1. ENE 325 Electromagnetic Fields and Waves Lecture 10 Time-Varying Fields and Maxwell’s Equations

2. Magnetic boundary conditions B n1 = B n2 and Inductance and mutual inductance self inductance L is defined as the ratio of flux linkage to the current generating the flux, henrys or Wb/A. mutual inductance M, where M 12 = M 21. Review

3. Time-Varying fields and Maxwell’s equations Concept The electric field E is produced by the change in the magnetic field B. The magnetic field B is produced by the change in the electric field E.

4. Faraday’s law V where emf = electromotive force that may establish a current in a suitable closed circuit and is a voltage that arises from conductors moving in static or changing magnetic fields. is arisen from 1. the change of flux in a closed path 2. the moving closed path in a stationary magnetic field 3. both 1 and 2 For the N number of loops, v.

5. emf in the closed loop is not zero a) direction of the induced current b) emf

6. Changing flux in a stationary path (transformer emf) From apply Stokes’ theorem, So we have 1 st Maxwell’s equation

7. Transformer To transform AC voltages and currents between a pair of windings in magnetic circuits With Faraday’s law, we have Since the term is the same for both voltages, so we get

8. Ex1 Assume, prove the 1 st Maxwell’s equation.

9. Changing flux in a moving closed path (1) a conductor moves in a uniform magnetic field. The sign of emf determines the direction of the induced current.

10. Changing flux in a moving closed path (2) Examine in a different point of view So we get V.

11. Changing flux in a moving closed path (3) Combing both effects yields

12. Ex2 Let mT, find a)flux passing through the surface z = 0, 0 < x < 20 m, and 0 < y < 3 m at t = 1  S. b) value of closed line integral around the surface specified above at t = 1  S.

13. Ex3 A moving conductor is located on the conducting rail as shown at time t = 0, a) find emf when the conductor is at rest at x = 0.05 m and T.

14. b) find emf when the conductor is moving with the speed m/s.

15. Displacement current (1) The next Maxwell’s equation can be found in terms of time- changing electric field From a steady magnetic field, From the equation of continuity, therefore this is impossible!

16. Displacement current (2) Another term must be added to make the equation valid. 2 nd Maxwell’s equation In a non-conductive medium,

17. Displacement current (3) We can show the displacement current as The more general Ampere’s circuital law:

18. Where is the displacement current from? Consider a simple current loop, let emf = V o cos  t

19. Ex4 Determine the magnitude of for the following situations: a) a) in the air near the antenna that radiates V/m.

20. b) a pair of 100 cm 2 area plates separated by a 1.0 mm thick layer of lossy dielectric characterized by  r = 50 and  = 1.0  10 -4 S/m given the voltage across plates V(t) = 1.0cos(2  10 3 t) V.