Understanding Induced Electric Fields and Eddy Currents

1. 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. Back emf. A current in a coil of wire produces an emf that opposes the original current.

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

3. 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.”

4. 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, and the equation on the left is incomplete.)

5. 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.  Is the dielectric constant of the medium in between the capacitor plates. In the diagram, with an air-filled capacitor,  = 1.

6. We define the displacement current to be ds  is NOT emf! 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.

7. 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 displacement of charge, and therefore a current.

8. Homework Hints Displacement current: where  is the relative dielectric constant; not emf

9. Homework Hints Displacement current density is calculated the same as conventional current density This hint is not needed every semester.

10. 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:

11. Congratulations! It is my great honor to pronounce you fully qualified to wear… The Missouri S&T Society of Physics Student T-Shirt! …with all the rights and privileges thereof.* This will not be tested on the exam. *Some day I might figure out what these rights and privileges are. Contact me if you want to buy an SPS t-shirt for the member price of $10.