272 - Fall 14 - von Doetinchem - 188

15 Displacement current and Maxwell's equations Electric flux increases while conducting current is decreasing Charge on capacitor: Charging capacitor current changes: Phys272 - Fall 14 - von Doetinchem - 189

16 Displacement current and Maxwell's equations Discrepancy from last slides can be resolved by having the change in conducting current translate into a change of electric flux Ampere's law becomes: Displacement current density: In this sense the displacement current is going through the capacitor Phys272 - Fall 14 - von Doetinchem - 190

17 The reality of displacement current Physical significance of displacement current? This magnetic field can me measured and has a real physical meaning Phys272 - Fall 14 - von Doetinchem - 191

18 Maxwell's equations of electromagnetism Gauss's law for electric fields (surface integral) Electric field is related to total charge in an enclosed surface Electric charges are sources of magnetic fields Gauss's law for magnetism (surface integral) No magnetic monopoles exist magnetic flux through closed surface is always zero Phys272 - Fall 14 - von Doetinchem - 192

19 Maxwell's equations of electromagnetism Ampere's law (line integral) Conducting and displacement current act as sources of magnetic fields Faraday's law (line integral) A changing magnetic field or magnetic flux induces an electric field Phys272 - Fall 14 - von Doetinchem - 193

20 Maxwell's equations of electromagnetism Electric field in Maxwell's equation is a superposition of the conservative part from the electrostatic field caused by a charge distribution (does not contribute to line integral in Faraday's law) The non-conservative part caused by induced currents (does not contribute to surface integral in Gauss's law as it is not caused by static charges) Time-varying field of either kind induce field of the other kind Starting point for electromagnetic wave discussion physical basis for light, X-ray, etc. Phys272 - Fall 14 - von Doetinchem - 194

21 Additional material Phys272 - Fall 14 - von Doetinchem - 195

22 Induced electric fields aolenoid with 500 turns, A=4.0cm2, current in windings is increasing with 100A/s Phys272 - Fall 14 - von Doetinchem - 196

23 Eddy currents solid metal ring placed on iron core whose base is wrapped in wire when DC current is passed through the wire, a magnetic field is formed in the iron core this sudden magnetic field induces a current in the metal ring, which in turn creates another magnetic field that opposes the original field ring briefly jumps upwards cut in the ring cannot form current inside will not jump ring is cooled in liquid nitrogen resistance of the metal is lowered more current to flow. ring jump jumps higher magnetic field curves away at the top of the iron coil with DC power ring will never fly off the top When AC current is passed through wire ring flies off the top of the iron core. current lags the emf by 90 degrees in inductors forces on the ring are always pointing upwards Phys272 - Fall 14 - von Doetinchem - 197

24 Conducting and displacement current Rod of pure silicon is carrying a current. Electric field varies sinusoidal with time. Phys272 - Fall 14 - von Doetinchem - 198

25 Conducting and displacement current Rod of pure silicon is carrying a current. Electric field varies sinusoidal with time. Phys272 - Fall 14 - von Doetinchem - 199