March 20. Physics 272. Spring Prof. Philip von Doetinchem

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1 Physics 272 March 20 Spring Prof. Philip von Doetinchem Phys272 - Spring 14 - von Doetinchem - 129

2 Summary No magnetic forces act on the charges in the stationary U part, but sliding rod creates potential difference (source of emf) Phys272 - Spring 14 - von Doetinchem - 130

3 Bar on inclined plane Phys272 - Spring 14 - von Doetinchem - 131

4 Bar on inclined plane Phys272 - Spring 14 - von Doetinchem - 132

5 Bar on inclined plane Phys272 - Spring 14 - von Doetinchem - 133

6 Bar on inclined plane Phys272 - Spring 14 - von Doetinchem - 134

7 Summary Only the magnetic flux through the loop is changing Phys272 - Spring 14 - von Doetinchem - 135

8 Summary What does the electric field look like? Line integral has to be negative when magnetic flux is increasing (Lenz's law) Phys272 - Spring 14 - von Doetinchem - 136

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 - Spring 14 - von Doetinchem - 137

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 - Spring 14 - von Doetinchem - 141

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 - Spring 14 - von Doetinchem - 142

12 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 - Spring 14 - von Doetinchem - 143

13 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 - Spring 14 - von Doetinchem - 144

14 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 - Spring 14 - von Doetinchem - 145

15 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 - Spring 14 - von Doetinchem - 146

16 Displacement current and Maxwell's equations Electric flux increases while conducting current is decreasing Charge on capacitor: Charging capacitor current changes: Phys272 - Spring 14 - von Doetinchem - 147

17 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 - Spring 14 - von Doetinchem - 148

18 The reality of displacement current Physical significance of displacement current? This magnetic field can me measured and has a real physical meaning Phys272 - Spring 14 - von Doetinchem - 149

19 Conducting and displacement current Rod of pure silicon is carrying a current. Electric field varies sinusoidal with time. Phys272 - Spring 14 - von Doetinchem - 150

20 Conducting and displacement current Rod of pure silicon is carrying a current. Electric field varies sinusoidal with time. Phys272 - Spring 14 - von Doetinchem - 151

21 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 - Spring 14 - von Doetinchem - 152

22 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 - Spring 14 - von Doetinchem - 153

23 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 - Spring 14 - von Doetinchem - 154

24 Review Faraday's law: induced emf is the negative of the time rate of change of magnetic flux through a loop Lenz's law: induced current/emf tends to cancel the change that caused it. Motional emf: moving conductor in magnetic field causes motional emf Induced electric fields: emf induced by a changing magnetic flux through a stationary conductor induced electric field of non-conservative nature Phys272 - Spring 14 - von Doetinchem - 155

25 Review Maxwell's equations Surface integrals: Line integrals: Phys272 - Spring 14 - von Doetinchem - 156

26 Discussion The angular speed of the loop is doubled, then the frequency with which the induced current changes direction doubles, and the maximum emf also doubles. Why? Does the torque required to turn the loop change? Phys272 - Spring 14 - von Doetinchem - 157

27 Discussion The angular speed of the loop is doubled, then the frequency with which the induced current changes direction doubles, and the maximum emf also doubles. Why? Does the torque required to turn the loop change? when angular speed is doubled the rate of change of the flux doubles and this causes the induced emf and induced current to double torque required is proportional to the current in the loop, so the torque also doubles Phys272 - Spring 14 - von Doetinchem - 158

28 Discussion An airplane flies over Antarctica, where the magnetic field of the earth mostly directed upward away from the ground. As viewed by a passenger facing toward the front of the plane, is the left or the right wingtip at higher potential? Does the answer depend on the direction the plane is flying? Phys272 - Spring 14 - von Doetinchem - 159

29 Discussion An airplane flies over Antarctica, where the magnetic field of the earth mostly directed upward away from the ground. As viewed by a passenger facing toward the front of the plane, is the left or the right wingtip at higher potential? Does the answer depend on the direction the plane is flying? positive charge collects toward the right; the righthand wing tip is at higher potential does not depend on the direction the plane is flying Phys272 - Spring 14 - von Doetinchem - 160

30 Discussion Would it be appropriate to ask how much energy an electron gains during a complete trip around the wire loop with a certain induced current? Would it be appropriate to ask what potential difference the electron moves through during such a trip? Phys272 - Spring 14 - von Doetinchem - 161

31 Discussion 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 - Spring 14 - von Doetinchem - 162

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