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

2. Review Magnetic boundary conditions Bn1 = Bn2 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 M12 = M21.

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
1st 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 1st 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
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.
2nd 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 = Vocost

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

20. b) a pair of 100 cm2 area plates separated by a 1
b) a pair of 100 cm2 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103t) V.