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1. http://www.lab-initio.com/250dpi/nz396.jpg

2. Today’s agenda: Induced Electric Fields. You must understand how a changing magnetic flux induces an electric field, and be able to calculate induced electric fields. Eddy Currents. You must understand how induced electric fields give rise to circulating currents called “eddy currents.” Displacement Current and Maxwell’s Equations. Displacement currents explain how current can flow “through” a capacitor, and how a time-varying electric field can induce a magnetic field. Leftovers. Any assorted topics I need to “cover” before finishing magnetism.

3. Time-Varying Magnetic Fields and Induced Electric Fields A Changing Magnetic Flux Produces an Electric Field? This suggests that a changing magnetic flux produces an electric field. This is true not just in conductors, but any-where in space where there is a changing magnetic flux.

4.       r       The previous slide uses an equation (Mr. Ed’s) valid only for a uniform electric field. Let’s see what a more general analysis gives us. Consider the conducting loop of radius r around (but not in) a region where the magnetic field is into the page and increasing (e.g., a solenoid). Put your pens and pencils down and just listen for a few minutes! This could be a wire loop around the outside of a solenoid. The charged particles in the conductor are not in a magnetic field, so they experience no magnetic force. But the changing magnetic flux induces an emf around the loop.

5.       r       The induced emf causes a counter-clockwise current (charges move). E E I But the magnetic field did not accelerate the charged particles (they aren’t in it). Therefore, there must be a tangential electric field around the loop. r E E B is increasing. The work done moving a charged particle once around the loop is. The sign is positive because the particle’s kinetic energy increases.

6.       r       We can look at work from a different point of view. ds E E I The electric field exerts a force qE on the charged particle. The instantaneous displacement is always parallel to this force. r E E Thus, the work done by the electric field in moving a charged particle once around the loop is. The sign is positive because the particle’s displacement and the force are always parallel.

7.       r       Summarizing… ds E E I r E E Combining and generalizing…

8.             Generalizing still further… ds E E The loop of wire was just a convenient way for us to visualize the effect of the changing magnetic field. I r E E The electric field exists whether or not the loop is present. A changing magnetic flux gives rise to an electric field. Was there anything in this discussion that bothered you?

9.       - +       ds E E I r E E This should bother you: where are the + and – charges in this picture? Answer: there are no + and – charges. Instead, there are electric field lines that form continuous, closed loops. Huh?

10.             But wait…there’s more! E A potential energy can be defined only for a conservative force.* A potential energy is a single-valued function. If this electric field E is due to a conservative force, then the potential energy of a charged particle must be unchanged when it goes once around the loop. *The work done by the force is independent of path.

11.             But the work done is I and F E r Work depends on the path! If we tried to define a potential energy, it would not be single-valued: U is not single-valued! We can’t define a U for this E! (*%&^#!)

12. One or two of you might not have followed the discussion on the previous 9 slides. Did I confuse anybody? You can start taking notes again, if you want.

13. Induced Electric Fields: a summary of the key ideas A changing magnetic flux induces an electric field, as given by Faraday’s Law: This is a different kind of electric field than the one you are familiar with; it is not the electrostatic field caused by the presence of stationary charged particles. Unlike the electrostatic electric field, this “new” electric field is nonconservative. “conservative,” or “Coulomb” “nonconservative”

14. Stated slightly differently: we have “discovered” two different ways to generate an electric field. Coulomb Electric Field “Faraday” Electric Field Both “kinds” of electric fields are part of Maxwell’s Equations. Both “kinds” of electric fields exert forces on charged particles. The Coulomb force is conservative, the “Faraday” force is not.

15. Direction of Induced Electric Fields The direction of E is in the direction a positively charged particle would be accelerated by the changing flux. Use Lenz’s Law to determine the direction the changing magnetic flux would cause a current to flow. That is the direction of E.

16. Example—details to be shown at the board A long thin solenoid has 500 turns per meter and a radius of 3.0 cm. The current is decreasing at a steady rate of 50 A/s. What is the magnitude of the induced electric field near the center of the solenoid 1.0 cm from the axis of the solenoid?

17. Some Revolutionary Applications of Faraday’s Law  Magnetic Tape Readers   Phonograph Cartridges   Electric Guitar Pickup Coils  Ground Fault Interruptors  Alternators  Generators  Transformers  Electric Motors

18. Application of Faraday’s Law (MAE Plasma Lab) From Meeks and Rovey, Phys. Plasmas 19, 052505 (2012); doi: 10.1063/1.4717731. Online at http://dx.doi.org/10.1063/1.4717731.T “The theta-pinch concept is one of the most widely used inductive plasma source designs ever developed. It has established a workhorse reputation within many research circles, including thin films and material surface processing, fusion, high-power space propulsion, and academia, filling the role of not only a simply constructed plasma source but also that of a key component… “Theta-pinch devices utilize relatively simple coil geometry to induce electromagnetic fields and create plasma… “This process is illustrated in Figure 1(a), which shows a cut-away of typical theta-pinch operation during an initial current rise. “FIG. 1. (a) Ideal theta-pinch field topology for an increasing current, I.”

19. The Big Picture You have now learned Gauss’s Law for both electricity and magnetism… You have now learned Gauss’s Law for both electricity and magnetism… and Faraday’s Law of Induction: These equations can also be written in differential form:

20. The S&T Society of Physics Student T-Shirt! You are ¾ of the way to being qualified to wear… This will not be tested on the exam.

21. Today’s agenda: Induced Electric Fields. You must understand how a changing magnetic flux induces an electric field, and be able to calculate induced electric fields. Eddy Currents. You must understand how induced electric fields give rise to circulating currents called “eddy currents.” Displacement Current and Maxwell’s Equations. Displacement currents explain how current can flow “through” a capacitor, and how a time-varying electric field can induce a magnetic field. Leftovers. Any assorted topics I need to “cover” before Spring Break.

22. Eddy Currents You have seen how a changing magnetic field can induce a “swirling” current in a conductor (the beginning of this lecture). If a conductor and a magnetic field are in relative motion, the magnetic force on charged particles in the conductor causes circulating currents. These currents are called “eddy currents.” Eddy currents give rise to magnetic fields that oppose any external change in the magnetic field.

23. Eddy Currents Eddy currents are useful in generators, microphones, and roller coaster brakes (among other things). However, the I2R heating from eddy currents causes energy loss, so if you don’t want energy loss, you probably think eddy currents are “bad.”

24. Eddy Current Demos cylinders falling through a tube magnetic “guillotine” hopping coil coil launcher

25. Conceptual Example: Induction Stove An ac current in a coil in the stove top produces a changing magnetic field at the bottom of a metal pan. The changing magnetic field gives rise to a current in the bottom of the pan. Because the pan has resistance, the current heats the pan. If the coil in the stove has low resistance it doesn’t get hot but the pan does. An insulator won’t heat up on an induction stove. Remember the controversy about cancer from power lines a few years back? Careful studies showed no harmful effect. Nevertheless, some believe induction stoves are hazardous.

26. “Quiz” time (maybe for points, maybe not!)

27. Today’s agenda: Induced Electric Fields. You must understand how a changing magnetic flux induces an electric field, and be able to calculate induced electric fields. Eddy Currents. You must understand how induced electric fields give rise to circulating currents called “eddy currents.” Displacement Current and Maxwell’s Equations. Displacement currents explain how current can flow “through” a capacitor, and how a time-varying electric field can induce a magnetic field. Leftovers. Any assorted topics I need to “cover” before we leave magnetism.

28. Displacement Current Apply Ampere’s Law to a charging capacitor. ds IC IC + - -q +q

29. The shape of the surface used for Ampere’s Law shouldn’t matter, as long as the “path” is the same. Imagine a soup bowl surface, with the + plate resting near the bottom of the bowl. ds IC IC + - -q +q Apply Ampere’s Law to the charging capacitor. The integral is zero because no current passes through the “bowl.”

30. Hold it right there pal! You can’t have it both ways. Which is it that you want? (The equation on the right is actually incorrect.)

31. As the capacitor charges, the electric field between the plates changes. ds E IC IC + - As the charge and electric field change, the electric flux changes. -q +q This term has units of current.

32. We define the displacement current to be ds The changing electric flux through the “bowl” surface is equivalent to the current IC through the flat surface. E IC IC + - -q +q The generalized (“always correct”) form of Ampere’s Law is then Magnetic fields are produced by both conduction currents and time varying electric fields.

33. The “stuff” inside the gray boxes serves as your official starting equation for the displacement current ID.  is the relative dielectric constant; not emf. In a vacuum, replace  with 0. ID Why “displacement?” If you put an insulator in between the plates of the capacitor, the atoms of the insulator are “stretched” because the electric field makes the protons “want” to go one way and the electrons the other. The process of “stretching” the atom involves movement of charge, and therefore a current.

34. Homework Hints For problem 9.45a, recall that For problem 9.45c, displacement current density is calculated the same as conventional current density

35. The Big Picture You have now learned Gauss’s Law for both electricity and magnetism, Faraday’s Law of Induction… You have now learned Gauss’s Law for both electricity and magnetism, Faraday’s Law of Induction, and the generalized form of Ampere’s Law: These equations can also be written in differential form:

36. Congratulations! It is my great honor to pronounce you fully qualified to wear… The S&T Society of Physics Student T-Shirt! …with all the rights and privileges thereof. This will not be tested on the exam.

37. Today’s agenda: Induced Electric Fields. You must understand how a changing magnetic flux induces an electric field, and be able to calculate induced electric fields. Eddy Currents. You must understand how induced electric fields give rise to circulating currents called “eddy currents.” Displacement Current and Maxwell’s Equations. Displacement currents explain how current can flow “through” a capacitor, and how a time-varying electric field can induce a magnetic field. Leftovers. Any assorted topics I need to “cover” before Exam 3.

38. back emf (also known as “counter emf”) (if time permits) A changing magnetic field in wire produces a current. A constant magnetic field does not. An electrical current produces a magnetic field, which by Lenz’s law, opposes the change in flux which produced the current in the first place. http://campus.murraystate.edu/tsm/tsm118/Ch7/Ch7_4/Ch7_4.htm

39. The effect is “like” that of friction. The counter emf is “like” friction that opposes the original change of current. Motors have many coils of wire, and thus generate a large counter emf when they are running. Good—keeps the motor from “running away.” Bad—”robs” you of energy.

40. If your house lights dim when an appliance starts up, that’s because the appliance is drawing lots of current and not producing a counter emf. When the appliance reaches operating speed, the counter emf reduces the current flow and the lights “undim.” Motors have design speeds their engineers expect them to run at. If the motor runs at a lower speed, there is less-than-expected counter emf, and the motor can draw more-than-expected current. If a motor is jammed or overloaded and slows or stops, it can draw enough current to melt the windings and burn out. Or even burn up.

41. leftover: torque on generator coils In lecture 15, I derived an expression for the torque on a current-carrying coil in a magnetic field. FR The source of external mechanical energy provides this torque.  IR B   Recall that for linear motion,  IL A FL For rotational motion,