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 + - -q +q ICIC ICIC ds Apply Ampere’s Law to a charging capacitor.

3. Apply Ampere’s Law to the charging capacitor. 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. The integral is zero because no current passes through the “bowl.” + - -q +q ICIC ICIC ds

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. + - -q +q ICIC ICIC ds As the capacitor charges, the electric field between the plates changes. E As the charge and electric field change, the electric flux changes. 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. + - -q +q ICIC ICIC ds We define the displacement current to be E The changing electric flux through the “bowl” surface is equivalent to the current I C through the flat surface. 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 I D.  is the relative dielectric constant; not emf. In a vacuum, replace  with  0. IDID 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 For problem 9.45a, recall that For problem 9.45c, displacement current density is calculated the same as conventional current density Not applicable Spring 2015.

10. The Big Picture These equations can also be written in differential form: 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:

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